Optimal. Leaf size=184 \[ -\frac{i \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2\ 2^{2/3} a^{2/3} d}+\frac{3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4\ 2^{2/3} a^{2/3} d}+\frac{i \log (\cos (c+d x))}{4\ 2^{2/3} a^{2/3} d}-\frac{x}{4\ 2^{2/3} a^{2/3}}+\frac{3 i}{4 d (a+i a \tan (c+d x))^{2/3}} \]
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Rubi [A] time = 0.109878, antiderivative size = 184, 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 = {3479, 3481, 57, 617, 204, 31} \[ -\frac{i \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2\ 2^{2/3} a^{2/3} d}+\frac{3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4\ 2^{2/3} a^{2/3} d}+\frac{i \log (\cos (c+d x))}{4\ 2^{2/3} a^{2/3} d}-\frac{x}{4\ 2^{2/3} a^{2/3}}+\frac{3 i}{4 d (a+i a \tan (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 3479
Rule 3481
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{(a+i a \tan (c+d x))^{2/3}} \, dx &=\frac{3 i}{4 d (a+i a \tan (c+d x))^{2/3}}+\frac{\int \sqrt [3]{a+i a \tan (c+d x)} \, dx}{2 a}\\ &=\frac{3 i}{4 d (a+i a \tan (c+d x))^{2/3}}-\frac{i \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{2 d}\\ &=-\frac{x}{4\ 2^{2/3} a^{2/3}}+\frac{i \log (\cos (c+d x))}{4\ 2^{2/3} a^{2/3} d}+\frac{3 i}{4 d (a+i a \tan (c+d x))^{2/3}}-\frac{(3 i) \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\ 2^{2/3} a^{2/3} d}-\frac{(3 i) \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 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=-\frac{x}{4\ 2^{2/3} a^{2/3}}+\frac{i \log (\cos (c+d x))}{4\ 2^{2/3} a^{2/3} d}+\frac{3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4\ 2^{2/3} a^{2/3} d}+\frac{3 i}{4 d (a+i a \tan (c+d x))^{2/3}}+\frac{(3 i) \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^{2/3} a^{2/3} d}\\ &=-\frac{x}{4\ 2^{2/3} a^{2/3}}-\frac{i \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\ 2^{2/3} a^{2/3} d}+\frac{i \log (\cos (c+d x))}{4\ 2^{2/3} a^{2/3} d}+\frac{3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4\ 2^{2/3} a^{2/3} d}+\frac{3 i}{4 d (a+i a \tan (c+d x))^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.939973, size = 290, normalized size = 1.58 \[ \frac{i \left (3 \left (1+e^{2 i (c+d x)}\right )^{2/3}+2 e^{\frac{4}{3} i (c+d x)} \log \left (1-\frac{e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}\right )-e^{\frac{4}{3} i (c+d x)} \log \left (\frac{\left (1+e^{2 i (c+d x)}\right )^{2/3}+e^{\frac{2}{3} i (c+d x)} \sqrt [3]{1+e^{2 i (c+d x)}}+e^{\frac{4}{3} i (c+d x)}}{\left (1+e^{2 i (c+d x)}\right )^{2/3}}\right )-2 \sqrt{3} e^{\frac{4}{3} i (c+d x)} \tan ^{-1}\left (\frac{1+\frac{2 e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right )\right )}{4\ 2^{2/3} d \left (1+e^{2 i (c+d x)}\right )^{2/3} \left (\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 158, normalized size = 0.9 \begin{align*}{\frac{{\frac{3\,i}{4}}}{d} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{2}{3}}}}+{\frac{{\frac{i}{4}}\sqrt [3]{2}}{d}\ln \left ( \sqrt [3]{a+ia\tan \left ( dx+c \right ) }-\sqrt [3]{2}\sqrt [3]{a} \right ){a}^{-{\frac{2}{3}}}}-{\frac{{\frac{i}{8}}\sqrt [3]{2}}{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 ){a}^{-{\frac{2}{3}}}}-{\frac{{\frac{i}{4}}\sqrt [3]{2}\sqrt{3}}{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 ){a}^{-{\frac{2}{3}}}} \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 = 1.69366, size = 940, normalized size = 5.11 \begin{align*} \frac{{\left (8 \, a d \left (-\frac{i}{32 \, a^{2} d^{3}}\right )^{\frac{1}{3}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (4 i \, a d \left (-\frac{i}{32 \, a^{2} d^{3}}\right )^{\frac{1}{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 (4 i \, \sqrt{3} a d - 4 \, a d\right )} \left (-\frac{i}{32 \, a^{2} d^{3}}\right )^{\frac{1}{3}} e^{\left (2 i \, d x + 2 i \, c\right )} \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 \,{\left (\sqrt{3} a d + i \, a d\right )} \left (-\frac{i}{32 \, a^{2} d^{3}}\right )^{\frac{1}{3}}\right ) +{\left (-4 i \, \sqrt{3} a d - 4 \, a d\right )} \left (-\frac{i}{32 \, a^{2} d^{3}}\right )^{\frac{1}{3}} e^{\left (2 i \, d x + 2 i \, c\right )} \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 \,{\left (\sqrt{3} a d - i \, a d\right )} \left (-\frac{i}{32 \, a^{2} d^{3}}\right )^{\frac{1}{3}}\right ) + 2^{\frac{1}{3}} \left (\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac{1}{3}}{\left (3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (\frac{2}{3} i \, d x + \frac{2}{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{1}{\left (i a \tan{\left (c + d x \right )} + a\right )^{\frac{2}{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 \frac{1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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