Optimal. Leaf size=282 \[ -\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{i \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}+\frac{3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{x}{4 \sqrt [3]{2} \sqrt [3]{a}} \]
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Rubi [A] time = 0.442492, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {3560, 3595, 3592, 3527, 3481, 55, 617, 204, 31} \[ -\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{i \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}+\frac{3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{x}{4 \sqrt [3]{2} \sqrt [3]{a}} \]
Antiderivative was successfully verified.
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Rule 3560
Rule 3595
Rule 3592
Rule 3527
Rule 3481
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\tan ^4(c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx &=\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{3 \int \frac{\tan ^2(c+d x) \left (3 a-\frac{1}{3} i a \tan (c+d x)\right )}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{8 a}\\ &=-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{9 \int \tan (c+d x) (a+i a \tan (c+d x))^{2/3} \left (\frac{20 i a^2}{3}+\frac{52}{9} a^2 \tan (c+d x)\right ) \, dx}{16 a^3}\\ &=-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}+\frac{9 \int (a+i a \tan (c+d x))^{2/3} \left (-\frac{52 a^2}{9}+\frac{20}{3} i a^2 \tan (c+d x)\right ) \, dx}{16 a^3}\\ &=-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}+\frac{\int (a+i a \tan (c+d x))^{2/3} \, dx}{2 a}\\ &=-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}-\frac{i \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt [3]{a+x}} \, dx,x,i a \tan (c+d x)\right )}{2 d}\\ &=-\frac{x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\frac{i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 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 d}-\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 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=-\frac{x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\frac{i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{3 i \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{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}-\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 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=-\frac{x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\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 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{3 i \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{15 i \tan ^2(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}\\ \end{align*}
Mathematica [C] time = 1.23264, size = 125, normalized size = 0.44 \[ \frac{15 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right ) (\tan (c+d x)-i)+3 i \sec ^3(c+d x) (2 i \sin (c+d x)+7 i \sin (3 (c+d x))+37 \cos (c+d x)+12 \cos (3 (c+d x)))}{40 d \sqrt [3]{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 227, normalized size = 0.8 \begin{align*}{\frac{{\frac{3\,i}{8}}}{d{a}^{3}} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{8}{3}}}}-{\frac{{\frac{6\,i}{5}}}{{a}^{2}d} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{5}{3}}}}+{\frac{3\,i}{ad} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}}+{\frac{{\frac{3\,i}{2}}}{d}{\frac{1}{\sqrt [3]{a+ia\tan \left ( dx+c \right ) }}}}+{\frac{{\frac{i}{4}}{2}^{{\frac{2}{3}}}}{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{{\frac{i}{8}}{2}^{{\frac{2}{3}}}}{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{{\frac{i}{4}}\sqrt{3}{2}^{{\frac{2}{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 ){\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 = 1.74475, size = 1453, normalized size = 5.15 \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 \frac{\tan ^{4}{\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 )^{4}}{{\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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