Optimal. Leaf size=67 \[ \frac{i a \tan ^3(c+d x)}{3 d}+\frac{a \tan ^2(c+d x)}{2 d}-\frac{i a \tan (c+d x)}{d}+\frac{a \log (\cos (c+d x))}{d}+i a x \]
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Rubi [A] time = 0.0601947, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3528, 3525, 3475} \[ \frac{i a \tan ^3(c+d x)}{3 d}+\frac{a \tan ^2(c+d x)}{2 d}-\frac{i a \tan (c+d x)}{d}+\frac{a \log (\cos (c+d x))}{d}+i a x \]
Antiderivative was successfully verified.
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Rule 3528
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \tan ^3(c+d x) (a+i a \tan (c+d x)) \, dx &=\frac{i a \tan ^3(c+d x)}{3 d}+\int \tan ^2(c+d x) (-i a+a \tan (c+d x)) \, dx\\ &=\frac{a \tan ^2(c+d x)}{2 d}+\frac{i a \tan ^3(c+d x)}{3 d}+\int \tan (c+d x) (-a-i a \tan (c+d x)) \, dx\\ &=i a x-\frac{i a \tan (c+d x)}{d}+\frac{a \tan ^2(c+d x)}{2 d}+\frac{i a \tan ^3(c+d x)}{3 d}-a \int \tan (c+d x) \, dx\\ &=i a x+\frac{a \log (\cos (c+d x))}{d}-\frac{i a \tan (c+d x)}{d}+\frac{a \tan ^2(c+d x)}{2 d}+\frac{i a \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.153555, size = 74, normalized size = 1.1 \[ \frac{i a \tan ^3(c+d x)}{3 d}+\frac{i a \tan ^{-1}(\tan (c+d x))}{d}-\frac{i a \tan (c+d x)}{d}+\frac{a \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 75, normalized size = 1.1 \begin{align*}{\frac{-ia\tan \left ( dx+c \right ) }{d}}+{\frac{{\frac{i}{3}}a \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{a\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}+{\frac{ia\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.9417, size = 80, normalized size = 1.19 \begin{align*} -\frac{-2 i \, a \tan \left (d x + c\right )^{3} - 3 \, a \tan \left (d x + c\right )^{2} - 6 i \,{\left (d x + c\right )} a + 3 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 i \, a \tan \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.20169, size = 348, normalized size = 5.19 \begin{align*} \frac{18 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 18 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 3 \,{\left (a e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8 \, a}{3 \,{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.6724, size = 126, normalized size = 1.88 \begin{align*} \frac{a \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{\frac{6 a e^{- 2 i c} e^{4 i d x}}{d} + \frac{6 a e^{- 4 i c} e^{2 i d x}}{d} + \frac{8 a e^{- 6 i c}}{3 d}}{e^{6 i d x} + 3 e^{- 2 i c} e^{4 i d x} + 3 e^{- 4 i c} e^{2 i d x} + e^{- 6 i c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.85808, size = 211, normalized size = 3.15 \begin{align*} \frac{3 \, a e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 9 \, a e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 9 \, a e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 18 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 3 \, a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8 \, a}{3 \,{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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