Optimal. Leaf size=92 \[ \frac{3}{8 a^3 d (1+i \tan (c+d x))}+\frac{i x}{8 a^3}+\frac{i \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac{1}{8 a d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.129724, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3546, 3540, 3526, 8} \[ \frac{3}{8 a^3 d (1+i \tan (c+d x))}+\frac{i x}{8 a^3}+\frac{i \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac{1}{8 a d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3546
Rule 3540
Rule 3526
Rule 8
Rubi steps
\begin{align*} \int \frac{\tan ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=\frac{i \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac{i \int \frac{\tan ^2(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{2 a}\\ &=\frac{i \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac{1}{8 a d (a+i a \tan (c+d x))^2}-\frac{i \int \frac{a-2 i a \tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{4 a^3}\\ &=\frac{3}{8 a^3 d (1+i \tan (c+d x))}+\frac{i \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac{1}{8 a d (a+i a \tan (c+d x))^2}+\frac{i \int 1 \, dx}{8 a^3}\\ &=\frac{i x}{8 a^3}+\frac{3}{8 a^3 d (1+i \tan (c+d x))}+\frac{i \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac{1}{8 a d (a+i a \tan (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 0.235596, size = 91, normalized size = 0.99 \[ -\frac{\sec ^3(c+d x) (27 \sin (c+d x)+12 i d x \sin (3 (c+d x))-2 \sin (3 (c+d x))-9 i \cos (c+d x)+2 (6 d x-i) \cos (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 97, normalized size = 1.1 \begin{align*}{\frac{{\frac{i}{6}}}{d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{{\frac{7\,i}{8}}}{d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{5}{8\,d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+{\frac{\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{16\,d{a}^{3}}}-{\frac{\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{16\,d{a}^{3}}} \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 = 2.19763, size = 161, normalized size = 1.75 \begin{align*} \frac{{\left (12 i \, d x e^{\left (6 i \, d x + 6 i \, c\right )} + 18 \, e^{\left (4 i \, d x + 4 i \, c\right )} - 9 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2\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 = 1.86485, size = 158, normalized size = 1.72 \begin{align*} \begin{cases} \frac{\left (4608 a^{6} d^{2} e^{10 i c} e^{- 2 i d x} - 2304 a^{6} d^{2} e^{8 i c} e^{- 4 i d x} + 512 a^{6} d^{2} e^{6 i c} e^{- 6 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{\left (i e^{6 i c} - 3 i e^{4 i c} + 3 i e^{2 i c} - i\right ) e^{- 6 i c}}{8 a^{3}} - \frac{i}{8 a^{3}}\right ) & \text{otherwise} \end{cases} + \frac{i x}{8 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.84574, size = 109, normalized size = 1.18 \begin{align*} \frac{\frac{6 \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} - \frac{6 \, \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} - \frac{11 \, \tan \left (d x + c\right )^{3} + 51 i \, \tan \left (d x + c\right )^{2} + 75 \, \tan \left (d x + c\right ) - 29 i}{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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