Integrand size = 49, antiderivative size = 586 \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=-\frac {(i A+B-i C) (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{7/2} f}-\frac {(B-i (A-C)) (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{7/2} f}-\frac {2 \left (2 a^3 b B d+3 a^4 C d+b^4 (5 B c+3 A d)+2 a b^3 (5 A c-5 c C-4 B d)-a^2 b^2 (5 B c+7 A d-13 C d)\right ) \sqrt {c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (2 a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (10 B c+(8 A+C) d)+a^2 b^4 \left (45 A c^2-45 c^2 C-90 B c d-49 A d^2+58 C d^2\right )-a^3 b^3 \left (50 c (A-C) d+B \left (15 c^2-39 d^2\right )\right )+a b^5 \left (70 c (A-C) d+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (3 c C+4 B d)-3 A \left (5 c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 b^2 \left (a^2+b^2\right )^3 (b c-a d) f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}} \]
-(I*A+B-I*C)*(c-I*d)^(3/2)*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)^(7/2)/f-(B-I*(A-C))*(c+I*d)^(3 /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)^(7/2)/f-2/15*(2*a^5*b*B*d^2+3*a^6*C*d^2+a^4*b^2*d*(10 *B*c+(8*A+C)*d)+a^2*b^4*(45*A*c^2-49*A*d^2-90*B*c*d-45*C*c^2+58*C*d^2)-a^3 *b^3*(50*c*(A-C)*d+B*(15*c^2-39*d^2))+a*b^5*(70*c*(A-C)*d+B*(45*c^2-23*d^2 ))+b^6*(5*c*(4*B*d+3*C*c)-3*A*(5*c^2-d^2)))*(c+d*tan(f*x+e))^(1/2)/b^2/(a^ 2+b^2)^3/(-a*d+b*c)/f/(a+b*tan(f*x+e))^(1/2)-2/15*(2*a^3*b*B*d+3*a^4*C*d+b ^4*(3*A*d+5*B*c)+2*a*b^3*(5*A*c-4*B*d-5*C*c)-a^2*b^2*(7*A*d+5*B*c-13*C*d)) *(c+d*tan(f*x+e))^(1/2)/b^2/(a^2+b^2)^2/f/(a+b*tan(f*x+e))^(3/2)-2/5*(A*b^ 2-a*(B*b-C*a))*(c+d*tan(f*x+e))^(3/2)/b/(a^2+b^2)/f/(a+b*tan(f*x+e))^(5/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3134\) vs. \(2(586)=1172\).
Time = 9.50 (sec) , antiderivative size = 3134, normalized size of antiderivative = 5.35 \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Result too large to show} \]
Integrate[((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x] ^2))/(a + b*Tan[e + f*x])^(7/2),x]
-((C*(c + d*Tan[e + f*x])^(3/2))/(b*f*(a + b*Tan[e + f*x])^(5/2))) - (-1/4 *((3*b*c*C - 2*b*B*d - 3*a*C*d)*Sqrt[c + d*Tan[e + f*x]])/(b*f*(a + b*Tan[ e + f*x])^(5/2)) - ((-2*((b^2*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - a*(-1/4*(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d))) + 2*b^3*(2*c*(A - C)*d + B* (c^2 - d^2))))*Sqrt[c + d*Tan[e + f*x]])/(5*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(5/2)) - (2*((-2*(b^2*(((2*b^2*d - (5*a*(b*c - a*d))/2)*( 8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)) )/4 + ((-5*b*c)/2 + (a*d)/2)*(-1/4*(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d))) + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2 )))) - a*((5*b*(b*c - a*d)*((b*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - (b*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 - 2*a*b^2*(2*c*(A - C)*d + B*(c^ 2 - d^2))))/2 - 2*a*d*((b^2*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - a*(-1/4*(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d))) + 2*b^3*(2*c*(A - C)*d + B*(c ^2 - d^2))))))*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)) - (2*((-15*b^2*(b*c - a*d)^2*(((3*a^2*A*b*c^2 - A* b^3*c^2 - a^3*B*c^2 + 3*a*b^2*B*c^2 - 3*a^2*b*c^2*C + b^3*c^2*C - 2*a^3*A* c*d + 6*a*A*b^2*c*d - 6*a^2*b*B*c*d + 2*b^3*B*c*d + 2*a^3*c*C*d - 6*a*b...
Time = 4.67 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.16, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {3042, 4128, 27, 3042, 4128, 27, 3042, 4132, 27, 3042, 4099, 3042, 4098, 104, 221}
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.
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(a+b \tan (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {c+d \tan (e+f x)} \left (-\left (\left (-3 C a^2-2 b B a+2 A b^2-5 b^2 C\right ) d \tan ^2(e+f x)\right )-5 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (5 b c-3 a d)+A b (5 a c+3 b d)\right )}{2 (a+b \tan (e+f x))^{5/2}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-\left (\left (-3 C a^2-2 b B a+2 A b^2-5 b^2 C\right ) d \tan ^2(e+f x)\right )-5 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (5 b c-3 a d)+A b (5 a c+3 b d)\right )}{(a+b \tan (e+f x))^{5/2}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-\left (\left (-3 C a^2-2 b B a+2 A b^2-5 b^2 C\right ) d \tan (e+f x)^2\right )-5 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (5 b c-3 a d)+A b (5 a c+3 b d)\right )}{(a+b \tan (e+f x))^{5/2}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\frac {2 \int \frac {-15 ((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-C c-B d))) \tan (e+f x) b^2+(3 a c+b d) ((b B-a C) (5 b c-3 a d)+A b (5 a c+3 b d)) b+d \left (3 C d a^4+2 b B d a^3+2 b^2 (5 B c+4 A d-C d) a^2-2 b^3 (10 A c-10 C c-11 B d) a-b^4 (10 B c+3 (4 A-5 C) d)\right ) \tan ^2(e+f x)-(3 b c-a d) \left (3 C d a^3+2 b B d a^2-5 b^2 (B c-2 C d) a+A b^2 (5 b c-7 a d)-5 b^3 (c C+B d)\right )}{2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 b \left (a^2+b^2\right )}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+2 a^3 b B d-a^2 b^2 (7 A d+5 B c-13 C d)+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-15 ((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-C c-B d))) \tan (e+f x) b^2+(3 a c+b d) ((b B-a C) (5 b c-3 a d)+A b (5 a c+3 b d)) b+d \left (3 C d a^4+2 b B d a^3+2 b^2 (5 B c+4 A d-C d) a^2-2 b^3 (10 A c-10 C c-11 B d) a-b^4 (10 B c+3 (4 A-5 C) d)\right ) \tan ^2(e+f x)-(3 b c-a d) \left (3 C d a^3+2 b B d a^2-5 b^2 (B c-2 C d) a+A b^2 (5 b c-7 a d)-5 b^3 (c C+B d)\right )}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 b \left (a^2+b^2\right )}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+2 a^3 b B d-a^2 b^2 (7 A d+5 B c-13 C d)+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-15 ((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-C c-B d))) \tan (e+f x) b^2+(3 a c+b d) ((b B-a C) (5 b c-3 a d)+A b (5 a c+3 b d)) b+d \left (3 C d a^4+2 b B d a^3+2 b^2 (5 B c+4 A d-C d) a^2-2 b^3 (10 A c-10 C c-11 B d) a-b^4 (10 B c+3 (4 A-5 C) d)\right ) \tan (e+f x)^2-(3 b c-a d) \left (3 C d a^3+2 b B d a^2-5 b^2 (B c-2 C d) a+A b^2 (5 b c-7 a d)-5 b^3 (c C+B d)\right )}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 b \left (a^2+b^2\right )}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+2 a^3 b B d-a^2 b^2 (7 A d+5 B c-13 C d)+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {\frac {-\frac {2 \int \frac {15 \left (b^2 (b c-a d) \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^3-3 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^2 (b c-a d) \left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3+3 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x)\right )}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^6 C d^2+2 a^5 b B d^2+a^4 b^2 d (d (8 A+C)+10 B c)-a^3 b^3 \left (50 c d (A-C)+B \left (15 c^2-39 d^2\right )\right )+a^2 b^4 \left (45 A c^2-49 A d^2-90 B c d-45 c^2 C+58 C d^2\right )+a b^5 \left (70 c d (A-C)+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (4 B d+3 c C)-3 A \left (5 c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+2 a^3 b B d-a^2 b^2 (7 A d+5 B c-13 C d)+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {15 \int \frac {b^2 (b c-a d) \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^3-3 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^2 (b c-a d) \left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3+3 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^6 C d^2+2 a^5 b B d^2+a^4 b^2 d (d (8 A+C)+10 B c)-a^3 b^3 \left (50 c d (A-C)+B \left (15 c^2-39 d^2\right )\right )+a^2 b^4 \left (45 A c^2-49 A d^2-90 B c d-45 c^2 C+58 C d^2\right )+a b^5 \left (70 c d (A-C)+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (4 B d+3 c C)-3 A \left (5 c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+2 a^3 b B d-a^2 b^2 (7 A d+5 B c-13 C d)+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {15 \int \frac {b^2 (b c-a d) \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^3-3 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^2 (b c-a d) \left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3+3 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^6 C d^2+2 a^5 b B d^2+a^4 b^2 d (d (8 A+C)+10 B c)-a^3 b^3 \left (50 c d (A-C)+B \left (15 c^2-39 d^2\right )\right )+a^2 b^4 \left (45 A c^2-49 A d^2-90 B c d-45 c^2 C+58 C d^2\right )+a b^5 \left (70 c d (A-C)+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (4 B d+3 c C)-3 A \left (5 c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}}{3 b \left (a^2+b^2\right )}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+2 a^3 b B d-a^2 b^2 (7 A d+5 B c-13 C d)+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4099 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+2 a^3 b B d-a^2 b^2 (7 A d+5 B c-13 C d)+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^6 C d^2+2 a^5 b B d^2+a^4 b^2 d (d (8 A+C)+10 B c)-a^3 b^3 \left (50 c d (A-C)+B \left (15 c^2-39 d^2\right )\right )+a^2 b^4 \left (45 A c^2-49 A d^2-90 B c d-45 c^2 C+58 C d^2\right )+a b^5 \left (70 c d (A-C)+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (4 B d+3 c C)-3 A \left (5 c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}-\frac {15 \left (-\frac {1}{2} b^2 (a-i b)^3 (c+i d)^2 (A+i B-C) (b c-a d) \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx-\frac {1}{2} b^2 (a+i b)^3 (c-i d)^2 (A-i B-C) (b c-a d) \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}}{5 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+2 a^3 b B d-a^2 b^2 (7 A d+5 B c-13 C d)+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^6 C d^2+2 a^5 b B d^2+a^4 b^2 d (d (8 A+C)+10 B c)-a^3 b^3 \left (50 c d (A-C)+B \left (15 c^2-39 d^2\right )\right )+a^2 b^4 \left (45 A c^2-49 A d^2-90 B c d-45 c^2 C+58 C d^2\right )+a b^5 \left (70 c d (A-C)+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (4 B d+3 c C)-3 A \left (5 c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}-\frac {15 \left (-\frac {1}{2} b^2 (a-i b)^3 (c+i d)^2 (A+i B-C) (b c-a d) \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx-\frac {1}{2} b^2 (a+i b)^3 (c-i d)^2 (A-i B-C) (b c-a d) \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}}{5 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4098 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+2 a^3 b B d-a^2 b^2 (7 A d+5 B c-13 C d)+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^6 C d^2+2 a^5 b B d^2+a^4 b^2 d (d (8 A+C)+10 B c)-a^3 b^3 \left (50 c d (A-C)+B \left (15 c^2-39 d^2\right )\right )+a^2 b^4 \left (45 A c^2-49 A d^2-90 B c d-45 c^2 C+58 C d^2\right )+a b^5 \left (70 c d (A-C)+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (4 B d+3 c C)-3 A \left (5 c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}-\frac {15 \left (-\frac {b^2 (a-i b)^3 (c+i d)^2 (A+i B-C) (b c-a d) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}-\frac {b^2 (a+i b)^3 (c-i d)^2 (A-i B-C) (b c-a d) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}}{5 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+2 a^3 b B d-a^2 b^2 (7 A d+5 B c-13 C d)+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^6 C d^2+2 a^5 b B d^2+a^4 b^2 d (d (8 A+C)+10 B c)-a^3 b^3 \left (50 c d (A-C)+B \left (15 c^2-39 d^2\right )\right )+a^2 b^4 \left (45 A c^2-49 A d^2-90 B c d-45 c^2 C+58 C d^2\right )+a b^5 \left (70 c d (A-C)+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (4 B d+3 c C)-3 A \left (5 c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}-\frac {15 \left (-\frac {b^2 (a-i b)^3 (c+i d)^2 (A+i B-C) (b c-a d) \int \frac {1}{-i a+b+\frac {(i c-d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}-\frac {b^2 (a+i b)^3 (c-i d)^2 (A-i B-C) (b c-a d) \int \frac {1}{i a+b-\frac {(i c+d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}}{5 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+2 a^3 b B d-a^2 b^2 (7 A d+5 B c-13 C d)+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (3 a^6 C d^2+2 a^5 b B d^2+a^4 b^2 d (d (8 A+C)+10 B c)-a^3 b^3 \left (50 c d (A-C)+B \left (15 c^2-39 d^2\right )\right )+a^2 b^4 \left (45 A c^2-49 A d^2-90 B c d-45 c^2 C+58 C d^2\right )+a b^5 \left (70 c d (A-C)+B \left (45 c^2-23 d^2\right )\right )+b^6 \left (5 c (4 B d+3 c C)-3 A \left (5 c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}-\frac {15 \left (\frac {i b^2 (a+i b)^3 (c-i d)^{3/2} (A-i B-C) (b c-a d) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b}}-\frac {i b^2 (a-i b)^3 (c+i d)^{3/2} (A+i B-C) (b c-a d) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b}}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 b \left (a^2+b^2\right )}}{5 b \left (a^2+b^2\right )}\) |
Int[((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/( a + b*Tan[e + f*x])^(7/2),x]
(-2*(A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(3/2))/(5*b*(a^2 + b^2)*f *(a + b*Tan[e + f*x])^(5/2)) + ((-2*(2*a^3*b*B*d + 3*a^4*C*d + b^4*(5*B*c + 3*A*d) + 2*a*b^3*(5*A*c - 5*c*C - 4*B*d) - a^2*b^2*(5*B*c + 7*A*d - 13*C *d))*Sqrt[c + d*Tan[e + f*x]])/(3*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(3/ 2)) + ((-15*((I*(a + I*b)^3*b^2*(A - I*B - C)*(c - I*d)^(3/2)*(b*c - a*d)* ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d *Tan[e + f*x]])])/(Sqrt[a - I*b]*f) - (I*(a - I*b)^3*b^2*(A + I*B - C)*(c + I*d)^(3/2)*(b*c - a*d)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/ (Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*f)))/((a^2 + b^2 )*(b*c - a*d)) - (2*(2*a^5*b*B*d^2 + 3*a^6*C*d^2 + a^4*b^2*d*(10*B*c + (8* A + C)*d) + a^2*b^4*(45*A*c^2 - 45*c^2*C - 90*B*c*d - 49*A*d^2 + 58*C*d^2) - a^3*b^3*(50*c*(A - C)*d + B*(15*c^2 - 39*d^2)) + a*b^5*(70*c*(A - C)*d + B*(45*c^2 - 23*d^2)) + b^6*(5*c*(3*c*C + 4*B*d) - 3*A*(5*c^2 - d^2)))*Sq rt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x] ]))/(3*b*(a^2 + b^2)))/(5*b*(a^2 + b^2))
3.2.40.3.1 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_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[A^2/f Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e + f* x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(A + I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T an[e + f*x]), x], x] + Simp[(A - I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d *Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A , B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 0]
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.
\[\int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {7}{2}}}d x\]
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(7/2),x, algorithm="fricas")
\[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx \]
integrate((c+d*tan(f*x+e))**(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*ta n(f*x+e))**(7/2),x)
Integral((c + d*tan(e + f*x))**(3/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)* *2)/(a + b*tan(e + f*x))**(7/2), x)
Exception generated. \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Exception raised: ValueError} \]
integrate((c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(7/2),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. \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(7/2),x, algorithm="giac")
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Hanged} \]