Optimal. Leaf size=117 \[ -\frac{16 a^3 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 \left (a^3 \cot (c+d x)+a^3\right ) \sqrt{e \cot (c+d x)}}{3 d e}+\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d \sqrt{e}} \]
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Rubi [A] time = 0.1691, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3566, 3630, 3532, 208} \[ -\frac{16 a^3 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 \left (a^3 \cot (c+d x)+a^3\right ) \sqrt{e \cot (c+d x)}}{3 d e}+\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 3566
Rule 3630
Rule 3532
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+a \cot (c+d x))^3}{\sqrt{e \cot (c+d x)}} \, dx &=-\frac{2 \sqrt{e \cot (c+d x)} \left (a^3+a^3 \cot (c+d x)\right )}{3 d e}-\frac{2 \int \frac{-a^3 e-3 a^3 e \cot (c+d x)-4 a^3 e \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{3 e}\\ &=-\frac{16 a^3 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 \sqrt{e \cot (c+d x)} \left (a^3+a^3 \cot (c+d x)\right )}{3 d e}-\frac{2 \int \frac{3 a^3 e-3 a^3 e \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{3 e}\\ &=-\frac{16 a^3 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 \sqrt{e \cot (c+d x)} \left (a^3+a^3 \cot (c+d x)\right )}{3 d e}+\frac{\left (12 a^6 e\right ) \operatorname{Subst}\left (\int \frac{1}{18 a^6 e^2-e x^2} \, dx,x,\frac{3 a^3 e+3 a^3 e \cot (c+d x)}{\sqrt{e \cot (c+d x)}}\right )}{d}\\ &=\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{e}+\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d \sqrt{e}}-\frac{16 a^3 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 \sqrt{e \cot (c+d x)} \left (a^3+a^3 \cot (c+d x)\right )}{3 d e}\\ \end{align*}
Mathematica [C] time = 5.03914, size = 292, normalized size = 2.5 \[ -\frac{a^3 \sin (c+d x) (\cot (c+d x)+1)^3 \left (8 \cos ^2(c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )+18 \sin (2 (c+d x))+4 \cos ^2(c+d x)+3 \sqrt{2} \sin ^2(c+d x) \sqrt{\cot (c+d x)} \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-3 \sqrt{2} \sin ^2(c+d x) \sqrt{\cot (c+d x)} \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+6 \sqrt{2} \sin ^2(c+d x) \sqrt{\cot (c+d x)} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-6 \sqrt{2} \sin ^2(c+d x) \sqrt{\cot (c+d x)} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )}{6 d \sqrt{e \cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 379, normalized size = 3.2 \begin{align*} -{\frac{2\,{a}^{3}}{3\,d{e}^{2}} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-6\,{\frac{{a}^{3}\sqrt{e\cot \left ( dx+c \right ) }}{de}}+{\frac{{a}^{3}\sqrt{2}}{2\,de}\sqrt [4]{{e}^{2}}\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{{a}^{3}\sqrt{2}}{de}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{{a}^{3}\sqrt{2}}{de}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{{a}^{3}\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}^{3}\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}^{3}\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 [A] time = 1.68323, size = 863, normalized size = 7.38 \begin{align*} \left [\frac{3 \, \sqrt{2} a^{3} \sqrt{e} \log \left (-\frac{\sqrt{2} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}{\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) - 1\right )}}{\sqrt{e}} + 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right ) \sin \left (2 \, d x + 2 \, c\right ) - 2 \,{\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) + 9 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) + a^{3}\right )} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{3 \, d e \sin \left (2 \, d x + 2 \, c\right )}, -\frac{2 \,{\left (3 \, \sqrt{2} a^{3} e \sqrt{-\frac{1}{e}} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sqrt{-\frac{1}{e}}{\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \,{\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right ) \sin \left (2 \, d x + 2 \, c\right ) +{\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) + 9 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) + a^{3}\right )} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{3 \, d e \sin \left (2 \, d x + 2 \, c\right )}\right ] \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^{3} \left (\int \frac{1}{\sqrt{e \cot{\left (c + d x \right )}}}\, dx + \int \frac{3 \cot{\left (c + d x \right )}}{\sqrt{e \cot{\left (c + d x \right )}}}\, dx + \int \frac{3 \cot ^{2}{\left (c + d x \right )}}{\sqrt{e \cot{\left (c + d x \right )}}}\, dx + \int \frac{\cot ^{3}{\left (c + d x \right )}}{\sqrt{e \cot{\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 \frac{{\left (a \cot \left (d x + c\right ) + a\right )}^{3}}{\sqrt{e \cot \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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