Optimal. Leaf size=327 \[ -\frac{i \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} d}-\frac{i \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{5 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{i \log (\tan (c+d x))}{6 \sqrt [3]{a} d}-\frac{i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}-\frac{3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{x}{4 \sqrt [3]{2} \sqrt [3]{a}} \]
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Rubi [A] time = 0.577922, antiderivative size = 327, 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, 3596, 3600, 3481, 55, 617, 204, 31, 3599} \[ -\frac{i \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} d}-\frac{i \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{5 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{i \log (\tan (c+d x))}{6 \sqrt [3]{a} d}-\frac{i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}-\frac{3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{x}{4 \sqrt [3]{2} \sqrt [3]{a}} \]
Antiderivative was successfully verified.
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Rule 3561
Rule 3596
Rule 3600
Rule 3481
Rule 55
Rule 617
Rule 204
Rule 31
Rule 3599
Rubi steps
\begin{align*} \int \frac{\cot ^2(c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx &=-\frac{\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{\int \frac{\cot (c+d x) \left (-\frac{i a}{3}-\frac{4}{3} a \tan (c+d x)\right )}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{a}\\ &=-\frac{5 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{3 \int \cot (c+d x) (a+i a \tan (c+d x))^{2/3} \left (-\frac{2 i a^2}{9}-\frac{5}{9} a^2 \tan (c+d x)\right ) \, dx}{2 a^3}\\ &=-\frac{5 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{i \int \cot (c+d x) (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{2/3} \, dx}{3 a^2}-\frac{\int (a+i a \tan (c+d x))^{2/3} \, dx}{2 a}\\ &=-\frac{5 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{i \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+i a x}} \, dx,x,\tan (c+d x)\right )}{3 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{i \log (\tan (c+d x))}{6 \sqrt [3]{a} d}-\frac{5 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{i \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 )}{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{i \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} 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{i \log (\tan (c+d x))}{6 \sqrt [3]{a} d}-\frac{i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{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{5 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}+\frac{i \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 )}{\sqrt [3]{a} 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 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a} d}-\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{i \log (\tan (c+d x))}{6 \sqrt [3]{a} d}-\frac{i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{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{5 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [C] time = 1.30902, size = 179, normalized size = 0.55 \[ \frac{\csc (c+d x) \sec (c+d x) \left (3 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right ) (i \sin (2 (c+d x))+\cos (2 (c+d x))-1)+4 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right ) (i \sin (2 (c+d x))+\cos (2 (c+d x))-1)-20 i \sin (2 (c+d x))-8 \cos (2 (c+d x))-8\right )}{16 d \sqrt [3]{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.12, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cot \left ( dx+c \right ) \right ) ^{2}{\frac{1}{\sqrt [3]{a+ia\tan \left ( dx+c \right ) }}}}\, 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 = 1.82444, size = 2114, normalized size = 6.46 \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{\cot ^{2}{\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{\cot \left (d x + c\right )^{2}}{{\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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