Optimal. Leaf size=110 \[ \frac {A-i B}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {x (B+i A)}{8 a^3}+\frac {A+3 i B}{8 a d (a+i a \tan (c+d x))^2}-\frac {A+i B}{6 d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.16, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3590, 3526, 3479, 8} \[ \frac {A-i B}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {x (B+i A)}{8 a^3}+\frac {A+3 i B}{8 a d (a+i a \tan (c+d x))^2}-\frac {A+i B}{6 d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3479
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))^3} \, dx &=-\frac {A+i B}{6 d (a+i a \tan (c+d x))^3}-\frac {i \int \frac {a (A+i B)+2 a B \tan (c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{2 a^2}\\ &=-\frac {A+i B}{6 d (a+i a \tan (c+d x))^3}+\frac {A+3 i B}{8 a d (a+i a \tan (c+d x))^2}-\frac {(i A+B) \int \frac {1}{a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=-\frac {A+i B}{6 d (a+i a \tan (c+d x))^3}+\frac {A+3 i B}{8 a d (a+i a \tan (c+d x))^2}+\frac {A-i B}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {(i A+B) \int 1 \, dx}{8 a^3}\\ &=-\frac {(i A+B) x}{8 a^3}-\frac {A+i B}{6 d (a+i a \tan (c+d x))^3}+\frac {A+3 i B}{8 a d (a+i a \tan (c+d x))^2}+\frac {A-i B}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 1.55, size = 148, normalized size = 1.35 \[ \frac {(\cos (3 (c+d x))-i \sin (3 (c+d x))) (3 (A+3 i B) \cos (c+d x)-2 (6 i A d x+A+B (6 d x+i)) \cos (3 (c+d x))+9 i A \sin (c+d x)+2 i A \sin (3 (c+d x))+12 A d x \sin (3 (c+d x))-3 B \sin (c+d x)-2 B \sin (3 (c+d x))-12 i B d x \sin (3 (c+d x)))}{96 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 75, normalized size = 0.68 \[ \frac {{\left ({\left (-12 i \, A - 12 \, B\right )} d x e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, {\left (A + i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - 3 \, {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, A - 2 i \, B\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 130, normalized size = 1.18 \[ -\frac {\frac {6 \, {\left (A - i \, B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} - \frac {6 \, {\left (A - i \, B\right )} \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} - \frac {11 \, A \tan \left (d x + c\right )^{3} - 11 i \, B \tan \left (d x + c\right )^{3} - 45 i \, A \tan \left (d x + c\right )^{2} - 45 \, B \tan \left (d x + c\right )^{2} - 69 \, A \tan \left (d x + c\right ) + 21 i \, B \tan \left (d x + c\right ) + 19 i \, A + 3 \, B}{a^{3} {\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 203, normalized size = 1.85 \[ \frac {A \ln \left (\tan \left (d x +c \right )+i\right )}{16 d \,a^{3}}-\frac {i B \ln \left (\tan \left (d x +c \right )+i\right )}{16 d \,a^{3}}-\frac {A}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {3 i B}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {i A}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {B}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {i A}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}-\frac {B}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}-\frac {\ln \left (\tan \left (d x +c \right )-i\right ) A}{16 d \,a^{3}}+\frac {i \ln \left (\tan \left (d x +c \right )-i\right ) B}{16 d \,a^{3}} \]
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 = 6.50, size = 147, normalized size = 1.34 \[ \frac {\frac {A}{12\,a^3}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {A}{8\,a^3}-\frac {B\,1{}\mathrm {i}}{8\,a^3}\right )+\frac {B\,1{}\mathrm {i}}{12\,a^3}+\mathrm {tan}\left (c+d\,x\right )\,\left (-\frac {B}{8\,a^3}+\frac {A\,3{}\mathrm {i}}{8\,a^3}\right )}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}+1\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{16\,a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.68, size = 262, normalized size = 2.38 \[ \begin {cases} - \frac {\left (\left (512 A a^{6} d^{2} e^{6 i c} + 512 i B a^{6} d^{2} e^{6 i c}\right ) e^{- 6 i d x} + \left (768 A a^{6} d^{2} e^{8 i c} - 768 i B a^{6} d^{2} e^{8 i c}\right ) e^{- 4 i d x} + \left (- 1536 A a^{6} d^{2} e^{10 i c} - 1536 i B a^{6} d^{2} e^{10 i c}\right ) e^{- 2 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text {for}\: 24576 a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (- \frac {- i A - B}{8 a^{3}} + \frac {\left (- i A e^{6 i c} - i A e^{4 i c} + i A e^{2 i c} + i A - B e^{6 i c} + B e^{4 i c} + B e^{2 i c} - B\right ) e^{- 6 i c}}{8 a^{3}}\right ) & \text {otherwise} \end {cases} - \frac {x \left (i A + B\right )}{8 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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