Optimal. Leaf size=234 \[ \frac{3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2^{2/3} d}-\frac{3 (a+i a \tan (c+d x))^{4/3}}{28 a d}-\frac{18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac{\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{i \sqrt [3]{a} x}{2\ 2^{2/3}} \]
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Rubi [A] time = 0.292819, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3560, 3592, 3527, 3481, 57, 617, 204, 31} \[ \frac{3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2^{2/3} d}-\frac{3 (a+i a \tan (c+d x))^{4/3}}{28 a d}-\frac{18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac{\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{i \sqrt [3]{a} x}{2\ 2^{2/3}} \]
Antiderivative was successfully verified.
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Rule 3560
Rule 3592
Rule 3527
Rule 3481
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \tan ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx &=\frac{3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac{3 \int \tan (c+d x) \left (2 a+\frac{1}{3} i a \tan (c+d x)\right ) \sqrt [3]{a+i a \tan (c+d x)} \, dx}{7 a}\\ &=\frac{3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac{3 (a+i a \tan (c+d x))^{4/3}}{28 a d}-\frac{3 \int \sqrt [3]{a+i a \tan (c+d x)} \left (-\frac{i a}{3}+2 a \tan (c+d x)\right ) \, dx}{7 a}\\ &=-\frac{18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac{3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac{3 (a+i a \tan (c+d x))^{4/3}}{28 a d}+i \int \sqrt [3]{a+i a \tan (c+d x)} \, dx\\ &=-\frac{18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac{3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac{3 (a+i a \tan (c+d x))^{4/3}}{28 a d}+\frac{a \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i \sqrt [3]{a} x}{2\ 2^{2/3}}-\frac{\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac{3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac{3 (a+i a \tan (c+d x))^{4/3}}{28 a d}+\frac{\left (3 \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 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{i \sqrt [3]{a} x}{2\ 2^{2/3}}-\frac{\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{3 \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{18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac{3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac{3 (a+i a \tan (c+d x))^{4/3}}{28 a d}-\frac{\left (3 \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{i \sqrt [3]{a} x}{2\ 2^{2/3}}+\frac{\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{\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{3 \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{18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac{3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac{3 (a+i a \tan (c+d x))^{4/3}}{28 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 = 0.02, size = 198, normalized size = 0.9 \begin{align*} -{\frac{3}{7\,{a}^{2}d} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{7}{3}}}}+{\frac{3}{4\,ad} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}}}-3\,{\frac{\sqrt [3]{a+ia\tan \left ( dx+c \right ) }}{d}}-{\frac{\sqrt [3]{2}}{2\,d}\sqrt [3]{a}\ln \left ( \sqrt [3]{a+ia\tan \left ( dx+c \right ) }-\sqrt [3]{2}\sqrt [3]{a} \right ) }+{\frac{\sqrt [3]{2}}{4\,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{\sqrt [3]{2}\sqrt{3}}{2\,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 [B] time = 2.26488, size = 1202, normalized size = 5.14 \begin{align*} \text{result too large to display} \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]{a \left (i \tan{\left (c + d x \right )} + 1\right )} \tan ^{3}{\left (c + d x \right )}\, 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}} \tan \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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