Optimal. Leaf size=178 \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{i x}{4 \sqrt [3]{2} \sqrt [3]{a}} \]
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Rubi [A] time = 0.118031, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3526, 3481, 55, 617, 204, 31} \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{i x}{4 \sqrt [3]{2} \sqrt [3]{a}} \]
Antiderivative was successfully verified.
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Rule 3526
Rule 3481
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\tan (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx &=-\frac{3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{i \int (a+i a \tan (c+d x))^{2/3} \, dx}{2 a}\\ &=-\frac{3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt [3]{a+x}} \, dx,x,i a \tan (c+d x)\right )}{2 d}\\ &=\frac{i x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\frac{\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \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 )}{4 d}-\frac{3 \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 )}{4 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=\frac{i x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\frac{\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{3 \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 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=\frac{i x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\frac{\sqrt{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 )}{2 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.576638, size = 140, normalized size = 0.79 \[ -\frac{3 \left (e^{2 i d x} (\cos (c)+i \sin (c)) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )+2 i \sin (c) \left (-1+e^{2 i d x}\right )+2 \cos (c) \left (1+e^{2 i d x}\right )\right )}{4 d \sqrt [3]{a+i a \tan (c+d x)} \left (i \sin (c) \left (-1+e^{2 i d x}\right )+\cos (c) \left (1+e^{2 i d x}\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 154, normalized size = 0.9 \begin{align*} -{\frac{3}{2\,d}{\frac{1}{\sqrt [3]{a+ia\tan \left ( dx+c \right ) }}}}+{\frac{{2}^{{\frac{2}{3}}}}{4\,d}\ln \left ( \sqrt [3]{a+ia\tan \left ( dx+c \right ) }-\sqrt [3]{2}\sqrt [3]{a} \right ){\frac{1}{\sqrt [3]{a}}}}-{\frac{{2}^{{\frac{2}{3}}}}{8\,d}\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{1}{\sqrt [3]{a}}}}+{\frac{\sqrt{3}{2}^{{\frac{2}{3}}}}{4\,d}\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 ){\frac{1}{\sqrt [3]{a}}}} \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 = 1.76889, size = 996, normalized size = 5.6 \begin{align*} \frac{{\left (4 \, \left (\frac{1}{2}\right )^{\frac{1}{3}} a d \left (\frac{1}{a d^{3}}\right )^{\frac{1}{3}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-2 \, \left (\frac{1}{2}\right )^{\frac{2}{3}} a d^{2} \left (\frac{1}{a d^{3}}\right )^{\frac{2}{3}} + 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 )}\right ) + \left (\frac{1}{2}\right )^{\frac{1}{3}}{\left (2 i \, \sqrt{3} a d - 2 \, a d\right )} \left (\frac{1}{a d^{3}}\right )^{\frac{1}{3}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (\frac{1}{4} \, \left (\frac{1}{2}\right )^{\frac{2}{3}}{\left (4 i \, \sqrt{3} a d^{2} + 4 \, a d^{2}\right )} \left (\frac{1}{a d^{3}}\right )^{\frac{2}{3}} + 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 )}\right ) + \left (\frac{1}{2}\right )^{\frac{1}{3}}{\left (-2 i \, \sqrt{3} a d - 2 \, a d\right )} \left (\frac{1}{a d^{3}}\right )^{\frac{1}{3}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (\frac{1}{4} \, \left (\frac{1}{2}\right )^{\frac{2}{3}}{\left (-4 i \, \sqrt{3} a d^{2} + 4 \, a d^{2}\right )} \left (\frac{1}{a d^{3}}\right )^{\frac{2}{3}} + 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 )}\right ) - 6 \cdot 2^{\frac{2}{3}} \left (\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac{2}{3}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (\frac{4}{3} i \, d x + \frac{4}{3} i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a 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 \frac{\tan{\left (c + d x \right )}}{\sqrt [3]{a \left (i \tan{\left (c + d x \right )} + 1\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 \frac{\tan \left (d x + c\right )}{{\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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