Optimal. Leaf size=246 \[ \frac{a^2 e^{3/2} \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{\sqrt{2} d}-\frac{a^2 e^{3/2} \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{\sqrt{2} d}-\frac{\sqrt{2} a^2 e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d}+\frac{\sqrt{2} a^2 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{d}-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}-\frac{4 a^2 (e \cot (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.234511, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {3543, 12, 16, 3473, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{a^2 e^{3/2} \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{\sqrt{2} d}-\frac{a^2 e^{3/2} \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{\sqrt{2} d}-\frac{\sqrt{2} a^2 e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d}+\frac{\sqrt{2} a^2 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{d}-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}-\frac{4 a^2 (e \cot (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3543
Rule 12
Rule 16
Rule 3473
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2 \, dx &=-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\int 2 a^2 \cot (c+d x) (e \cot (c+d x))^{3/2} \, dx\\ &=-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\left (2 a^2\right ) \int \cot (c+d x) (e \cot (c+d x))^{3/2} \, dx\\ &=-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac{\left (2 a^2\right ) \int (e \cot (c+d x))^{5/2} \, dx}{e}\\ &=-\frac{4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}-\left (2 a^2 e\right ) \int \sqrt{e \cot (c+d x)} \, dx\\ &=-\frac{4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac{\left (2 a^2 e^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{e^2+x^2} \, dx,x,e \cot (c+d x)\right )}{d}\\ &=-\frac{4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac{\left (4 a^2 e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}\\ &=-\frac{4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}-\frac{\left (2 a^2 e^2\right ) \operatorname{Subst}\left (\int \frac{e-x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}+\frac{\left (2 a^2 e^2\right ) \operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}\\ &=-\frac{4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac{\left (a^2 e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}+2 x}{-e-\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d}+\frac{\left (a^2 e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}-2 x}{-e+\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d}+\frac{\left (a^2 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}+\frac{\left (a^2 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}\\ &=-\frac{4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac{a^2 e^{3/2} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d}-\frac{a^2 e^{3/2} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d}+\frac{\left (\sqrt{2} a^2 e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d}-\frac{\left (\sqrt{2} a^2 e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d}\\ &=-\frac{\sqrt{2} a^2 e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d}+\frac{\sqrt{2} a^2 e^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d}-\frac{4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac{a^2 e^{3/2} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d}-\frac{a^2 e^{3/2} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d}\\ \end{align*}
Mathematica [C] time = 0.370475, size = 52, normalized size = 0.21 \[ -\frac{2 a^2 (e \cot (c+d x))^{3/2} \left (-10 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )+3 \cot (c+d x)+10\right )}{15 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 213, normalized size = 0.9 \begin{align*} -{\frac{2\,{a}^{2}}{5\,de} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{4\,{a}^{2}}{3\,d} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}{e}^{2}\sqrt{2}}{2\,d}\ln \left ({ \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}+{\frac{{a}^{2}{e}^{2}\sqrt{2}}{d}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}-{\frac{{a}^{2}{e}^{2}\sqrt{2}}{d}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}} \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 [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int \left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}\, dx + \int 2 \left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}} \cot{\left (c + d x \right )}\, dx + \int \left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}} \cot ^{2}{\left (c + d x \right )}\, dx\right ) \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 (a \cot \left (d x + c\right ) + a\right )}^{2} \left (e \cot \left (d x + c\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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