Integrand size = 47, antiderivative size = 679 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {(i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 (c-i d)^{5/2} f}-\frac {(B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 (c+i d)^{5/2} f}-\frac {b^{3/2} \left (7 a^3 b B d-5 a^4 C d+b^4 (2 B c-5 A d)+a b^3 (4 A c-4 c C+3 B d)-a^2 b^2 (2 B c+(9 A+C) d)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{7/2} f}-\frac {d \left (2 b^2 c (c C-B d)-3 a b B \left (c^2+d^2\right )+a^2 \left (5 c^2 C-2 B c d+3 C d^2\right )+A \left (2 a^2 d^2+b^2 \left (3 c^2+5 d^2\right )\right )\right )}{3 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {d \left (2 a^3 d^2 \left (B c^2+2 c C d-B d^2\right )+2 b^3 c \left (2 c^3 C-3 B c^2 d-B d^3\right )-a b^2 \left (B c^4-4 c C d^3+3 B d^4\right )+a^2 b \left (5 c^4 C-6 B c^3 d+2 c^2 C d^2-2 B c d^3+C d^4\right )-A \left (4 a^3 c d^3+4 a b^2 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )-b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \]
-(I*A+B-I*C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(a-I*b)^2/(c-I* d)^(5/2)/f-(B-I*(A-C))*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(a+I* b)^2/(c+I*d)^(5/2)/f-b^(3/2)*(7*a^3*b*B*d-5*a^4*C*d+b^4*(-5*A*d+2*B*c)+a*b ^3*(4*A*c+3*B*d-4*C*c)-a^2*b^2*(2*B*c+(9*A+C)*d))*arctanh(b^(1/2)*(c+d*tan (f*x+e))^(1/2)/(-a*d+b*c)^(1/2))/(a^2+b^2)^2/(-a*d+b*c)^(7/2)/f-d*(2*a^3*d ^2*(B*c^2-B*d^2+2*C*c*d)+2*b^3*c*(-3*B*c^2*d-B*d^3+2*C*c^3)-a*b^2*(B*c^4+3 *B*d^4-4*C*c*d^3)+a^2*b*(-6*B*c^3*d-2*B*c*d^3+5*C*c^4+2*C*c^2*d^2+C*d^4)-A *(4*a^3*c*d^3+4*a*b^2*c*d^3-4*a^2*b*d^2*(2*c^2+d^2)-b^3*(c^4+10*c^2*d^2+5* d^4)))/(a^2+b^2)/(-a*d+b*c)^3/(c^2+d^2)^2/f/(c+d*tan(f*x+e))^(1/2)-1/3*d*( 2*b^2*c*(-B*d+C*c)-3*a*b*B*(c^2+d^2)+a^2*(-2*B*c*d+5*C*c^2+3*C*d^2)+A*(2*a ^2*d^2+b^2*(3*c^2+5*d^2)))/(a^2+b^2)/(-a*d+b*c)^2/(c^2+d^2)/f/(c+d*tan(f*x +e))^(3/2)+(-A*b^2+a*(B*b-C*a))/(a^2+b^2)/(-a*d+b*c)/f/(a+b*tan(f*x+e))/(c +d*tan(f*x+e))^(3/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(6052\) vs. \(2(679)=1358\).
Time = 6.84 (sec) , antiderivative size = 6052, normalized size of antiderivative = 8.91 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Result too large to show} \]
Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2* (c + d*Tan[e + f*x])^(5/2)),x]
Time = 7.03 (sec) , antiderivative size = 767, normalized size of antiderivative = 1.13, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.468, Rules used = {3042, 4132, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 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.
| \(\displaystyle \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (e+f x)+C \tan (e+f x)^2}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {\int \frac {5 A d b^2+5 \left (A b^2-a (b B-a C)\right ) d \tan ^2(e+f x)-2 a A (b c-a d)-(b B-a C) (2 b c+3 a d)+2 (A b-C b-a B) (b c-a d) \tan (e+f x)}{2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {5 A d b^2+5 \left (A b^2-a (b B-a C)\right ) d \tan ^2(e+f x)-2 a A (b c-a d)-(b B-a C) (2 b c+3 a d)+2 (A b-C b-a B) (b c-a d) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}dx}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {5 A d b^2+5 \left (A b^2-a (b B-a C)\right ) d \tan (e+f x)^2-2 a A (b c-a d)-(b B-a C) (2 b c+3 a d)+2 (A b-C b-a B) (b c-a d) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}dx}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {\frac {2 \int -\frac {3 \left (2 d^2 (A c-C c+B d) a^3-b (4 A+C) d \left (c^2+d^2\right ) a^2-b^2 \left (2 C c^3-B d c^2+4 C d^2 c-3 B d^3-2 A \left (c^3+2 d^2 c\right )\right ) a-b d \left (2 A d^2 a^2+\left (5 C c^2-2 B d c+3 C d^2\right ) a^2-3 b B \left (c^2+d^2\right ) a+2 b^2 c (c C-B d)+A b^2 \left (3 c^2+5 d^2\right )\right ) \tan ^2(e+f x)+b^3 (2 B c-5 A d) \left (c^2+d^2\right )+2 (b c-a d)^2 (b c C-b B d-A (b c+a d)+a (B c+C d)) \tan (e+f x)\right )}{2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}dx}{3 \left (c^2+d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {2 d^2 (A c-C c+B d) a^3-b (4 A+C) d \left (c^2+d^2\right ) a^2-b^2 \left (2 C c^3-B d c^2+4 C d^2 c-3 B d^3-2 A \left (c^3+2 d^2 c\right )\right ) a-b d \left (2 A d^2 a^2+\left (5 C c^2-2 B d c+3 C d^2\right ) a^2-3 b B \left (c^2+d^2\right ) a+2 b^2 c (c C-B d)+A b^2 \left (3 c^2+5 d^2\right )\right ) \tan ^2(e+f x)+b^3 (2 B c-5 A d) \left (c^2+d^2\right )+2 (b c-a d)^2 (a B c+b C c-b B d+a C d-A (b c+a d)) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}dx}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {2 d^2 (A c-C c+B d) a^3-b (4 A+C) d \left (c^2+d^2\right ) a^2-b^2 \left (2 C c^3-B d c^2+4 C d^2 c-3 B d^3-2 A \left (c^3+2 d^2 c\right )\right ) a-b d \left (2 A d^2 a^2+\left (5 C c^2-2 B d c+3 C d^2\right ) a^2-3 b B \left (c^2+d^2\right ) a+2 b^2 c (c C-B d)+A b^2 \left (3 c^2+5 d^2\right )\right ) \tan (e+f x)^2+b^3 (2 B c-5 A d) \left (c^2+d^2\right )+2 (b c-a d)^2 (a B c+b C c-b B d+a C d-A (b c+a d)) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}dx}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 A d^2 a^2+\left (5 C c^2-2 B d c+3 C d^2\right ) a^2-3 b B \left (c^2+d^2\right ) a+2 b^2 c (c C-B d)+A b^2 \left (3 c^2+5 d^2\right )\right )}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 \int \frac {2 d^3 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^4-2 b d^2 \left (3 C c^3-4 B d c^2+C d^2 c-A \left (3 c^2+d^2\right ) c-2 B d^3\right ) a^3+b^2 d \left (C c^4+4 C d^2 c^2-4 B d^3 c-C d^4-2 A \left (3 c^4+7 d^2 c^2+2 d^4\right )\right ) a^2-b^3 \left (2 C c^5+B d c^4+10 C d^2 c^3-6 B d^3 c^2+4 C d^4 c-3 B d^5-2 A \left (c^5+5 d^2 c^3+2 d^4 c\right )\right ) a+b^4 (2 B c-5 A d) \left (c^2+d^2\right )^2-b d \left (2 d^2 \left (B c^2+2 C d c-B d^2\right ) a^3+b \left (5 C c^4-6 B d c^3+2 C d^2 c^2-2 B d^3 c+C d^4\right ) a^2-b^2 \left (B c^4-4 C d^3 c+3 B d^4\right ) a+2 b^3 c \left (2 C c^3-3 B d c^2-B d^3\right )-A \left (-\left (\left (c^4+10 d^2 c^2+5 d^4\right ) b^3\right )+4 a c d^3 b^2-4 a^2 d^2 \left (2 c^2+d^2\right ) b+4 a^3 c d^3\right )\right ) \tan ^2(e+f x)-2 (b c-a d)^3 \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )-a B \left (c^2-d^2\right )-b \left (C c^2-2 B d c-C d^2\right )\right ) \tan (e+f x)}{2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{(b c-a d) \left (c^2+d^2\right )}-\frac {2 d \left (2 d^2 \left (B c^2+2 C d c-B d^2\right ) a^3+b \left (5 C c^4-6 B d c^3+2 C d^2 c^2-2 B d^3 c+C d^4\right ) a^2-b^2 \left (B c^4-4 C d^3 c+3 B d^4\right ) a+2 b^3 c \left (2 C c^3-3 B d c^2-B d^3\right )-A \left (-\left (\left (c^4+10 d^2 c^2+5 d^4\right ) b^3\right )+4 a c d^3 b^2-4 a^2 d^2 \left (2 c^2+d^2\right ) b+4 a^3 c d^3\right )\right )}{(b c-a d) \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}}{(b c-a d) \left (c^2+d^2\right )}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 A d^2 a^2+\left (5 C c^2-2 B d c+3 C d^2\right ) a^2-3 b B \left (c^2+d^2\right ) a+2 b^2 c (c C-B d)+A b^2 \left (3 c^2+5 d^2\right )\right )}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {\frac {\int \frac {2 d^3 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^4-2 b d^2 \left (3 C c^3-4 B d c^2+C d^2 c-A \left (3 c^2+d^2\right ) c-2 B d^3\right ) a^3+b^2 d \left (C c^4+4 C d^2 c^2-4 B d^3 c-C d^4-2 A \left (3 c^4+7 d^2 c^2+2 d^4\right )\right ) a^2-b^3 \left (2 C c^5+B d c^4+10 C d^2 c^3-6 B d^3 c^2+4 C d^4 c-3 B d^5-2 A \left (c^5+5 d^2 c^3+2 d^4 c\right )\right ) a+b^4 (2 B c-5 A d) \left (c^2+d^2\right )^2-b d \left (2 d^2 \left (B c^2+2 C d c-B d^2\right ) a^3+b \left (5 C c^4-6 B d c^3+2 C d^2 c^2-2 B d^3 c+C d^4\right ) a^2-b^2 \left (B c^4-4 C d^3 c+3 B d^4\right ) a+2 b^3 c \left (2 C c^3-3 B d c^2-B d^3\right )-A \left (-\left (\left (c^4+10 d^2 c^2+5 d^4\right ) b^3\right )+4 a c d^3 b^2-4 a^2 d^2 \left (2 c^2+d^2\right ) b+4 a^3 c d^3\right )\right ) \tan ^2(e+f x)-2 (b c-a d)^3 \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )-a B \left (c^2-d^2\right )-b \left (C c^2-2 B d c-C d^2\right )\right ) \tan (e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{(b c-a d) \left (c^2+d^2\right )}-\frac {2 d \left (2 d^2 \left (B c^2+2 C d c-B d^2\right ) a^3+b \left (5 C c^4-6 B d c^3+2 C d^2 c^2-2 B d^3 c+C d^4\right ) a^2-b^2 \left (B c^4-4 C d^3 c+3 B d^4\right ) a+2 b^3 c \left (2 C c^3-3 B d c^2-B d^3\right )-A \left (-\left (\left (c^4+10 d^2 c^2+5 d^4\right ) b^3\right )+4 a c d^3 b^2-4 a^2 d^2 \left (2 c^2+d^2\right ) b+4 a^3 c d^3\right )\right )}{(b c-a d) \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}}{(b c-a d) \left (c^2+d^2\right )}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 A d^2 a^2+\left (5 C c^2-2 B d c+3 C d^2\right ) a^2-3 b B \left (c^2+d^2\right ) a+2 b^2 c (c C-B d)+A b^2 \left (3 c^2+5 d^2\right )\right )}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {\frac {\int \frac {2 d^3 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^4-2 b d^2 \left (3 C c^3-4 B d c^2+C d^2 c-A \left (3 c^2+d^2\right ) c-2 B d^3\right ) a^3+b^2 d \left (C c^4+4 C d^2 c^2-4 B d^3 c-C d^4-2 A \left (3 c^4+7 d^2 c^2+2 d^4\right )\right ) a^2-b^3 \left (2 C c^5+B d c^4+10 C d^2 c^3-6 B d^3 c^2+4 C d^4 c-3 B d^5-2 A \left (c^5+5 d^2 c^3+2 d^4 c\right )\right ) a+b^4 (2 B c-5 A d) \left (c^2+d^2\right )^2-b d \left (2 d^2 \left (B c^2+2 C d c-B d^2\right ) a^3+b \left (5 C c^4-6 B d c^3+2 C d^2 c^2-2 B d^3 c+C d^4\right ) a^2-b^2 \left (B c^4-4 C d^3 c+3 B d^4\right ) a+2 b^3 c \left (2 C c^3-3 B d c^2-B d^3\right )-A \left (-\left (\left (c^4+10 d^2 c^2+5 d^4\right ) b^3\right )+4 a c d^3 b^2-4 a^2 d^2 \left (2 c^2+d^2\right ) b+4 a^3 c d^3\right )\right ) \tan (e+f x)^2-2 (b c-a d)^3 \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )-a B \left (c^2-d^2\right )-b \left (C c^2-2 B d c-C d^2\right )\right ) \tan (e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{(b c-a d) \left (c^2+d^2\right )}-\frac {2 d \left (2 d^2 \left (B c^2+2 C d c-B d^2\right ) a^3+b \left (5 C c^4-6 B d c^3+2 C d^2 c^2-2 B d^3 c+C d^4\right ) a^2-b^2 \left (B c^4-4 C d^3 c+3 B d^4\right ) a+2 b^3 c \left (2 C c^3-3 B d c^2-B d^3\right )-A \left (-\left (\left (c^4+10 d^2 c^2+5 d^4\right ) b^3\right )+4 a c d^3 b^2-4 a^2 d^2 \left (2 c^2+d^2\right ) b+4 a^3 c d^3\right )\right )}{(b c-a d) \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}}{(b c-a d) \left (c^2+d^2\right )}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 A d^2 a^2+\left (5 C c^2-2 B d c+3 C d^2\right ) a^2-3 b B \left (c^2+d^2\right ) a+2 b^2 c (c C-B d)+A b^2 \left (3 c^2+5 d^2\right )\right )}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {\frac {\frac {b^2 \left (-5 C d a^4+7 b B d a^3-b^2 (2 B c+(9 A+C) d) a^2+b^3 (4 A c-4 C c+3 B d) a+b^4 (2 B c-5 A d)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx \left (c^2+d^2\right )^2}{a^2+b^2}+\frac {\int -\frac {2 \left ((b c-a d)^3 \left (\left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right )-(b c-a d)^3 \left (-\left (\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^2\right )+2 b \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{(b c-a d) \left (c^2+d^2\right )}-\frac {2 d \left (2 d^2 \left (B c^2+2 C d c-B d^2\right ) a^3+b \left (5 C c^4-6 B d c^3+2 C d^2 c^2-2 B d^3 c+C d^4\right ) a^2-b^2 \left (B c^4-4 C d^3 c+3 B d^4\right ) a+2 b^3 c \left (2 C c^3-3 B d c^2-B d^3\right )-A \left (-\left (\left (c^4+10 d^2 c^2+5 d^4\right ) b^3\right )+4 a c d^3 b^2-4 a^2 d^2 \left (2 c^2+d^2\right ) b+4 a^3 c d^3\right )\right )}{(b c-a d) \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}}{(b c-a d) \left (c^2+d^2\right )}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 A d^2 a^2+\left (5 C c^2-2 B d c+3 C d^2\right ) a^2-3 b B \left (c^2+d^2\right ) a+2 b^2 c (c C-B d)+A b^2 \left (3 c^2+5 d^2\right )\right )}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {\frac {\frac {b^2 \left (c^2+d^2\right )^2 \left (-5 C d a^4+7 b B d a^3-b^2 (2 B c+(9 A+C) d) a^2+b^3 (4 A c-4 C c+3 B d) a+b^4 (2 B c-5 A d)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {2 \int \frac {(b c-a d)^3 \left (\left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right )-(b c-a d)^3 \left (-\left (\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^2\right )+2 b \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{(b c-a d) \left (c^2+d^2\right )}-\frac {2 d \left (2 d^2 \left (B c^2+2 C d c-B d^2\right ) a^3+b \left (5 C c^4-6 B d c^3+2 C d^2 c^2-2 B d^3 c+C d^4\right ) a^2-b^2 \left (B c^4-4 C d^3 c+3 B d^4\right ) a+2 b^3 c \left (2 C c^3-3 B d c^2-B d^3\right )-A \left (-\left (\left (c^4+10 d^2 c^2+5 d^4\right ) b^3\right )+4 a c d^3 b^2-4 a^2 d^2 \left (2 c^2+d^2\right ) b+4 a^3 c d^3\right )\right )}{(b c-a d) \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}}{(b c-a d) \left (c^2+d^2\right )}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 A d^2 a^2+\left (5 C c^2-2 B d c+3 C d^2\right ) a^2-3 b B \left (c^2+d^2\right ) a+2 b^2 c (c C-B d)+A b^2 \left (3 c^2+5 d^2\right )\right )}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {\frac {\frac {b^2 \left (c^2+d^2\right )^2 \left (-5 C d a^4+7 b B d a^3-b^2 (2 B c+(9 A+C) d) a^2+b^3 (4 A c-4 C c+3 B d) a+b^4 (2 B c-5 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {2 \int \frac {(b c-a d)^3 \left (\left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right )-(b c-a d)^3 \left (-\left (\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) a^2\right )+2 b \left (C c^2-2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{(b c-a d) \left (c^2+d^2\right )}-\frac {2 d \left (2 d^2 \left (B c^2+2 C d c-B d^2\right ) a^3+b \left (5 C c^4-6 B d c^3+2 C d^2 c^2-2 B d^3 c+C d^4\right ) a^2-b^2 \left (B c^4-4 C d^3 c+3 B d^4\right ) a+2 b^3 c \left (2 C c^3-3 B d c^2-B d^3\right )-A \left (-\left (\left (c^4+10 d^2 c^2+5 d^4\right ) b^3\right )+4 a c d^3 b^2-4 a^2 d^2 \left (2 c^2+d^2\right ) b+4 a^3 c d^3\right )\right )}{(b c-a d) \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}}{(b c-a d) \left (c^2+d^2\right )}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {2 d \left (2 a^3 d^2 \left (B c^2-B d^2+2 c C d\right )+a^2 b \left (-6 B c^3 d-2 B c d^3+5 c^4 C+2 c^2 C d^2+C d^4\right )-A \left (4 a^3 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )+4 a b^2 c d^3-\left (b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )-a b^2 \left (B c^4+3 B d^4-4 c C d^3\right )+2 b^3 c \left (-3 B c^2 d-B d^3+2 c^3 C\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^2 \left (c^2+d^2\right )^2 \left (-5 a^4 C d+7 a^3 b B d-a^2 b^2 (d (9 A+C)+2 B c)+a b^3 (4 A c+3 B d-4 c C)+b^4 (2 B c-5 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {2 \left (-\frac {1}{2} (a-i b)^2 (c-i d)^2 (A+i B-C) (b c-a d)^3 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {1}{2} (a+i b)^2 (c+i d)^2 (A-i B-C) (b c-a d)^3 \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {2 d \left (2 a^3 d^2 \left (B c^2-B d^2+2 c C d\right )+a^2 b \left (-6 B c^3 d-2 B c d^3+5 c^4 C+2 c^2 C d^2+C d^4\right )-A \left (4 a^3 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )+4 a b^2 c d^3-\left (b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )-a b^2 \left (B c^4+3 B d^4-4 c C d^3\right )+2 b^3 c \left (-3 B c^2 d-B d^3+2 c^3 C\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^2 \left (c^2+d^2\right )^2 \left (-5 a^4 C d+7 a^3 b B d-a^2 b^2 (d (9 A+C)+2 B c)+a b^3 (4 A c+3 B d-4 c C)+b^4 (2 B c-5 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {2 \left (-\frac {1}{2} (a-i b)^2 (c-i d)^2 (A+i B-C) (b c-a d)^3 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {1}{2} (a+i b)^2 (c+i d)^2 (A-i B-C) (b c-a d)^3 \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {2 d \left (2 a^3 d^2 \left (B c^2-B d^2+2 c C d\right )+a^2 b \left (-6 B c^3 d-2 B c d^3+5 c^4 C+2 c^2 C d^2+C d^4\right )-A \left (4 a^3 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )+4 a b^2 c d^3-\left (b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )-a b^2 \left (B c^4+3 B d^4-4 c C d^3\right )+2 b^3 c \left (-3 B c^2 d-B d^3+2 c^3 C\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^2 \left (c^2+d^2\right )^2 \left (-5 a^4 C d+7 a^3 b B d-a^2 b^2 (d (9 A+C)+2 B c)+a b^3 (4 A c+3 B d-4 c C)+b^4 (2 B c-5 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {2 \left (\frac {i (a-i b)^2 (c-i d)^2 (A+i B-C) (b c-a d)^3 \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {i (a+i b)^2 (c+i d)^2 (A-i B-C) (b c-a d)^3 \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {2 d \left (2 a^3 d^2 \left (B c^2-B d^2+2 c C d\right )+a^2 b \left (-6 B c^3 d-2 B c d^3+5 c^4 C+2 c^2 C d^2+C d^4\right )-A \left (4 a^3 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )+4 a b^2 c d^3-\left (b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )-a b^2 \left (B c^4+3 B d^4-4 c C d^3\right )+2 b^3 c \left (-3 B c^2 d-B d^3+2 c^3 C\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^2 \left (c^2+d^2\right )^2 \left (-5 a^4 C d+7 a^3 b B d-a^2 b^2 (d (9 A+C)+2 B c)+a b^3 (4 A c+3 B d-4 c C)+b^4 (2 B c-5 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {2 \left (\frac {i (a+i b)^2 (c+i d)^2 (A-i B-C) (b c-a d)^3 \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {i (a-i b)^2 (c-i d)^2 (A+i B-C) (b c-a d)^3 \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {2 d \left (2 a^3 d^2 \left (B c^2-B d^2+2 c C d\right )+a^2 b \left (-6 B c^3 d-2 B c d^3+5 c^4 C+2 c^2 C d^2+C d^4\right )-A \left (4 a^3 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )+4 a b^2 c d^3-\left (b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )-a b^2 \left (B c^4+3 B d^4-4 c C d^3\right )+2 b^3 c \left (-3 B c^2 d-B d^3+2 c^3 C\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^2 \left (c^2+d^2\right )^2 \left (-5 a^4 C d+7 a^3 b B d-a^2 b^2 (d (9 A+C)+2 B c)+a b^3 (4 A c+3 B d-4 c C)+b^4 (2 B c-5 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {2 \left (-\frac {(a-i b)^2 (c-i d)^2 (A+i B-C) (b c-a d)^3 \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}-\frac {(a+i b)^2 (c+i d)^2 (A-i B-C) (b c-a d)^3 \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {2 d \left (2 a^3 d^2 \left (B c^2-B d^2+2 c C d\right )+a^2 b \left (-6 B c^3 d-2 B c d^3+5 c^4 C+2 c^2 C d^2+C d^4\right )-A \left (4 a^3 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )+4 a b^2 c d^3-\left (b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )-a b^2 \left (B c^4+3 B d^4-4 c C d^3\right )+2 b^3 c \left (-3 B c^2 d-B d^3+2 c^3 C\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^2 \left (c^2+d^2\right )^2 \left (-5 a^4 C d+7 a^3 b B d-a^2 b^2 (d (9 A+C)+2 B c)+a b^3 (4 A c+3 B d-4 c C)+b^4 (2 B c-5 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {2 \left (-\frac {(a+i b)^2 (c+i d)^2 (A-i B-C) (b c-a d)^3 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}-\frac {(a-i b)^2 (c-i d)^2 (A+i B-C) (b c-a d)^3 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {2 d \left (2 a^3 d^2 \left (B c^2-B d^2+2 c C d\right )+a^2 b \left (-6 B c^3 d-2 B c d^3+5 c^4 C+2 c^2 C d^2+C d^4\right )-A \left (4 a^3 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )+4 a b^2 c d^3-\left (b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )-a b^2 \left (B c^4+3 B d^4-4 c C d^3\right )+2 b^3 c \left (-3 B c^2 d-B d^3+2 c^3 C\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^2 \left (c^2+d^2\right )^2 \left (-5 a^4 C d+7 a^3 b B d-a^2 b^2 (d (9 A+C)+2 B c)+a b^3 (4 A c+3 B d-4 c C)+b^4 (2 B c-5 A d)\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f \left (a^2+b^2\right )}-\frac {2 \left (-\frac {(a+i b)^2 (c+i d)^2 (A-i B-C) (b c-a d)^3 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}-\frac {(a-i b)^2 (c-i d)^2 (A+i B-C) (b c-a d)^3 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {2 d \left (2 a^3 d^2 \left (B c^2-B d^2+2 c C d\right )+a^2 b \left (-6 B c^3 d-2 B c d^3+5 c^4 C+2 c^2 C d^2+C d^4\right )-A \left (4 a^3 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )+4 a b^2 c d^3-\left (b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )-a b^2 \left (B c^4+3 B d^4-4 c C d^3\right )+2 b^3 c \left (-3 B c^2 d-B d^3+2 c^3 C\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^2 \left (c^2+d^2\right )^2 \left (-5 a^4 C d+7 a^3 b B d-a^2 b^2 (d (9 A+C)+2 B c)+a b^3 (4 A c+3 B d-4 c C)+b^4 (2 B c-5 A d)\right ) \int \frac {1}{a+\frac {b (c+d \tan (e+f x))}{d}-\frac {b c}{d}}d\sqrt {c+d \tan (e+f x)}}{d f \left (a^2+b^2\right )}-\frac {2 \left (-\frac {(a+i b)^2 (c+i d)^2 (A-i B-C) (b c-a d)^3 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}-\frac {(a-i b)^2 (c-i d)^2 (A+i B-C) (b c-a d)^3 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+5 c^2 C+3 C d^2\right )-3 a b B \left (c^2+d^2\right )+A b^2 \left (3 c^2+5 d^2\right )+2 b^2 c (c C-B d)\right )}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {2 d \left (2 a^3 d^2 \left (B c^2-B d^2+2 c C d\right )+a^2 b \left (-6 B c^3 d-2 B c d^3+5 c^4 C+2 c^2 C d^2+C d^4\right )-A \left (4 a^3 c d^3-4 a^2 b d^2 \left (2 c^2+d^2\right )+4 a b^2 c d^3-\left (b^3 \left (c^4+10 c^2 d^2+5 d^4\right )\right )\right )-a b^2 \left (B c^4+3 B d^4-4 c C d^3\right )+2 b^3 c \left (-3 B c^2 d-B d^3+2 c^3 C\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 b^{3/2} \left (c^2+d^2\right )^2 \left (-5 a^4 C d+7 a^3 b B d-a^2 b^2 (d (9 A+C)+2 B c)+a b^3 (4 A c+3 B d-4 c C)+b^4 (2 B c-5 A d)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right ) \sqrt {b c-a d}}-\frac {2 \left (-\frac {(a+i b)^2 (c+i d)^2 (A-i B-C) (b c-a d)^3 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}-\frac {(a-i b)^2 (c-i d)^2 (A+i B-C) (b c-a d)^3 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\) |
Int[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*(c + d *Tan[e + f*x])^(5/2)),x]
-((A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])* (c + d*Tan[e + f*x])^(3/2))) - ((2*d*(2*a^2*A*d^2 + 2*b^2*c*(c*C - B*d) - 3*a*b*B*(c^2 + d^2) + A*b^2*(3*c^2 + 5*d^2) + a^2*(5*c^2*C - 2*B*c*d + 3*C *d^2)))/(3*(b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (((-2*( -(((a + I*b)^2*(A - I*B - C)*(c + I*d)^2*(b*c - a*d)^3*ArcTan[Tan[e + f*x] /Sqrt[c - I*d]])/(Sqrt[c - I*d]*f)) - ((a - I*b)^2*(A + I*B - C)*(c - I*d) ^2*(b*c - a*d)^3*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f)))/( a^2 + b^2) - (2*b^(3/2)*(c^2 + d^2)^2*(7*a^3*b*B*d - 5*a^4*C*d + b^4*(2*B* c - 5*A*d) + a*b^3*(4*A*c - 4*c*C + 3*B*d) - a^2*b^2*(2*B*c + (9*A + C)*d) )*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2 )*Sqrt[b*c - a*d]*f))/((b*c - a*d)*(c^2 + d^2)) - (2*d*(2*a^3*d^2*(B*c^2 + 2*c*C*d - B*d^2) + 2*b^3*c*(2*c^3*C - 3*B*c^2*d - B*d^3) - a*b^2*(B*c^4 - 4*c*C*d^3 + 3*B*d^4) + a^2*b*(5*c^4*C - 6*B*c^3*d + 2*c^2*C*d^2 - 2*B*c*d ^3 + C*d^4) - A*(4*a^3*c*d^3 + 4*a*b^2*c*d^3 - 4*a^2*b*d^2*(2*c^2 + d^2) - b^3*(c^4 + 10*c^2*d^2 + 5*d^4))))/((b*c - a*d)*(c^2 + d^2)*f*Sqrt[c + d*T an[e + f*x]]))/((b*c - a*d)*(c^2 + d^2)))/(2*(a^2 + b^2)*(b*c - a*d))
3.2.27.3.1 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_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
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])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*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] & & !GtQ[n, 0] && !LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(67569\) vs. \(2(639)=1278\).
Time = 0.35 (sec) , antiderivative size = 67570, normalized size of antiderivative = 99.51
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(67570\) |
| default | \(\text {Expression too large to display}\) | \(67570\) |
int((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(5 /2),x,method=_RETURNVERBOSE)
Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+ e))^(5/2),x, algorithm="fricas")
\[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}}{\left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Integral((A + B*tan(e + f*x) + C*tan(e + f*x)**2)/((a + b*tan(e + f*x))**2 *(c + d*tan(e + f*x))**(5/2)), x)
Exception generated. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: ValueError} \]
integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+ e))^(5/2),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(a*d-b*c>0)', see `assume?` for m ore detail
Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+ e))^(5/2),x, algorithm="giac")
Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Hanged} \]