Integrand size = 49, antiderivative size = 597 \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=-\frac {(i A+B-i C) \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{7/2} f}-\frac {(B-i (A-C)) \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{7/2} f}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac {2 \left (4 a^3 b B d+a^4 C d+b^4 (5 B c+A d)+2 a b^3 (5 A c-5 c C-3 B d)-a^2 b^2 (5 B c+9 A d-11 C d)\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))^{3/2}}+\frac {2 \left (8 a^5 b B d^2+2 a^6 C d^2-a^4 b^2 d (25 B c+33 A d-39 C d)-a^2 b^4 \left (45 A c^2-45 c^2 C-90 B c d-29 A d^2+23 C d^2\right )+a^3 b^3 \left (80 c (A-C) d+B \left (15 c^2-49 d^2\right )\right )-a b^5 \left (40 c (A-C) d+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (3 c C+B d)-A \left (15 c^2+2 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)}} \] Output:
-(I*A+B-I*C)*(c-I*d)^(1/2)*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)^(1 /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-2/5*(A*b^2-a*(B*b-C*a))*(c+d*tan(f*x+e))^(1/2 )/b/(a^2+b^2)/f/(a+b*tan(f*x+e))^(5/2)-2/15*(4*a^3*b*B*d+a^4*C*d+b^4*(A*d+ 5*B*c)+2*a*b^3*(5*A*c-3*B*d-5*C*c)-a^2*b^2*(9*A*d+5*B*c-11*C*d))*(c+d*tan( f*x+e))^(1/2)/b/(a^2+b^2)^2/(-a*d+b*c)/f/(a+b*tan(f*x+e))^(3/2)+2/15*(8*a^ 5*b*B*d^2+2*a^6*C*d^2-a^4*b^2*d*(33*A*d+25*B*c-39*C*d)-a^2*b^4*(45*A*c^2-2 9*A*d^2-90*B*c*d-45*C*c^2+23*C*d^2)+a^3*b^3*(80*c*(A-C)*d+B*(15*c^2-49*d^2 ))-a*b^5*(40*c*(A-C)*d+B*(45*c^2-3*d^2))-b^6*(5*c*(B*d+3*C*c)-A*(15*c^2+2* d^2)))*(c+d*tan(f*x+e))^(1/2)/b/(a^2+b^2)^3/(-a*d+b*c)^2/f/(a+b*tan(f*x+e) )^(1/2)
Time = 6.93 (sec) , antiderivative size = 1109, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=-\frac {C \sqrt {c+d \tan (e+f x)}}{2 b f (a+b \tan (e+f x))^{5/2}}-\frac {-\frac {2 \left (\frac {1}{2} b^2 (-4 A b c+5 b c C-a C d)-a \left (-2 b^2 (B c+(A-C) d)-\frac {1}{2} a (b c C-4 b B d-a C d)\right )\right ) \sqrt {c+d \tan (e+f x)}}{5 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{5/2}}-\frac {2 \left (-\frac {2 \left (b^2 (b c-a d) \left (a^2 C d+b^2 (5 B c+A d)+a b (5 A c-5 c C-B d)\right )-a \left (a \left (4 A b^2-4 a b B-a^2 C-5 b^2 C\right ) d (b c-a d)-5 b^2 (b c-a d) (A b c-a B c-b c C-a A d-b B d+a C d)\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (-\frac {15 b (b c-a d)^2 \left (\frac {(i a-b)^3 (A-i B-C) \sqrt {-c+i 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 )}{\sqrt {-a+i b}}-\frac {(i a+b)^3 (A+i B-C) \sqrt {c+i 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 )}{\sqrt {a+i b}}\right )}{2 \left (a^2+b^2\right ) f}-\frac {2 \left (b^2 \left ((b c-a d) \left (b^2 d-\frac {3}{2} a (b c-a d)\right ) \left (a^2 C d+b^2 (5 B c+A d)+a b (5 A c-5 c C-B d)\right )+\left (-\frac {3 b c}{2}+\frac {a d}{2}\right ) \left (a \left (4 A b^2-4 a b B-a^2 C-5 b^2 C\right ) d (b c-a d)-5 b^2 (b c-a d) (A b c-a B c-b c C-a A d-b B d+a C d)\right )\right )-a \left (\frac {3}{2} b (b c-a d) \left (b \left (4 A b^2-4 a b B-a^2 C-5 b^2 C\right ) d (b c-a d)+5 a b (b c-a d) (A b c-a B c-b c C-a A d-b B d+a C d)+b (b c-a d) \left (a^2 C d+b^2 (5 B c+A d)+a b (5 A c-5 c C-B d)\right )\right )-a d \left (b^2 (b c-a d) \left (a^2 C d+b^2 (5 B c+A d)+a b (5 A c-5 c C-B d)\right )-a \left (a \left (4 A b^2-4 a b B-a^2 C-5 b^2 C\right ) d (b c-a d)-5 b^2 (b c-a d) (A b c-a B c-b c C-a A d-b B d+a C d)\right )\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)}}\right )}{3 \left (a^2+b^2\right ) (b c-a d)}\right )}{5 \left (a^2+b^2\right ) (b c-a d)}}{2 b} \] Input:
Integrate[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2 ))/(a + b*Tan[e + f*x])^(7/2),x]
Output:
-1/2*(C*Sqrt[c + d*Tan[e + f*x]])/(b*f*(a + b*Tan[e + f*x])^(5/2)) - ((-2* ((b^2*(-4*A*b*c + 5*b*c*C - a*C*d))/2 - a*(-2*b^2*(B*c + (A - C)*d) - (a*( b*c*C - 4*b*B*d - a*C*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*(b*c - a*d)*(a^2*C*d + b^2*(5*B*c + A*d) + a*b*(5*A*c - 5*c*C - B*d)) - a*(a*(4*A*b^2 - 4*a*b* B - a^2*C - 5*b^2*C)*d*(b*c - a*d) - 5*b^2*(b*c - a*d)*(A*b*c - a*B*c - b* c*C - a*A*d - b*B*d + a*C*d)))*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*(b*c - a*d)^2*(((I*a - b)^3*(A - I*B - C)*Sqrt[-c + I*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] - ( (I*a + b)^3*(A + I*B - C)*Sqrt[c + I*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]))/ (2*(a^2 + b^2)*f) - (2*(b^2*((b*c - a*d)*(b^2*d - (3*a*(b*c - a*d))/2)*(a^ 2*C*d + b^2*(5*B*c + A*d) + a*b*(5*A*c - 5*c*C - B*d)) + ((-3*b*c)/2 + (a* d)/2)*(a*(4*A*b^2 - 4*a*b*B - a^2*C - 5*b^2*C)*d*(b*c - a*d) - 5*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))) - a*((3*b*(b*c - a*d)*(b*(4*A*b^2 - 4*a*b*B - a^2*C - 5*b^2*C)*d*(b*c - a*d) + 5*a*b*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d) + b*(b*c - a*d)*(a^2 *C*d + b^2*(5*B*c + A*d) + a*b*(5*A*c - 5*c*C - B*d))))/2 - a*d*(b^2*(b*c - a*d)*(a^2*C*d + b^2*(5*B*c + A*d) + a*b*(5*A*c - 5*c*C - B*d)) - a*(a...
Time = 7.46 (sec) , antiderivative size = 693, 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, 4132, 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.
| \(\displaystyle \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(a+b \tan (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {2 \int \frac {-\left (\left (-C a^2-4 b B a+4 A b^2-5 b^2 C\right ) d \tan ^2(e+f x)\right )-5 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (5 b c-a d)+A b (5 a c+b d)}{2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-\left (\left (-C a^2-4 b B a+4 A b^2-5 b^2 C\right ) d \tan ^2(e+f x)\right )-5 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (5 b c-a d)+A b (5 a c+b d)}{(a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-\left (\left (-C a^2-4 b B a+4 A b^2-5 b^2 C\right ) d \tan (e+f x)^2\right )-5 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (5 b c-a d)+A b (5 a c+b d)}{(a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {2 \int \frac {2 d \left (C d a^4+4 b B d a^3-b^2 (5 B c+9 A d-11 C d) a^2+2 b^3 (5 A c-5 C c-3 B d) a+b^4 (5 B c+A d)\right ) \tan ^2(e+f x)+15 b (b c-a d) \left (-\left ((B c+(A-C) d) a^2\right )+2 b (A c-C c-B d) a+b^2 (B c+(A-C) d)\right ) \tan (e+f x)+2 \left (b^2 d-\frac {3}{2} a (b c-a d)\right ) ((b B-a C) (5 b c-a d)+A b (5 a c+b d))+(3 b c-a d) \left (C d a^3+4 b B d a^2-5 b^2 (B c-2 C d) a+A b^2 (5 b c-9 a d)-5 b^3 (c C+B d)\right )}{2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+4 a^3 b B d-a^2 b^2 (9 A d+5 B c-11 C d)+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {2 d \left (C d a^4+4 b B d a^3-b^2 (5 B c+9 A d-11 C d) a^2+2 b^3 (5 A c-5 C c-3 B d) a+b^4 (5 B c+A d)\right ) \tan ^2(e+f x)+15 b (b c-a d) \left (-\left ((B c+(A-C) d) a^2\right )+2 b (A c-C c-B d) a+b^2 (B c+(A-C) d)\right ) \tan (e+f x)+2 \left (b^2 d-\frac {3}{2} a (b c-a d)\right ) ((b B-a C) (5 b c-a d)+A b (5 a c+b d))+(3 b c-a d) \left (C d a^3+4 b B d a^2-5 b^2 (B c-2 C d) a+A b^2 (5 b c-9 a d)-5 b^3 (c C+B d)\right )}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+4 a^3 b B d-a^2 b^2 (9 A d+5 B c-11 C d)+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {2 d \left (C d a^4+4 b B d a^3-b^2 (5 B c+9 A d-11 C d) a^2+2 b^3 (5 A c-5 C c-3 B d) a+b^4 (5 B c+A d)\right ) \tan (e+f x)^2+15 b (b c-a d) \left (-\left ((B c+(A-C) d) a^2\right )+2 b (A c-C c-B d) a+b^2 (B c+(A-C) d)\right ) \tan (e+f x)+2 \left (b^2 d-\frac {3}{2} a (b c-a d)\right ) ((b B-a C) (5 b c-a d)+A b (5 a c+b d))+(3 b c-a d) \left (C d a^3+4 b B d a^2-5 b^2 (B c-2 C d) a+A b^2 (5 b c-9 a d)-5 b^3 (c C+B d)\right )}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+4 a^3 b B d-a^2 b^2 (9 A d+5 B c-11 C d)+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {-\frac {2 \int \frac {15 \left (b (b c-a d)^2 \left ((A c-C c-B d) a^3+3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a-b^3 (B c+(A-C) d)\right )-b (b c-a d)^2 \left (-\left ((B c+(A-C) d) a^3\right )+3 b (A c-C c-B d) a^2+3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x)\right )}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (2 a^6 C d^2+8 a^5 b B d^2-a^4 b^2 d (33 A d+25 B c-39 C d)+a^3 b^3 \left (80 c d (A-C)+B \left (15 c^2-49 d^2\right )\right )-a^2 b^4 \left (45 A c^2-29 A d^2-90 B c d-45 c^2 C+23 C d^2\right )-a b^5 \left (40 c d (A-C)+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (B d+3 c C)-A \left (15 c^2+2 d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+4 a^3 b B d-a^2 b^2 (9 A d+5 B c-11 C d)+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {15 \int \frac {b (b c-a d)^2 \left ((A c-C c-B d) a^3+3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a-b^3 (B c+(A-C) d)\right )-b (b c-a d)^2 \left (-\left ((B c+(A-C) d) a^3\right )+3 b (A c-C c-B d) a^2+3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (2 a^6 C d^2+8 a^5 b B d^2-a^4 b^2 d (33 A d+25 B c-39 C d)+a^3 b^3 \left (80 c d (A-C)+B \left (15 c^2-49 d^2\right )\right )-a^2 b^4 \left (45 A c^2-29 A d^2-90 B c d-45 c^2 C+23 C d^2\right )-a b^5 \left (40 c d (A-C)+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (B d+3 c C)-A \left (15 c^2+2 d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+4 a^3 b B d-a^2 b^2 (9 A d+5 B c-11 C d)+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {15 \int \frac {b (b c-a d)^2 \left ((A c-C c-B d) a^3+3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a-b^3 (B c+(A-C) d)\right )-b (b c-a d)^2 \left (-\left ((B c+(A-C) d) a^3\right )+3 b (A c-C c-B d) a^2+3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (2 a^6 C d^2+8 a^5 b B d^2-a^4 b^2 d (33 A d+25 B c-39 C d)+a^3 b^3 \left (80 c d (A-C)+B \left (15 c^2-49 d^2\right )\right )-a^2 b^4 \left (45 A c^2-29 A d^2-90 B c d-45 c^2 C+23 C d^2\right )-a b^5 \left (40 c d (A-C)+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (B d+3 c C)-A \left (15 c^2+2 d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+4 a^3 b B d-a^2 b^2 (9 A d+5 B c-11 C d)+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4099 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+4 a^3 b B d-a^2 b^2 (9 A d+5 B c-11 C d)+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2}}-\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (2 a^6 C d^2+8 a^5 b B d^2-a^4 b^2 d (33 A d+25 B c-39 C d)+a^3 b^3 \left (80 c d (A-C)+B \left (15 c^2-49 d^2\right )\right )-a^2 b^4 \left (45 A c^2-29 A d^2-90 B c d-45 c^2 C+23 C d^2\right )-a b^5 \left (40 c d (A-C)+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (B d+3 c C)-A \left (15 c^2+2 d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}-\frac {15 \left (\frac {1}{2} b (a-i b)^3 (c+i d) (A+i B-C) (b c-a d)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} b (a+i b)^3 (c-i d) (A-i B-C) (b c-a d)^2 \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+4 a^3 b B d-a^2 b^2 (9 A d+5 B c-11 C d)+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2}}-\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (2 a^6 C d^2+8 a^5 b B d^2-a^4 b^2 d (33 A d+25 B c-39 C d)+a^3 b^3 \left (80 c d (A-C)+B \left (15 c^2-49 d^2\right )\right )-a^2 b^4 \left (45 A c^2-29 A d^2-90 B c d-45 c^2 C+23 C d^2\right )-a b^5 \left (40 c d (A-C)+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (B d+3 c C)-A \left (15 c^2+2 d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}-\frac {15 \left (\frac {1}{2} b (a-i b)^3 (c+i d) (A+i B-C) (b c-a d)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} b (a+i b)^3 (c-i d) (A-i B-C) (b c-a d)^2 \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4098 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+4 a^3 b B d-a^2 b^2 (9 A d+5 B c-11 C d)+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2}}-\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (2 a^6 C d^2+8 a^5 b B d^2-a^4 b^2 d (33 A d+25 B c-39 C d)+a^3 b^3 \left (80 c d (A-C)+B \left (15 c^2-49 d^2\right )\right )-a^2 b^4 \left (45 A c^2-29 A d^2-90 B c d-45 c^2 C+23 C d^2\right )-a b^5 \left (40 c d (A-C)+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (B d+3 c C)-A \left (15 c^2+2 d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}-\frac {15 \left (\frac {b (a-i b)^3 (c+i d) (A+i B-C) (b c-a d)^2 \int \frac {1}{(i \tan (e+f x)+1) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}+\frac {b (a+i b)^3 (c-i d) (A-i B-C) (b c-a d)^2 \int \frac {1}{(1-i \tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+4 a^3 b B d-a^2 b^2 (9 A d+5 B c-11 C d)+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2}}-\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (2 a^6 C d^2+8 a^5 b B d^2-a^4 b^2 d (33 A d+25 B c-39 C d)+a^3 b^3 \left (80 c d (A-C)+B \left (15 c^2-49 d^2\right )\right )-a^2 b^4 \left (45 A c^2-29 A d^2-90 B c d-45 c^2 C+23 C d^2\right )-a b^5 \left (40 c d (A-C)+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (B d+3 c C)-A \left (15 c^2+2 d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}-\frac {15 \left (\frac {b (a-i b)^3 (c+i d) (A+i B-C) (b c-a d)^2 \int \frac {1}{-i a+b+\frac {(i c-d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}+\frac {b (a+i b)^3 (c-i d) (A-i B-C) (b c-a d)^2 \int \frac {1}{i a+b-\frac {(i c+d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^4 C d+4 a^3 b B d-a^2 b^2 (9 A d+5 B c-11 C d)+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2}}-\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (2 a^6 C d^2+8 a^5 b B d^2-a^4 b^2 d (33 A d+25 B c-39 C d)+a^3 b^3 \left (80 c d (A-C)+B \left (15 c^2-49 d^2\right )\right )-a^2 b^4 \left (45 A c^2-29 A d^2-90 B c d-45 c^2 C+23 C d^2\right )-a b^5 \left (40 c d (A-C)+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (B d+3 c C)-A \left (15 c^2+2 d^2\right )\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)}}-\frac {15 \left (\frac {i b (a-i b)^3 \sqrt {c+i d} (A+i B-C) (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b}}-\frac {i b (a+i b)^3 \sqrt {c-i d} (A-i B-C) (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b}}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 b \left (a^2+b^2\right )}\) |
Input:
Int[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(7/2),x]
Output:
(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*Tan[e + f*x]])/(5*b*(a^2 + b^2)*f*( a + b*Tan[e + f*x])^(5/2)) + ((-2*(4*a^3*b*B*d + a^4*C*d + b^4*(5*B*c + A* d) + 2*a*b^3*(5*A*c - 5*c*C - 3*B*d) - a^2*b^2*(5*B*c + 9*A*d - 11*C*d))*S qrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]) ^(3/2)) - ((-15*(((-I)*(a + I*b)^3*b*(A - I*B - C)*Sqrt[c - I*d]*(b*c - a* d)^2*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*(A + I*B - C)* Sqrt[c + I*d]*(b*c - a*d)^2*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*(8*a^5*b*B*d^2 + 2*a^6*C*d^2 - a^4*b^2*d*(25*B*c + 33*A*d - 39*C*d) - a^2*b^4*(45*A*c^2 - 45*c^2*C - 90*B*c*d - 29*A*d^2 + 23 *C*d^2) + a^3*b^3*(80*c*(A - C)*d + B*(15*c^2 - 49*d^2)) - a*b^5*(40*c*(A - C)*d + B*(45*c^2 - 3*d^2)) - b^6*(5*c*(3*c*C + B*d) - A*(15*c^2 + 2*d^2) ))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]))/(3*(a^2 + b^2)*(b*c - a*d)))/(5*b*(a^2 + b^2))
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.
\[\int \frac {\sqrt {c +d \tan \left (f x +e \right )}\, \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {7}{2}}}d x\]
Input:
int((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e) )^(7/2),x)
Output:
int((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e) )^(7/2),x)
Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(7/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\int \frac {\sqrt {c + d \tan {\left (e + f x \right )}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((c+d*tan(f*x+e))**(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*ta n(f*x+e))**(7/2),x)
Output:
Integral(sqrt(c + d*tan(e + f*x))*(A + B*tan(e + f*x) + C*tan(e + f*x)**2) /(a + b*tan(e + f*x))**(7/2), x)
Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(7/2),x, algorithm="maxima")
Output:
Timed out
\[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\int { \frac {{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} \sqrt {d \tan \left (f x + e\right ) + c}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(7/2),x, algorithm="giac")
Output:
integrate((C*tan(f*x + e)^2 + B*tan(f*x + e) + A)*sqrt(d*tan(f*x + e) + c) /(b*tan(f*x + e) + a)^(7/2), x)
Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Hanged} \] Input:
int(((c + d*tan(e + f*x))^(1/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/( a + b*tan(e + f*x))^(7/2),x)
Output:
\text{Hanged}
\[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {too large to display} \] Input:
int((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e) )^(7/2),x)
Output:
(4*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**2*a*b*c *d**3 + 16*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)* *2*b**3*d**3 - 4*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**2*b**2*c**2*d**2 + 10*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)*a**2*c*d**3 + 40*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f *x)*b + a)*tan(e + f*x)*a*b**2*d**3 - 12*sqrt(tan(e + f*x)*d + c)*sqrt(tan (e + f*x)*b + a)*tan(e + f*x)*a*b*c**2*d**2 - 8*sqrt(tan(e + f*x)*d + c)*s qrt(tan(e + f*x)*b + a)*tan(e + f*x)*b**3*c*d**2 + 2*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)*b**2*c**3*d + 30*sqrt(tan(e + f* x)*d + c)*sqrt(tan(e + f*x)*b + a)*a**2*b*d**3 + 10*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*a**2*c**2*d**2 - 20*sqrt(tan(e + f*x)*d + c)*s qrt(tan(e + f*x)*b + a)*a*b**2*c*d**2 - 16*sqrt(tan(e + f*x)*d + c)*sqrt(t an(e + f*x)*b + a)*a*b*c**3*d + 6*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f* x)*b + a)*b**3*c**2*d + 6*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a )*b**2*c**4 + 15*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*ta n(e + f*x))/(tan(e + f*x)**5*b**4*d + 4*tan(e + f*x)**4*a*b**3*d + tan(e + f*x)**4*b**4*c + 6*tan(e + f*x)**3*a**2*b**2*d + 4*tan(e + f*x)**3*a*b**3 *c + 4*tan(e + f*x)**2*a**3*b*d + 6*tan(e + f*x)**2*a**2*b**2*c + tan(e + f*x)*a**4*d + 4*tan(e + f*x)*a**3*b*c + a**4*c),x)*tan(e + f*x)**3*a**4*b* *3*d**4*f - 30*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*t...