Optimal. Leaf size=124 \[ \frac {17 B+i A}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {x (A-i B)}{8 a^3}+\frac {(-B+i A) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {-7 B+i A}{24 a d (a+i a \tan (c+d x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.29, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3595, 3590, 3526, 8} \[ \frac {17 B+i A}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {x (A-i B)}{8 a^3}+\frac {(-B+i A) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {-7 B+i A}{24 a d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3526
Rule 3590
Rule 3595
Rubi steps
\begin {align*} \int \frac {\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx &=\frac {(i A-B) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac {\int \frac {\tan (c+d x) (2 a (i A-B)-a (A-5 i B) \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=\frac {(i A-B) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {i A-7 B}{24 a d (a+i a \tan (c+d x))^2}+\frac {i \int \frac {a^2 (i A-7 B)-2 a^2 (A-5 i B) \tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{12 a^4}\\ &=\frac {(i A-B) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {i A-7 B}{24 a d (a+i a \tan (c+d x))^2}+\frac {i A+17 B}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {(A-i B) \int 1 \, dx}{8 a^3}\\ &=-\frac {(A-i B) x}{8 a^3}+\frac {(i A-B) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {i A-7 B}{24 a d (a+i a \tan (c+d x))^2}+\frac {i A+17 B}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.22, size = 147, normalized size = 1.19 \[ \frac {\sec ^3(c+d x) (-9 (A-i B) \cos (c+d x)+2 (-6 i A d x+A-6 B d x+i B) \cos (3 (c+d x))-3 i A \sin (c+d x)-2 i A \sin (3 (c+d x))+12 A d x \sin (3 (c+d x))-27 B \sin (c+d x)+2 B \sin (3 (c+d x))-12 i B d x \sin (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.77, size = 78, normalized size = 0.63 \[ -\frac {{\left (12 \, {\left (A - i \, B\right )} d x e^{\left (6 i \, d x + 6 i \, c\right )} - {\left (6 i \, A + 18 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (3 i \, A - 9 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i \, A - 2 \, B\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.04, size = 131, normalized size = 1.06 \[ -\frac {\frac {6 \, {\left (-i \, A - B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} + \frac {6 \, {\left (i \, A + B\right )} \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} + \frac {11 i \, A \tan \left (d x + c\right )^{3} + 11 \, B \tan \left (d x + c\right )^{3} + 45 \, A \tan \left (d x + c\right )^{2} + 51 i \, B \tan \left (d x + c\right )^{2} - 21 i \, A \tan \left (d x + c\right ) + 75 \, B \tan \left (d x + c\right ) - 3 \, A - 29 i \, B}{a^{3} {\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.24, size = 203, normalized size = 1.64 \[ -\frac {B \ln \left (\tan \left (d x +c \right )+i\right )}{16 d \,a^{3}}-\frac {i A \ln \left (\tan \left (d x +c \right )+i\right )}{16 d \,a^{3}}+\frac {A}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {i B}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {A}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}-\frac {7 i B}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}+\frac {i \ln \left (\tan \left (d x +c \right )-i\right ) A}{16 d \,a^{3}}+\frac {\ln \left (\tan \left (d x +c \right )-i\right ) B}{16 d \,a^{3}}-\frac {3 i A}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {5 B}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}} \]
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 = 6.52, size = 111, normalized size = 0.90 \[ \frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {7\,B}{8\,a^3}+\frac {A\,1{}\mathrm {i}}{8\,a^3}\right )+\frac {A\,1{}\mathrm {i}}{12\,a^3}+\frac {5\,B}{12\,a^3}-\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {A}{8\,a^3}-\frac {B\,9{}\mathrm {i}}{8\,a^3}\right )}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}+1\right )}+\frac {x\,\left (B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.64, size = 264, normalized size = 2.13 \[ \begin {cases} - \frac {\left (\left (512 i A a^{6} d^{2} e^{6 i c} - 512 B a^{6} d^{2} e^{6 i c}\right ) e^{- 6 i d x} + \left (- 768 i A a^{6} d^{2} e^{8 i c} + 2304 B a^{6} d^{2} e^{8 i c}\right ) e^{- 4 i d x} + \left (- 1536 i A a^{6} d^{2} e^{10 i c} - 4608 B a^{6} d^{2} e^{10 i c}\right ) e^{- 2 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text {for}\: 24576 a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (- \frac {- A + i B}{8 a^{3}} + \frac {\left (- A e^{6 i c} + A e^{4 i c} + A e^{2 i c} - A + i B e^{6 i c} - 3 i B e^{4 i c} + 3 i B e^{2 i c} - i B\right ) e^{- 6 i c}}{8 a^{3}}\right ) & \text {otherwise} \end {cases} - \frac {x \left (A - i B\right )}{8 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________