Optimal. Leaf size=117 \[ \frac{16 a^3}{3 d e^2 \sqrt{e \cot (c+d x)}}-\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d e^{5/2}}+\frac{2 \left (a^3 \cot (c+d x)+a^3\right )}{3 d e (e \cot (c+d x))^{3/2}} \]
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Rubi [A] time = 0.188038, 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 = {3565, 3628, 3532, 208} \[ \frac{16 a^3}{3 d e^2 \sqrt{e \cot (c+d x)}}-\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d e^{5/2}}+\frac{2 \left (a^3 \cot (c+d x)+a^3\right )}{3 d e (e \cot (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3565
Rule 3628
Rule 3532
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{5/2}} \, dx &=\frac{2 \left (a^3+a^3 \cot (c+d x)\right )}{3 d e (e \cot (c+d x))^{3/2}}-\frac{2 \int \frac{-4 a^3 e^2-3 a^3 e^2 \cot (c+d x)-a^3 e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2}} \, dx}{3 e^3}\\ &=\frac{16 a^3}{3 d e^2 \sqrt{e \cot (c+d x)}}+\frac{2 \left (a^3+a^3 \cot (c+d x)\right )}{3 d e (e \cot (c+d x))^{3/2}}-\frac{2 \int \frac{-3 a^3 e^3+3 a^3 e^3 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{3 e^5}\\ &=\frac{16 a^3}{3 d e^2 \sqrt{e \cot (c+d x)}}+\frac{2 \left (a^3+a^3 \cot (c+d x)\right )}{3 d e (e \cot (c+d x))^{3/2}}+\frac{\left (12 a^6 e\right ) \operatorname{Subst}\left (\int \frac{1}{18 a^6 e^6-e x^2} \, dx,x,\frac{-3 a^3 e^3-3 a^3 e^3 \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 e^{5/2}}+\frac{16 a^3}{3 d e^2 \sqrt{e \cot (c+d x)}}+\frac{2 \left (a^3+a^3 \cot (c+d x)\right )}{3 d e (e \cot (c+d x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 6.11614, size = 417, normalized size = 3.56 \[ -\frac{2 \cos ^3(c+d x) \cot (c+d x) (a \cot (c+d x)+a)^3 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )}{3 d (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3}+\frac{6 \sin (c+d x) \cos ^2(c+d x) (a \cot (c+d x)+a)^3 \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},-\cot ^2(c+d x)\right )}{d (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3}+\frac{2 \sin ^2(c+d x) \cos (c+d x) (a \cot (c+d x)+a)^3 \text{Hypergeometric2F1}\left (-\frac{3}{4},1,\frac{1}{4},-\cot ^2(c+d x)\right )}{3 d (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3}+\frac{3 \sin ^3(c+d x) \cot ^{\frac{5}{2}}(c+d x) (a \cot (c+d x)+a)^3 \left (\sqrt{2} \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\sqrt{2} \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+2 \left (\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )\right )}{4 d (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 388, normalized size = 3.3 \begin{align*} -{\frac{{a}^{3}\sqrt{2}}{2\,d{e}^{3}}\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}}{d{e}^{3}}\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}}{d{e}^{3}}\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{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{1}{\sqrt [4]{{e}^{2}}}}}+{\frac{{a}^{3}\sqrt{2}}{d{e}^{2}}\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{e}^{2}}\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{2\,{a}^{3}}{3\,de} \left ( e\cot \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}+6\,{\frac{{a}^{3}}{d{e}^{2}\sqrt{e\cot \left ( dx+c \right ) }}} \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 = 1.59294, size = 917, normalized size = 7.84 \begin{align*} \left [\frac{\frac{3 \, \sqrt{2}{\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e\right )} \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 )}{\sqrt{e}} - 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 \,{\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\right )}}, \frac{2 \,{\left (3 \, \sqrt{2}{\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e\right )} \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 ) -{\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 \,{\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\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}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{5}{2}}}\, dx + \int \frac{3 \cot{\left (c + d x \right )}}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{5}{2}}}\, dx + \int \frac{3 \cot ^{2}{\left (c + d x \right )}}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{5}{2}}}\, dx + \int \frac{\cot ^{3}{\left (c + d x \right )}}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{5}{2}}}\, 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}}{\left (e \cot \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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