Optimal. Leaf size=159 \[ \frac {5 A-29 i B}{48 a^4 d (1+i \tan (c+d x))}-\frac {A-13 i B}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {x (B+i A)}{16 a^4}+\frac {(-B+i A) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.47, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3595, 3590, 3526, 8} \[ \frac {5 A-29 i B}{48 a^4 d (1+i \tan (c+d x))}-\frac {A-13 i B}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {x (B+i A)}{16 a^4}+\frac {(-B+i A) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3526
Rule 3590
Rule 3595
Rubi steps
\begin {align*} \int \frac {\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx &=\frac {(i A-B) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}-\frac {\int \frac {\tan ^2(c+d x) (3 a (i A-B)-a (A-7 i B) \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx}{8 a^2}\\ &=\frac {(i A-B) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {\int \frac {\tan (c+d x) \left (-4 a^2 (A+5 i B)-8 a^2 (i A+4 B) \tan (c+d x)\right )}{(a+i a \tan (c+d x))^2} \, dx}{48 a^4}\\ &=-\frac {A-13 i B}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {(i A-B) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3}-\frac {i \int \frac {4 a^3 (A-13 i B)-16 a^3 (i A+4 B) \tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{96 a^6}\\ &=-\frac {A-13 i B}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {(i A-B) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {5 A-29 i B}{48 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {(i A+B) \int 1 \, dx}{16 a^4}\\ &=\frac {(i A+B) x}{16 a^4}-\frac {A-13 i B}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {(i A-B) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {5 A-29 i B}{48 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.53, size = 158, normalized size = 0.99 \[ \frac {\sec ^4(c+d x) (16 (A-4 i B) \cos (2 (c+d x))+3 (8 i A d x+A+8 B d x+i B) \cos (4 (c+d x))+32 i A \sin (2 (c+d x))-3 i A \sin (4 (c+d x))-24 A d x \sin (4 (c+d x))+32 B \sin (2 (c+d x))+24 i B d x \sin (4 (c+d x))+3 B \sin (4 (c+d x))+36 i B)}{384 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 87, normalized size = 0.55 \[ \frac {{\left ({\left (24 i \, A + 24 \, B\right )} d x e^{\left (8 i \, d x + 8 i \, c\right )} + 24 \, {\left (A - 2 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 36 i \, B e^{\left (4 i \, d x + 4 i \, c\right )} - 8 \, {\left (A + 2 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 3 \, A + 3 i \, B\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{384 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.63, size = 153, normalized size = 0.96 \[ -\frac {\frac {12 \, {\left (A - i \, B\right )} \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{4}} - \frac {12 \, {\left (A - i \, B\right )} \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{4}} + \frac {25 \, A \tan \left (d x + c\right )^{4} - 25 i \, B \tan \left (d x + c\right )^{4} - 124 i \, A \tan \left (d x + c\right )^{3} + 260 \, B \tan \left (d x + c\right )^{3} - 54 \, A \tan \left (d x + c\right )^{2} - 522 i \, B \tan \left (d x + c\right )^{2} - 4 i \, A \tan \left (d x + c\right ) - 388 \, B \tan \left (d x + c\right ) - 7 \, A + 103 i \, B}{a^{4} {\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.24, size = 244, normalized size = 1.53 \[ -\frac {A \ln \left (\tan \left (d x +c \right )+i\right )}{32 d \,a^{4}}+\frac {i B \ln \left (\tan \left (d x +c \right )+i\right )}{32 d \,a^{4}}-\frac {5 i A}{12 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {7 B}{12 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {i A}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}-\frac {15 B}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}-\frac {7 A}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {17 i B}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {A}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}+\frac {i B}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}+\frac {\ln \left (\tan \left (d x +c \right )-i\right ) A}{32 d \,a^{4}}-\frac {i \ln \left (\tan \left (d x +c \right )-i\right ) B}{32 d \,a^{4}} \]
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 = 6.62, size = 178, normalized size = 1.12 \[ \frac {\frac {A}{12\,a^4}+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-\frac {15\,B}{16\,a^4}+\frac {A\,1{}\mathrm {i}}{16\,a^4}\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {A}{4\,a^4}-\frac {B\,7{}\mathrm {i}}{4\,a^4}\right )-\frac {B\,1{}\mathrm {i}}{3\,a^4}+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {61\,B}{48\,a^4}+\frac {A\,13{}\mathrm {i}}{48\,a^4}\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4-{\mathrm {tan}\left (c+d\,x\right )}^3\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,4{}\mathrm {i}+1\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{32\,a^4\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a^4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.00, size = 304, normalized size = 1.91 \[ \begin {cases} \frac {\left (294912 i B a^{12} d^{3} e^{16 i c} e^{- 4 i d x} + \left (24576 A a^{12} d^{3} e^{12 i c} + 24576 i B a^{12} d^{3} e^{12 i c}\right ) e^{- 8 i d x} + \left (- 65536 A a^{12} d^{3} e^{14 i c} - 131072 i B a^{12} d^{3} e^{14 i c}\right ) e^{- 6 i d x} + \left (196608 A a^{12} d^{3} e^{18 i c} - 393216 i B a^{12} d^{3} e^{18 i c}\right ) e^{- 2 i d x}\right ) e^{- 20 i c}}{3145728 a^{16} d^{4}} & \text {for}\: 3145728 a^{16} d^{4} e^{20 i c} \neq 0 \\x \left (- \frac {i A + B}{16 a^{4}} + \frac {\left (i A e^{8 i c} - 2 i A e^{6 i c} + 2 i A e^{2 i c} - i A + B e^{8 i c} - 4 B e^{6 i c} + 6 B e^{4 i c} - 4 B e^{2 i c} + B\right ) e^{- 8 i c}}{16 a^{4}}\right ) & \text {otherwise} \end {cases} - \frac {x \left (- i A - B\right )}{16 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________