Optimal. Leaf size=342 \[ \frac{a^{2/3} (3 B+2 i A) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} d}-\frac{\sqrt{3} a^{2/3} (B+i A) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{2} d}-\frac{a^{2/3} (3 B+2 i A) \log (\tan (c+d x))}{6 d}+\frac{a^{2/3} (3 B+2 i A) \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac{3 a^{2/3} (B+i A) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac{a^{2/3} (B+i A) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac{a^{2/3} x (A-i B)}{2 \sqrt [3]{2}}-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d} \]
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Rubi [A] time = 0.590007, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3598, 3600, 3481, 55, 617, 204, 31, 3599} \[ \frac{a^{2/3} (3 B+2 i A) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} d}-\frac{\sqrt{3} a^{2/3} (B+i A) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{2} d}-\frac{a^{2/3} (3 B+2 i A) \log (\tan (c+d x))}{6 d}+\frac{a^{2/3} (3 B+2 i A) \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac{3 a^{2/3} (B+i A) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac{a^{2/3} (B+i A) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac{a^{2/3} x (A-i B)}{2 \sqrt [3]{2}}-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d} \]
Antiderivative was successfully verified.
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Rule 3598
Rule 3600
Rule 3481
Rule 55
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))^{2/3} (A+B \tan (c+d x)) \, dx &=-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}+\frac{\int \cot (c+d x) (a+i a \tan (c+d x))^{2/3} \left (\frac{1}{3} a (2 i A+3 B)-\frac{1}{3} a A \tan (c+d x)\right ) \, dx}{a}\\ &=-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}+(-A+i B) \int (a+i a \tan (c+d x))^{2/3} \, dx+\frac{(2 i A+3 B) \int \cot (c+d x) (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{2/3} \, dx}{3 a}\\ &=-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}+\frac{(a (i A+B)) \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt [3]{a+x}} \, dx,x,i a \tan (c+d x)\right )}{d}+\frac{(a (2 i A+3 B)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+i a x}} \, dx,x,\tan (c+d x)\right )}{3 d}\\ &=\frac{a^{2/3} (A-i B) x}{2 \sqrt [3]{2}}-\frac{a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac{a^{2/3} (2 i A+3 B) \log (\tan (c+d x))}{6 d}-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}+\frac{\left (3 a^{2/3} (i A+B)\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 \sqrt [3]{2} d}-\frac{(3 a (i A+B)) \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 )}{2 d}-\frac{\left (a^{2/3} (2 i A+3 B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}+\frac{(a (2 i A+3 B)) \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{a^{2/3} (A-i B) x}{2 \sqrt [3]{2}}-\frac{a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac{a^{2/3} (2 i A+3 B) \log (\tan (c+d x))}{6 d}+\frac{a^{2/3} (2 i A+3 B) \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac{3 a^{2/3} (i A+B) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}+\frac{\left (3 a^{2/3} (i A+B)\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 )}{\sqrt [3]{2} d}-\frac{\left (a^{2/3} (2 i A+3 B)\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{a^{2/3} (A-i B) x}{2 \sqrt [3]{2}}+\frac{a^{2/3} (2 i A+3 B) \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{\sqrt{3} a^{2/3} (i A+B) \tan ^{-1}\left (\frac{1+\frac{2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{2} d}-\frac{a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac{a^{2/3} (2 i A+3 B) \log (\tan (c+d x))}{6 d}+\frac{a^{2/3} (2 i A+3 B) \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac{3 a^{2/3} (i A+B) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}\\ \end{align*}
Mathematica [F] time = 6.24911, size = 0, normalized size = 0. \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.211, 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{2}{3}}} \left ( A+B\tan \left ( dx+c \right ) \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 = 2.00936, size = 2824, normalized size = 8.26 \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 (B \tan \left (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{2}{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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