Optimal. Leaf size=156 \[ -\frac{i \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2^{2/3} d}+\frac{3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}+\frac{i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{\sqrt [3]{a} x}{2\ 2^{2/3}} \]
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Rubi [A] time = 0.0740147, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {3481, 57, 617, 204, 31} \[ -\frac{i \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2^{2/3} d}+\frac{3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}+\frac{i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{\sqrt [3]{a} x}{2\ 2^{2/3}} \]
Antiderivative was successfully verified.
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Rule 3481
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \sqrt [3]{a+i a \tan (c+d x)} \, dx &=-\frac{(i a) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\sqrt [3]{a} x}{2\ 2^{2/3}}+\frac{i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{\left (3 i \sqrt [3]{a}\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^{2/3} d}-\frac{\left (3 i a^{2/3}\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 )}{2 \sqrt [3]{2} d}\\ &=-\frac{\sqrt [3]{a} x}{2\ 2^{2/3}}+\frac{i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac{3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}+\frac{\left (3 i \sqrt [3]{a}\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 )}{2^{2/3} d}\\ &=-\frac{\sqrt [3]{a} x}{2\ 2^{2/3}}-\frac{i \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1+\frac{2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{2^{2/3} d}+\frac{i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac{3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}\\ \end{align*}
Mathematica [A] time = 0.907809, size = 223, normalized size = 1.43 \[ -\frac{i e^{-\frac{2}{3} i (c+d x)} \sqrt [3]{1+e^{2 i (c+d x)}} \sqrt [3]{a+i a \tan (c+d x)} \left (-2 \log \left (1-\frac{e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}\right )+\log \left (\frac{\left (1+e^{2 i (c+d x)}\right )^{2/3}+e^{\frac{2}{3} i (c+d x)} \sqrt [3]{1+e^{2 i (c+d x)}}+e^{\frac{4}{3} i (c+d x)}}{\left (1+e^{2 i (c+d x)}\right )^{2/3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 138, normalized size = 0.9 \begin{align*}{\frac{{\frac{i}{2}}\sqrt [3]{2}}{d}\sqrt [3]{a}\ln \left ( \sqrt [3]{a+ia\tan \left ( dx+c \right ) }-\sqrt [3]{2}\sqrt [3]{a} \right ) }-{\frac{{\frac{i}{4}}\sqrt [3]{2}}{d}\sqrt [3]{a}\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{{\frac{i}{2}}\sqrt [3]{2}\sqrt{3}}{d}\sqrt [3]{a}\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 [A] time = 2.1549, size = 593, normalized size = 3.8 \begin{align*} \frac{1}{2} \,{\left (i \, \sqrt{3} - 1\right )} \left (-\frac{i \, a}{4 \, d^{3}}\right )^{\frac{1}{3}} \log \left (2^{\frac{1}{3}} \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{i \, a}{4 \, d^{3}}\right )^{\frac{1}{3}}\right ) + \frac{1}{2} \,{\left (-i \, \sqrt{3} - 1\right )} \left (-\frac{i \, a}{4 \, d^{3}}\right )^{\frac{1}{3}} \log \left (2^{\frac{1}{3}} \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{i \, a}{4 \, d^{3}}\right )^{\frac{1}{3}}\right ) + \left (-\frac{i \, a}{4 \, d^{3}}\right )^{\frac{1}{3}} \log \left (2^{\frac{1}{3}} \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 )} + 2 i \, d \left (-\frac{i \, a}{4 \, d^{3}}\right )^{\frac{1}{3}}\right ) \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 \sqrt [3]{i a \tan{\left (c + d x \right )} + a}\, 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{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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