Optimal. Leaf size=203 \[ \frac{i \sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}-\frac{3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac{a^{4/3} x}{2^{2/3}}-\frac{3 i (a+i a \tan (c+d x))^{7/3}}{7 a d}-\frac{3 i a \sqrt [3]{a+i a \tan (c+d x)}}{d} \]
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Rubi [A] time = 0.156506, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {3543, 3478, 3481, 57, 617, 204, 31} \[ \frac{i \sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}-\frac{3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac{a^{4/3} x}{2^{2/3}}-\frac{3 i (a+i a \tan (c+d x))^{7/3}}{7 a d}-\frac{3 i a \sqrt [3]{a+i a \tan (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 3543
Rule 3478
Rule 3481
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx &=-\frac{3 i (a+i a \tan (c+d x))^{7/3}}{7 a d}-\int (a+i a \tan (c+d x))^{4/3} \, dx\\ &=-\frac{3 i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{3 i (a+i a \tan (c+d x))^{7/3}}{7 a d}-(2 a) \int \sqrt [3]{a+i a \tan (c+d x)} \, dx\\ &=-\frac{3 i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{3 i (a+i a \tan (c+d x))^{7/3}}{7 a d}+\frac{\left (2 i a^2\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac{a^{4/3} x}{2^{2/3}}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{3 i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{3 i (a+i a \tan (c+d x))^{7/3}}{7 a d}+\frac{\left (3 i a^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}+\frac{\left (3 i a^{5/3}\right ) \operatorname{Subst}\left (\int \frac{1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{\sqrt [3]{2} d}\\ &=\frac{a^{4/3} x}{2^{2/3}}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{3 i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{3 i (a+i a \tan (c+d x))^{7/3}}{7 a d}-\frac{\left (3 i \sqrt [3]{2} a^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{d}\\ &=\frac{a^{4/3} x}{2^{2/3}}+\frac{i \sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{1+\frac{2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{d}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{3 i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{3 i (a+i a \tan (c+d x))^{7/3}}{7 a d}\\ \end{align*}
Mathematica [F] time = 180.004, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [A] time = 0.017, size = 182, normalized size = 0.9 \begin{align*}{\frac{-{\frac{3\,i}{7}}}{ad} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{7}{3}}}}-{\frac{3\,ia}{d}\sqrt [3]{a+ia\tan \left ( dx+c \right ) }}-{\frac{i\sqrt [3]{2}}{d}{a}^{{\frac{4}{3}}}\ln \left ( \sqrt [3]{a+ia\tan \left ( dx+c \right ) }-\sqrt [3]{2}\sqrt [3]{a} \right ) }+{\frac{{\frac{i}{2}}\sqrt [3]{2}}{d}{a}^{{\frac{4}{3}}}\ln \left ( \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}+\sqrt [3]{2}\sqrt [3]{a}\sqrt [3]{a+ia\tan \left ( dx+c \right ) }+{2}^{{\frac{2}{3}}}{a}^{{\frac{2}{3}}} \right ) }+{\frac{i\sqrt [3]{2}\sqrt{3}}{d}{a}^{{\frac{4}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ({{2}^{{\frac{2}{3}}}\sqrt [3]{a+ia\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [3]{a}}}}+1 \right ) } \right ) } \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: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82518, size = 1197, normalized size = 5.9 \begin{align*} \frac{2^{\frac{1}{3}}{\left (-66 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 84 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 42 i \, a\right )} \left (\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac{1}{3}} e^{\left (\frac{2}{3} i \, d x + \frac{2}{3} i \, c\right )} - 7 \,{\left ({\left (-i \, \sqrt{3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \,{\left (-i \, \sqrt{3} d + d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, \sqrt{3} d + d\right )} \left (\frac{2 i \, a^{4}}{d^{3}}\right )^{\frac{1}{3}} \log \left (\frac{2 \cdot 2^{\frac{1}{3}} a \left (\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac{1}{3}} e^{\left (\frac{2}{3} i \, d x + \frac{2}{3} i \, c\right )} +{\left (\sqrt{3} d + i \, d\right )} \left (\frac{2 i \, a^{4}}{d^{3}}\right )^{\frac{1}{3}}}{2 \, a}\right ) -{\left (7 \,{\left (i \, \sqrt{3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 14 \,{\left (i \, \sqrt{3} d + d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, \sqrt{3} d + 7 \, d\right )} \left (\frac{2 i \, a^{4}}{d^{3}}\right )^{\frac{1}{3}} \log \left (\frac{2 \cdot 2^{\frac{1}{3}} a \left (\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac{1}{3}} e^{\left (\frac{2}{3} i \, d x + \frac{2}{3} i \, c\right )} -{\left (\sqrt{3} d - i \, d\right )} \left (\frac{2 i \, a^{4}}{d^{3}}\right )^{\frac{1}{3}}}{2 \, a}\right ) + 14 \,{\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \left (\frac{2 i \, a^{4}}{d^{3}}\right )^{\frac{1}{3}} \log \left (\frac{2^{\frac{1}{3}} a \left (\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac{1}{3}} e^{\left (\frac{2}{3} i \, d x + \frac{2}{3} i \, c\right )} - i \, \left (\frac{2 i \, a^{4}}{d^{3}}\right )^{\frac{1}{3}} d}{a}\right )}{14 \,{\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, 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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{4}{3}} \tan \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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