Optimal. Leaf size=315 \[ -\frac{4 i a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} d}+\frac{i \sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}-\frac{2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac{2 i a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}-\frac{3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac{a^{4/3} x}{2^{2/3}}+\frac{i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d} \]
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Rubi [A] time = 0.529468, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {3561, 3594, 3600, 3481, 57, 617, 204, 31, 3599} \[ -\frac{4 i a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} d}+\frac{i \sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}-\frac{2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac{2 i a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}-\frac{3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac{a^{4/3} x}{2^{2/3}}+\frac{i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d} \]
Antiderivative was successfully verified.
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Rule 3561
Rule 3594
Rule 3600
Rule 3481
Rule 57
Rule 617
Rule 204
Rule 31
Rule 3599
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx &=-\frac{\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}+\frac{\int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \left (\frac{4 i a}{3}+\frac{1}{3} a \tan (c+d x)\right ) \, dx}{a}\\ &=\frac{i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}+\frac{3 \int \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)} \left (\frac{4 i a^2}{9}-\frac{2}{9} a^2 \tan (c+d x)\right ) \, dx}{a}\\ &=\frac{i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}+\frac{4}{3} i \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt [3]{a+i a \tan (c+d x)} \, dx-(2 a) \int \sqrt [3]{a+i a \tan (c+d x)} \, dx\\ &=\frac{i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}+\frac{\left (4 i a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x (a+i a x)^{2/3}} \, dx,x,\tan (c+d x)\right )}{3 d}+\frac{\left (2 i a^2\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac{a^{4/3} x}{2^{2/3}}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac{i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}-\frac{\left (2 i a^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}+\frac{\left (3 i a^{4/3}\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^{2/3} d}-\frac{\left (2 i a^{5/3}\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}+\frac{\left (3 i a^{5/3}\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 )}{\sqrt [3]{2} d}\\ &=\frac{a^{4/3} x}{2^{2/3}}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac{2 i a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}-\frac{3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}+\frac{i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}+\frac{\left (4 i a^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{d}-\frac{\left (3 i \sqrt [3]{2} a^{4/3}\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 )}{d}\\ &=\frac{a^{4/3} x}{2^{2/3}}-\frac{4 i a^{4/3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} d}+\frac{i \sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{1+\frac{2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{d}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac{2 i a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}-\frac{3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}+\frac{i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}\\ \end{align*}
Mathematica [A] time = 7.54937, size = 587, normalized size = 1.86 \[ \frac{i \sqrt [3]{e^{i d x}} e^{-\frac{1}{3} i (5 c+2 d x)} \sqrt [3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt [3]{1+e^{2 i (c+d x)}} (a+i a \tan (c+d x))^{4/3} \left (-6 \log \left (1-\frac{e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}\right )+4\ 2^{2/3} \log \left (1-\frac{\sqrt [3]{2} e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}\right )+3 \log \left (\frac{\left (1+e^{2 i (c+d x)}\right )^{2/3}+e^{\frac{2}{3} i (c+d x)} \sqrt [3]{1+e^{2 i (c+d x)}}+e^{\frac{4}{3} i (c+d x)}}{\left (1+e^{2 i (c+d x)}\right )^{2/3}}\right )-2\ 2^{2/3} \log \left (\frac{\left (1+e^{2 i (c+d x)}\right )^{2/3}+\sqrt [3]{2} e^{\frac{2}{3} i (c+d x)} \sqrt [3]{1+e^{2 i (c+d x)}}+2^{2/3} e^{\frac{4}{3} i (c+d x)}}{\left (1+e^{2 i (c+d x)}\right )^{2/3}}\right )+6 \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right )-4\ 2^{2/3} \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{2} e^{\frac{2}{3} i (c+d x)}}{\sqrt [3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right )\right )}{3\ 2^{2/3} d \sec ^{\frac{4}{3}}(c+d x) (\cos (d x)+i \sin (d x))^{4/3}}+\frac{\cos (c+d x) (a+i a \tan (c+d x))^{4/3} (\csc (c) (\cos (c)-i \sin (c)) \sin (d x) \csc (c+d x)+\cot (c) (-\cos (c)+i \sin (c)))}{d (\cos (d x)+i \sin (d x))} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.123, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{2} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}}\, dx \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 = 1.90179, size = 1785, normalized size = 5.67 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{4}{3}} \cot \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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