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 = 0.164954, antiderivative size = 110, normalized size of antiderivative = 1., 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 3590
Rule 3526
Rule 3479
Rule 8
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*}
Mathematica [A] time = 1.30899, 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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Maple [B] time = 0.031, size = 203, normalized size = 1.9 \begin{align*}{\frac{-{\frac{i}{6}}A}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{B}{6\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{\ln \left ( \tan \left ( dx+c \right ) -i \right ) A}{16\,{a}^{3}d}}+{\frac{{\frac{i}{16}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) B}{{a}^{3}d}}-{\frac{A}{8\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{{\frac{3\,i}{8}}B}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{{\frac{i}{8}}A}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{B}{8\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{A\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{16\,{a}^{3}d}}-{\frac{{\frac{i}{16}}B\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{{a}^{3}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44393, size = 216, normalized size = 1.96 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.55575, size = 260, normalized size = 2.36 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38145, size = 176, normalized size = 1.6 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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