Optimal. Leaf size=148 \[ \frac {A+7 i B}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {x (-7 B+i A)}{8 a^3}-\frac {i B \log (\cos (c+d x))}{a^3 d}+\frac {(-B+i A) \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(A+3 i B) \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.37, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3595, 3589, 3475, 12, 3526, 8} \[ \frac {A+7 i B}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {x (-7 B+i A)}{8 a^3}-\frac {i B \log (\cos (c+d x))}{a^3 d}+\frac {(-B+i A) \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(A+3 i B) \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 3475
Rule 3526
Rule 3589
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))^3} \, dx &=\frac {(i A-B) \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac {\int \frac {\tan ^2(c+d x) (3 a (i A-B)+6 i a B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=\frac {(i A-B) \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(A+3 i B) \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}+\frac {\int \frac {\tan (c+d x) \left (-6 a^2 (A+3 i B)-24 a^2 B \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4}\\ &=\frac {(i A-B) \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(A+3 i B) \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}-\frac {i \int -\frac {6 a^3 (i A-7 B) \tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{24 a^5}+\frac {(i B) \int \tan (c+d x) \, dx}{a^3}\\ &=-\frac {i B \log (\cos (c+d x))}{a^3 d}+\frac {(i A-B) \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(A+3 i B) \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}-\frac {(A+7 i B) \int \frac {\tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=-\frac {i B \log (\cos (c+d x))}{a^3 d}+\frac {(i A-B) \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(A+3 i B) \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}+\frac {A+7 i B}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {(i A-7 B) \int 1 \, dx}{8 a^3}\\ &=\frac {(i A-7 B) x}{8 a^3}-\frac {i B \log (\cos (c+d x))}{a^3 d}+\frac {(i A-B) \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(A+3 i B) \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}+\frac {A+7 i B}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 1.49, size = 178, normalized size = 1.20 \[ \frac {\sec ^3(c+d x) ((-51 B+9 i A) \cos (c+d x)-2 \cos (3 (c+d x)) (6 A d x-i A-48 B \log (\cos (c+d x))+42 i B d x+B)-27 A \sin (c+d x)+2 A \sin (3 (c+d x))-12 i A d x \sin (3 (c+d x))-81 i B \sin (c+d x)+2 i B \sin (3 (c+d x))+84 B d x \sin (3 (c+d x))+96 i B \sin (3 (c+d x)) \log (\cos (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 103, normalized size = 0.70 \[ \frac {{\left ({\left (12 i \, A - 180 \, B\right )} d x e^{\left (6 i \, d x + 6 i \, c\right )} - 96 i \, B e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 6 \, {\left (3 \, A + 11 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - 3 \, {\left (3 \, A + 5 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, A + 2 i \, B\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.40, size = 130, normalized size = 0.88 \[ \frac {\frac {6 \, {\left (A + 15 i \, B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} - \frac {6 \, {\left (A - i \, B\right )} \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}} - \frac {11 \, A \tan \left (d x + c\right )^{3} + 165 i \, B \tan \left (d x + c\right )^{3} + 51 i \, A \tan \left (d x + c\right )^{2} + 291 \, B \tan \left (d x + c\right )^{2} + 75 \, A \tan \left (d x + c\right ) - 171 i \, B \tan \left (d x + c\right ) - 29 i \, A - 29 \, B}{a^{3} {\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 203, normalized size = 1.37 \[ -\frac {A \ln \left (\tan \left (d x +c \right )+i\right )}{16 d \,a^{3}}+\frac {i B \ln \left (\tan \left (d x +c \right )+i\right )}{16 d \,a^{3}}+\frac {\ln \left (\tan \left (d x +c \right )-i\right ) A}{16 d \,a^{3}}+\frac {15 i \ln \left (\tan \left (d x +c \right )-i\right ) B}{16 d \,a^{3}}+\frac {17 B}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}-\frac {7 i A}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}+\frac {5 A}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {7 i B}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {i A}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {B}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [B] time = 6.64, size = 146, normalized size = 0.99 \[ \frac {\frac {5\,A}{12\,a^3}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {7\,A}{8\,a^3}+\frac {B\,17{}\mathrm {i}}{8\,a^3}\right )+\frac {B\,17{}\mathrm {i}}{12\,a^3}+\mathrm {tan}\left (c+d\,x\right )\,\left (-\frac {27\,B}{8\,a^3}+\frac {A\,9{}\mathrm {i}}{8\,a^3}\right )}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}+1\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{16\,a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,15{}\mathrm {i}\right )}{16\,a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.42, size = 303, normalized size = 2.05 \[ - \frac {i B \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{a^{3} d} + \begin {cases} - \frac {\left (\left (- 512 A a^{6} d^{2} e^{6 i c} - 512 i B a^{6} d^{2} e^{6 i c}\right ) e^{- 6 i d x} + \left (2304 A a^{6} d^{2} e^{8 i c} + 3840 i B a^{6} d^{2} e^{8 i c}\right ) e^{- 4 i d x} + \left (- 4608 A a^{6} d^{2} e^{10 i c} - 16896 i B a^{6} d^{2} e^{10 i c}\right ) e^{- 2 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text {for}\: 24576 a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (- \frac {i A - 15 B}{8 a^{3}} - \frac {\left (- i A e^{6 i c} + 3 i A e^{4 i c} - 3 i A e^{2 i c} + i A + 15 B e^{6 i c} - 11 B e^{4 i c} + 5 B e^{2 i c} - B\right ) e^{- 6 i c}}{8 a^{3}}\right ) & \text {otherwise} \end {cases} - \frac {x \left (- i A + 15 B\right )}{8 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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