Integrand size = 49, antiderivative size = 946 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{9/2}} \, dx=-\frac {(i A+B-i C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{9/2} f}-\frac {(B-i (A-C)) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{9/2} f}-\frac {2 \left (6 a^5 b B d^2+15 a^6 C d^2+a^4 b^2 d (14 B c+8 A d+37 C d)+3 a^2 b^4 \left (35 A c^2-35 c^2 C-70 B c d-39 A d^2+54 C d^2\right )-a^3 b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right )+a b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right )+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^3 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (6 a^7 b B d^3+15 a^8 C d^3+2 a^6 b^2 d^2 (7 B c+4 A d+26 C d)-2 a b^7 \left (210 A c^3-210 c^3 C-525 B c^2 d-406 A c d^2+406 c C d^2+88 B d^3\right )-a^4 b^4 \left (105 B c^3+525 A c^2 d-525 c^2 C d-749 B c d^2-311 A d^3+221 C d^3\right )+2 a^2 b^6 \left (315 B c^3+875 A c^2 d-875 c^2 C d-812 B c d^2-261 A d^3+291 C d^3\right )+2 a^5 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right )-b^8 \left (5 d \left (49 A c^2-49 c^2 C-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )-2 a^3 b^5 \left (210 c^3 C+700 B c^2 d-798 c C d^2-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 b^3 \left (a^2+b^2\right )^4 (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (2 a^3 b B d+5 a^4 C d+b^4 (7 B c+5 A d)+2 a b^3 (7 A c-7 c C-6 B d)-a^2 b^2 (7 B c+9 A d-19 C d)\right ) (c+d \tan (e+f x))^{3/2}}{35 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{5/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}} \]
-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh((c-I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a -I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))/(a-I*b)^(9/2)/f-(B-I*(A-C))*(c+I*d)^(5 /2)*arctanh((c+I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a+I*b)^(1/2)/(c+d*tan(f* x+e))^(1/2))/(a+I*b)^(9/2)/f-2/105*(6*a^7*b*B*d^3+15*a^8*C*d^3+2*a^6*b^2*d ^2*(4*A*d+7*B*c+26*C*d)-2*a*b^7*(210*A*c^3-406*A*c*d^2-525*B*c^2*d+88*B*d^ 3-210*C*c^3+406*C*c*d^2)-a^4*b^4*(525*A*c^2*d-311*A*d^3+105*B*c^3-749*B*c* d^2-525*C*c^2*d+221*C*d^3)+2*a^2*b^6*(875*A*c^2*d-261*A*d^3+315*B*c^3-812* B*c*d^2-875*C*c^2*d+291*C*d^3)+2*a^5*b^3*d*(56*c*(A-C)*d+B*(35*c^2-12*d^2) )-b^8*(5*d*(49*A*c^2-3*A*d^2-49*C*c^2)+7*B*(15*c^3-23*c*d^2))-2*a^3*b^5*(2 10*c^3*C+700*B*c^2*d-798*C*c*d^2-317*B*d^3-42*A*(5*c^3-19*c*d^2)))*(c+d*ta n(f*x+e))^(1/2)/b^3/(a^2+b^2)^4/(-a*d+b*c)/f/(a+b*tan(f*x+e))^(1/2)-2/105* (6*a^5*b*B*d^2+15*a^6*C*d^2+a^4*b^2*d*(8*A*d+14*B*c+37*C*d)+3*a^2*b^4*(35* A*c^2-39*A*d^2-70*B*c*d-35*C*c^2+54*C*d^2)-a^3*b^3*(98*c*(A-C)*d+B*(35*c^2 -75*d^2))+a*b^5*(182*c*(A-C)*d+B*(105*c^2-71*d^2))+b^6*(7*c*(8*B*d+5*C*c)- 5*A*(7*c^2-3*d^2)))*(c+d*tan(f*x+e))^(1/2)/b^3/(a^2+b^2)^3/f/(a+b*tan(f*x+ e))^(3/2)-2/35*(2*a^3*b*B*d+5*a^4*C*d+b^4*(5*A*d+7*B*c)+2*a*b^3*(7*A*c-6*B *d-7*C*c)-a^2*b^2*(9*A*d+7*B*c-19*C*d))*(c+d*tan(f*x+e))^(3/2)/b^2/(a^2+b^ 2)^2/f/(a+b*tan(f*x+e))^(5/2)-2/7*(A*b^2-a*(B*b-C*a))*(c+d*tan(f*x+e))^(5/ 2)/b/(a^2+b^2)/f/(a+b*tan(f*x+e))^(7/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(10121\) vs. \(2(946)=1892\).
Time = 56.13 (sec) , antiderivative size = 10121, normalized size of antiderivative = 10.70 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{9/2}} \, dx=\text {Result too large to show} \]
Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x] ^2))/(a + b*Tan[e + f*x])^(9/2),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(a+b \tan (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {2 \int \frac {(c+d \tan (e+f x))^{3/2} \left (-\left (\left (-5 C a^2-2 b B a+2 A b^2-7 b^2 C\right ) d \tan ^2(e+f x)\right )-7 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (7 b c-5 a d)+A b (7 a c+5 b d)\right )}{2 (a+b \tan (e+f x))^{7/2}}dx}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (-\left (\left (-5 C a^2-2 b B a+2 A b^2-7 b^2 C\right ) d \tan ^2(e+f x)\right )-7 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (7 b c-5 a d)+A b (7 a c+5 b d)\right )}{(a+b \tan (e+f x))^{7/2}}dx}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (-\left (\left (-5 C a^2-2 b B a+2 A b^2-7 b^2 C\right ) d \tan (e+f x)^2\right )-7 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (7 b c-5 a d)+A b (7 a c+5 b d)\right )}{(a+b \tan (e+f x))^{7/2}}dx}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\frac {2 \int \frac {\sqrt {c+d \tan (e+f x)} \left (-35 ((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-C c-B d))) \tan (e+f x) b^2+(5 a c+3 b d) ((b B-a C) (7 b c-5 a d)+A b (7 a c+5 b d)) b+d \left (15 C d a^4+6 b B d a^3+2 b^2 (7 B c+4 A d+11 C d) a^2-2 b^3 (14 A c-14 C c-17 B d) a-b^4 (14 B c+5 (4 A-7 C) d)\right ) \tan ^2(e+f x)-(5 b c-3 a d) \left (5 C d a^3+2 b B d a^2-7 b^2 (B c-2 C d) a+A b^2 (7 b c-9 a d)-7 b^3 (c C+B d)\right )\right )}{2 (a+b \tan (e+f x))^{5/2}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 (c+d \tan (e+f x))^{3/2} \left (5 a^4 C d+2 a^3 b B d-a^2 b^2 (9 A d+7 B c-19 C d)+2 a b^3 (7 A c-6 B d-7 c C)+b^4 (5 A d+7 B c)\right )}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-35 ((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-C c-B d))) \tan (e+f x) b^2+(5 a c+3 b d) ((b B-a C) (7 b c-5 a d)+A b (7 a c+5 b d)) b+d \left (15 C d a^4+6 b B d a^3+2 b^2 (7 B c+4 A d+11 C d) a^2-2 b^3 (14 A c-14 C c-17 B d) a-b^4 (14 B c+5 (4 A-7 C) d)\right ) \tan ^2(e+f x)-(5 b c-3 a d) \left (5 C d a^3+2 b B d a^2-7 b^2 (B c-2 C d) a+A b^2 (7 b c-9 a d)-7 b^3 (c C+B d)\right )\right )}{(a+b \tan (e+f x))^{5/2}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 (c+d \tan (e+f x))^{3/2} \left (5 a^4 C d+2 a^3 b B d-a^2 b^2 (9 A d+7 B c-19 C d)+2 a b^3 (7 A c-6 B d-7 c C)+b^4 (5 A d+7 B c)\right )}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-35 ((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-C c-B d))) \tan (e+f x) b^2+(5 a c+3 b d) ((b B-a C) (7 b c-5 a d)+A b (7 a c+5 b d)) b+d \left (15 C d a^4+6 b B d a^3+2 b^2 (7 B c+4 A d+11 C d) a^2-2 b^3 (14 A c-14 C c-17 B d) a-b^4 (14 B c+5 (4 A-7 C) d)\right ) \tan (e+f x)^2-(5 b c-3 a d) \left (5 C d a^3+2 b B d a^2-7 b^2 (B c-2 C d) a+A b^2 (7 b c-9 a d)-7 b^3 (c C+B d)\right )\right )}{(a+b \tan (e+f x))^{5/2}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 (c+d \tan (e+f x))^{3/2} \left (5 a^4 C d+2 a^3 b B d-a^2 b^2 (9 A d+7 B c-19 C d)+2 a b^3 (7 A c-6 B d-7 c C)+b^4 (5 A d+7 B c)\right )}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {15 C d^3 a^6+6 b B d^3 a^5+b^2 d^2 (14 B c+8 A d+37 C d) a^4-b^3 \left (105 C c^3+245 B d c^2-203 C d^2 c-75 B d^3-7 A \left (15 c^3-29 c d^2\right )\right ) a^3+3 b^4 \left (35 B \left (3 c^3-5 c d^2\right )+d \left (245 A c^2-245 C c^2-39 A d^2+54 C d^2\right )\right ) a^2-b^5 \left (315 A c^3-315 C c^3-735 B d c^2-497 A d^2 c+497 C d^2 c+71 B d^3\right ) a+d \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4+2 b^3 \left (56 c (A-C) d+5 B \left (7 c^2-3 d^2\right )\right ) a^3-3 b^4 \left (70 A c^2-70 C c^2-140 B d c-66 A d^2+51 C d^2\right ) a^2-2 b^5 \left (224 c (A-C) d+B \left (105 c^2-122 d^2\right )\right ) a+b^6 \left (70 A c^2-70 C c^2-154 B d c-90 A d^2+105 C d^2\right )\right ) \tan ^2(e+f x)-b^6 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )+105 b^3 \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+3 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-3 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) \tan (e+f x)}{2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {15 C d^3 a^6+6 b B d^3 a^5+b^2 d^2 (14 B c+8 A d+37 C d) a^4-b^3 \left (105 C c^3+245 B d c^2-203 C d^2 c-75 B d^3-7 A \left (15 c^3-29 c d^2\right )\right ) a^3+3 b^4 \left (35 B \left (3 c^3-5 c d^2\right )+d \left (245 A c^2-245 C c^2-39 A d^2+54 C d^2\right )\right ) a^2-b^5 \left (315 A c^3-315 C c^3-735 B d c^2-497 A d^2 c+497 C d^2 c+71 B d^3\right ) a+d \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4+2 b^3 \left (56 c (A-C) d+5 B \left (7 c^2-3 d^2\right )\right ) a^3-3 b^4 \left (70 A c^2-70 C c^2-140 B d c-66 A d^2+51 C d^2\right ) a^2-2 b^5 \left (224 c (A-C) d+B \left (105 c^2-122 d^2\right )\right ) a+b^6 \left (70 A c^2-70 C c^2-154 B d c-90 A d^2+105 C d^2\right )\right ) \tan ^2(e+f x)-b^6 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )-105 b^3 \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3\right )+3 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+3 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {15 C d^3 a^6+6 b B d^3 a^5+b^2 d^2 (14 B c+8 A d+37 C d) a^4-b^3 \left (105 C c^3+245 B d c^2-203 C d^2 c-75 B d^3-7 A \left (15 c^3-29 c d^2\right )\right ) a^3+3 b^4 \left (35 B \left (3 c^3-5 c d^2\right )+d \left (245 A c^2-245 C c^2-39 A d^2+54 C d^2\right )\right ) a^2-b^5 \left (315 A c^3-315 C c^3-735 B d c^2-497 A d^2 c+497 C d^2 c+71 B d^3\right ) a+d \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4+2 b^3 \left (56 c (A-C) d+5 B \left (7 c^2-3 d^2\right )\right ) a^3-3 b^4 \left (70 A c^2-70 C c^2-140 B d c-66 A d^2+51 C d^2\right ) a^2-2 b^5 \left (224 c (A-C) d+B \left (105 c^2-122 d^2\right )\right ) a+b^6 \left (70 A c^2-70 C c^2-154 B d c-90 A d^2+105 C d^2\right )\right ) \tan (e+f x)^2-b^6 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )-105 b^3 \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3\right )+3 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+3 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {2 \int \frac {105 \left (b^3 (b c-a d) \left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^4-4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right )-b^3 (b c-a d) \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4+4 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)\right )}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^4+4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right )-b^3 (b c-a d) \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4\right )+4 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^3+6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^4-4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right )-b^3 (b c-a d) \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4+4 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^4+4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right )-b^3 (b c-a d) \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4\right )+4 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^3+6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^4-4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right )-b^3 (b c-a d) \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4+4 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^4+4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right )-b^3 (b c-a d) \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4\right )+4 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^3+6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^4-4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right )-b^3 (b c-a d) \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4+4 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^4+4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right )-b^3 (b c-a d) \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4\right )+4 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^3+6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^4-4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right )-b^3 (b c-a d) \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4+4 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^4+4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right )-b^3 (b c-a d) \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4\right )+4 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^3+6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^4-4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right )-b^3 (b c-a d) \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4+4 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^4+4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right )-b^3 (b c-a d) \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4\right )+4 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^3+6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^4-4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right )-b^3 (b c-a d) \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4+4 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^4+4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right )-b^3 (b c-a d) \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4\right )+4 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^3+6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^4-4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right )-b^3 (b c-a d) \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4+4 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^4+4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right )-b^3 (b c-a d) \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4\right )+4 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^3+6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^4-4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right )-b^3 (b c-a d) \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4+4 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^4+4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right )-b^3 (b c-a d) \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4\right )+4 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^3+6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^4-4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right )-b^3 (b c-a d) \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4+4 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3-6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left (56 c (A-C) d+B \left (35 c^2-12 d^2\right )\right ) a^5-b^4 \left (105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right ) a^4-2 b^5 \left (210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left (5 c^3-19 c d^2\right )\right ) a^3+2 b^6 \left (315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right ) a^2-2 b^7 \left (210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right ) a-b^8 \left (5 d \left (49 A c^2-49 C c^2-3 A d^2\right )+7 B \left (15 c^3-23 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {105 \int -\frac {b^3 (b c-a d) \left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^4+4 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+6 b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-4 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^4 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right )-b^3 (b c-a d) \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^4\right )+4 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^3+6 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+4 b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^4 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}-\frac {2 \left (15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left (98 c (A-C) d+B \left (35 c^2-75 d^2\right )\right ) a^3+3 b^4 \left (35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right ) a^2+b^5 \left (182 c (A-C) d+B \left (105 c^2-71 d^2\right )\right ) a+b^6 \left (7 c (5 c C+8 B d)-5 A \left (7 c^2-3 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}}{7 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{7 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{7/2}}\) |
Int[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/( a + b*Tan[e + f*x])^(9/2),x]
3.2.46.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim p[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* (n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b *(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ [a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Timed out.
\[\int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {9}{2}}}d x\]
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{9/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(9/2),x, algorithm="fricas")
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{9/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))**(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*ta n(f*x+e))**(9/2),x)
Exception generated. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{9/2}} \, dx=\text {Exception raised: ValueError} \]
integrate((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(9/2),x, algorithm="maxima")
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(((2*b*d+2*a*c)^2>0)', see `assum e?` for mo
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{9/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(9/2),x, algorithm="giac")
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{9/2}} \, dx=\text {Hanged} \]