Optimal. Leaf size=205 \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{4 \sqrt [3]{2} a^{4/3} d}+\frac{3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}+\frac{\log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac{i x}{8 \sqrt [3]{2} a^{4/3}}+\frac{3}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{3}{8 d (a+i a \tan (c+d x))^{4/3}} \]
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Rubi [A] time = 0.154235, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {3526, 3479, 3481, 55, 617, 204, 31} \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{4 \sqrt [3]{2} a^{4/3} d}+\frac{3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}+\frac{\log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac{i x}{8 \sqrt [3]{2} a^{4/3}}+\frac{3}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{3}{8 d (a+i a \tan (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Rule 3526
Rule 3479
Rule 3481
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\tan (c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx &=-\frac{3}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac{i \int \frac{1}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{2 a}\\ &=-\frac{3}{8 d (a+i a \tan (c+d x))^{4/3}}+\frac{3}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{i \int (a+i a \tan (c+d x))^{2/3} \, dx}{4 a^2}\\ &=-\frac{3}{8 d (a+i a \tan (c+d x))^{4/3}}+\frac{3}{4 a 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 )}{4 a d}\\ &=\frac{i x}{8 \sqrt [3]{2} a^{4/3}}+\frac{\log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}-\frac{3}{8 d (a+i a \tan (c+d x))^{4/3}}+\frac{3}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}-\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 )}{8 \sqrt [3]{2} a^{4/3} d}+\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 )}{8 a d}\\ &=\frac{i x}{8 \sqrt [3]{2} a^{4/3}}+\frac{\log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac{3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac{3}{8 d (a+i a \tan (c+d x))^{4/3}}+\frac{3}{4 a 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 )}{4 \sqrt [3]{2} a^{4/3} d}\\ &=\frac{i x}{8 \sqrt [3]{2} a^{4/3}}+\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 )}{4 \sqrt [3]{2} a^{4/3} d}+\frac{\log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac{3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac{3}{8 d (a+i a \tan (c+d x))^{4/3}}+\frac{3}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.598711, size = 130, normalized size = 0.63 \[ \frac{3 i \sec ^2(c+d x) \left (\, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-2 i \sin (2 (c+d x))-\cos (2 (c+d x))-1\right )}{16 a d (\tan (c+d x)-i) \sqrt [3]{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 176, normalized size = 0.9 \begin{align*}{\frac{{2}^{{\frac{2}{3}}}}{8\,d}\ln \left ( \sqrt [3]{a+ia\tan \left ( dx+c \right ) }-\sqrt [3]{2}\sqrt [3]{a} \right ){a}^{-{\frac{4}{3}}}}-{\frac{{2}^{{\frac{2}{3}}}}{16\,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{4}{3}}}}+{\frac{\sqrt{3}{2}^{{\frac{2}{3}}}}{8\,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{4}{3}}}}-{\frac{3}{8\,d} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{4}{3}}}}+{\frac{3}{4\,ad}{\frac{1}{\sqrt [3]{a+ia\tan \left ( dx+c \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 = 1.80625, size = 1096, normalized size = 5.35 \begin{align*} \frac{{\left (32 \, \left (\frac{1}{2}\right )^{\frac{1}{3}} a^{2} d \left (\frac{1}{a^{4} d^{3}}\right )^{\frac{1}{3}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-2 \, \left (\frac{1}{2}\right )^{\frac{2}{3}} a^{3} d^{2} \left (\frac{1}{a^{4} 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 (16 i \, \sqrt{3} a^{2} d - 16 \, a^{2} d\right )} \left (\frac{1}{a^{4} d^{3}}\right )^{\frac{1}{3}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac{1}{16} \, \left (\frac{1}{2}\right )^{\frac{2}{3}}{\left (16 i \, \sqrt{3} a^{3} d^{2} + 16 \, a^{3} d^{2}\right )} \left (\frac{1}{a^{4} 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 (-16 i \, \sqrt{3} a^{2} d - 16 \, a^{2} d\right )} \left (\frac{1}{a^{4} d^{3}}\right )^{\frac{1}{3}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac{1}{16} \, \left (\frac{1}{2}\right )^{\frac{2}{3}}{\left (-16 i \, \sqrt{3} a^{3} d^{2} + 16 \, a^{3} d^{2}\right )} \left (\frac{1}{a^{4} 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 ) + 12 \cdot 2^{\frac{2}{3}} \left (\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac{2}{3}}{\left (3 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, 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 (-4 i \, d x - 4 i \, c\right )}}{128 \, a^{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 \frac{\tan{\left (c + d x \right )}}{\left (a \left (i \tan{\left (c + d x \right )} + 1\right )\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 \frac{\tan \left (d x + c\right )}{{\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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