Optimal. Leaf size=76 \[ \frac {A+3 i B}{4 a^2 d (1+i \tan (c+d x))}-\frac {x (B+i A)}{4 a^2}-\frac {A+i B}{4 d (a+i a \tan (c+d x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3590, 3526, 8} \[ \frac {A+3 i B}{4 a^2 d (1+i \tan (c+d x))}-\frac {x (B+i A)}{4 a^2}-\frac {A+i B}{4 d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3526
Rule 3590
Rubi steps
\begin {align*} \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx &=-\frac {A+i B}{4 d (a+i a \tan (c+d x))^2}-\frac {i \int \frac {a (A+i B)+2 a B \tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{2 a^2}\\ &=\frac {A+3 i B}{4 a^2 d (1+i \tan (c+d x))}-\frac {A+i B}{4 d (a+i a \tan (c+d x))^2}-\frac {(i A+B) \int 1 \, dx}{4 a^2}\\ &=-\frac {(i A+B) x}{4 a^2}+\frac {A+3 i B}{4 a^2 d (1+i \tan (c+d x))}-\frac {A+i B}{4 d (a+i a \tan (c+d x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.59, size = 92, normalized size = 1.21 \[ \frac {\sec ^2(c+d x) ((-4 A d x-i A+4 i B d x+B) \sin (2 (c+d x))+(4 i A d x+A+B (4 d x+i)) \cos (2 (c+d x))-4 i B)}{16 a^2 d (\tan (c+d x)-i)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.65, size = 55, normalized size = 0.72 \[ \frac {{\left ({\left (-4 i \, A - 4 \, B\right )} d x e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, B e^{\left (2 i \, d x + 2 i \, c\right )} - A - i \, B\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{16 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.47, size = 109, normalized size = 1.43 \[ \frac {\frac {2 \, {\left (A - i \, B\right )} \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{2}} - \frac {2 \, {\left (A - i \, B\right )} \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{2}} + \frac {3 \, A \tan \left (d x + c\right )^{2} - 3 i \, B \tan \left (d x + c\right )^{2} - 10 i \, A \tan \left (d x + c\right ) + 6 \, B \tan \left (d x + c\right ) - 3 \, A - 5 i \, B}{a^{2} {\left (\tan \left (d x + c\right ) - i\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.20, size = 162, normalized size = 2.13 \[ \frac {A \ln \left (\tan \left (d x +c \right )+i\right )}{8 d \,a^{2}}-\frac {i B \ln \left (\tan \left (d x +c \right )+i\right )}{8 d \,a^{2}}+\frac {A}{4 d \,a^{2} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {i B}{4 d \,a^{2} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {i A}{4 d \,a^{2} \left (\tan \left (d x +c \right )-i\right )}+\frac {3 B}{4 d \,a^{2} \left (\tan \left (d x +c \right )-i\right )}-\frac {\ln \left (\tan \left (d x +c \right )-i\right ) A}{8 d \,a^{2}}+\frac {i \ln \left (\tan \left (d x +c \right )-i\right ) B}{8 d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.17, size = 106, normalized size = 1.39 \[ \frac {\frac {B}{2\,a^2}+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {A}{4\,a^2}+\frac {B\,3{}\mathrm {i}}{4\,a^2}\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,a^2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{8\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.37, size = 167, normalized size = 2.20 \[ \begin {cases} \frac {\left (16 i B a^{2} d e^{4 i c} e^{- 2 i d x} + \left (- 4 A a^{2} d e^{2 i c} - 4 i B a^{2} d e^{2 i c}\right ) e^{- 4 i d x}\right ) e^{- 6 i c}}{64 a^{4} d^{2}} & \text {for}\: 64 a^{4} d^{2} e^{6 i c} \neq 0 \\x \left (- \frac {- i A - B}{4 a^{2}} + \frac {\left (- i A e^{4 i c} + i A - B e^{4 i c} + 2 B e^{2 i c} - B\right ) e^{- 4 i c}}{4 a^{2}}\right ) & \text {otherwise} \end {cases} - \frac {x \left (i A + B\right )}{4 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________