Optimal. Leaf size=286 \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{\log (\tan (c+d x))}{2 \sqrt [3]{a} d}+\frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}-\frac{3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{i x}{4 \sqrt [3]{2} \sqrt [3]{a}} \]
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Rubi [A] time = 0.42829, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3562, 3479, 3481, 55, 617, 204, 31, 3596, 3599} \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}+\frac{3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{\log (\tan (c+d x))}{2 \sqrt [3]{a} d}+\frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}-\frac{3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{i x}{4 \sqrt [3]{2} \sqrt [3]{a}} \]
Antiderivative was successfully verified.
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Rule 3562
Rule 3479
Rule 3481
Rule 55
Rule 617
Rule 204
Rule 31
Rule 3596
Rule 3599
Rubi steps
\begin{align*} \int \frac{\cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx &=i \int \frac{1}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx-\frac{i \int \frac{\cot (c+d x) (i a+a \tan (c+d x))}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{a}\\ &=\frac{3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac{(3 i) \int \cot (c+d x) (a+i a \tan (c+d x))^{2/3} \left (\frac{2 i a^2}{3}+\frac{2}{3} a^2 \tan (c+d x)\right ) \, dx}{2 a^3}+\frac{i \int (a+i a \tan (c+d x))^{2/3} \, dx}{2 a}\\ &=\frac{3}{2 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 )}{2 d}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+i a x}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{i x}{4 \sqrt [3]{2} \sqrt [3]{a}}-\frac{\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{\log (\tan (c+d x))}{2 \sqrt [3]{a} d}+\frac{3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\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 )}{4 d}+\frac{3 \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 \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 \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{i x}{4 \sqrt [3]{2} \sqrt [3]{a}}-\frac{\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{\log (\tan (c+d x))}{2 \sqrt [3]{a} d}+\frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}-\frac{3 \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{3}{2 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 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} d}+\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 )}{2 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=-\frac{i x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\frac{\sqrt{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]{a} d}-\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 )}{2 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac{\log (\tan (c+d x))}{2 \sqrt [3]{a} d}+\frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}-\frac{3 \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{3}{2 d \sqrt [3]{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [C] time = 1.1233, size = 195, normalized size = 0.68 \[ \frac{3 \left (e^{2 i d x} (\cos (c)+i \sin (c)) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )-4 e^{2 i d x} (\cos (c)+i \sin (c)) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )+2 i \sin (c) \left (-1+e^{2 i d x}\right )+2 \cos (c) \left (1+e^{2 i d x}\right )\right )}{4 d \sqrt [3]{a+i a \tan (c+d x)} \left (i \sin (c) \left (-1+e^{2 i d x}\right )+\cos (c) \left (1+e^{2 i d x}\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.114, size = 0, normalized size = 0. \begin{align*} \int{\cot \left ( dx+c \right ){\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 = 1.81925, size = 1720, normalized size = 6.01 \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{\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 )}{{\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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