Optimal. Leaf size=175 \[ -\frac{i \sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}+\frac{3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}+\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{a^{4/3} x}{2^{2/3}}+\frac{3 i a \sqrt [3]{a+i a \tan (c+d x)}}{d} \]
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Rubi [A] time = 0.0937833, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {3478, 3481, 57, 617, 204, 31} \[ -\frac{i \sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}+\frac{3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}+\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{a^{4/3} x}{2^{2/3}}+\frac{3 i a \sqrt [3]{a+i a \tan (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 3478
Rule 3481
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \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}+(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{\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{\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{\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}\\ \end{align*}
Mathematica [A] time = 1.05573, size = 294, normalized size = 1.68 \[ \frac{i a e^{\frac{1}{3} i (c+d x)} \cos (c+d x) \sqrt [3]{a+i a \tan (c+d x)} \left (6 e^{\frac{2}{3} i (c+d x)}+2 \sqrt [3]{1+e^{2 i (c+d x)}} \log \left (1-\frac{e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}\right )-\sqrt [3]{1+e^{2 i (c+d x)}} \log \left (\frac{\left (1+e^{2 i (c+d x)}\right )^{2/3}+e^{\frac{2}{3} i (c+d x)} \sqrt [3]{1+e^{2 i (c+d x)}}+e^{\frac{4}{3} i (c+d x)}}{\left (1+e^{2 i (c+d x)}\right )^{2/3}}\right )-2 \sqrt{3} \sqrt [3]{1+e^{2 i (c+d x)}} \tan ^{-1}\left (\frac{1+\frac{2 e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right )\right )}{d \left (1+e^{2 i (c+d x)}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 159, normalized size = 0.9 \begin{align*}{\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 = 1.73662, size = 744, normalized size = 4.25 \begin{align*} \frac{6 i \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 (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 (-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 ) + 2 \, \left (-\frac{2 i \, a^{4}}{d^{3}}\right )^{\frac{1}{3}} d \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 )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (i a \tan{\left (c + d x \right )} + a\right )^{\frac{4}{3}}\, dx \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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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