Optimal. Leaf size=111 \[ -\frac{2 e^2 \sqrt{e \cot (c+d x)}}{a d}+\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{a d}-\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{\sqrt{2} a d} \]
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Rubi [A] time = 0.451404, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3566, 3653, 3532, 205, 3634, 63} \[ -\frac{2 e^2 \sqrt{e \cot (c+d x)}}{a d}+\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{a d}-\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{\sqrt{2} a d} \]
Antiderivative was successfully verified.
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Rule 3566
Rule 3653
Rule 3532
Rule 205
Rule 3634
Rule 63
Rubi steps
\begin{align*} \int \frac{(e \cot (c+d x))^{5/2}}{a+a \cot (c+d x)} \, dx &=-\frac{2 e^2 \sqrt{e \cot (c+d x)}}{a d}-\frac{2 \int \frac{\frac{a e^3}{2}+\frac{1}{2} a e^3 \cot (c+d x)+\frac{1}{2} a e^3 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{a}\\ &=-\frac{2 e^2 \sqrt{e \cot (c+d x)}}{a d}-\frac{\int \frac{\frac{a^2 e^3}{2}+\frac{1}{2} a^2 e^3 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{a^3}-\frac{1}{2} e^3 \int \frac{1+\cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx\\ &=-\frac{2 e^2 \sqrt{e \cot (c+d x)}}{a d}-\frac{e^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{2 d}+\frac{\left (a e^6\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} a^4 e^6-e x^2} \, dx,x,\frac{\frac{a^2 e^3}{2}-\frac{1}{2} a^2 e^3 \cot (c+d x)}{\sqrt{e \cot (c+d x)}}\right )}{2 d}\\ &=-\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{\sqrt{2} a d}-\frac{2 e^2 \sqrt{e \cot (c+d x)}}{a d}+\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{a+\frac{a x^2}{e}} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}\\ &=\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{a d}-\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{\sqrt{2} a d}-\frac{2 e^2 \sqrt{e \cot (c+d x)}}{a d}\\ \end{align*}
Mathematica [A] time = 0.847899, size = 110, normalized size = 0.99 \[ -\frac{(e \cot (c+d x))^{5/2} \left (4 \sqrt{\cot (c+d x)}+\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-2 \tan ^{-1}\left (\sqrt{\cot (c+d x)}\right )\right )}{2 a d \cot ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 394, normalized size = 3.6 \begin{align*} -2\,{\frac{{e}^{2}\sqrt{e\cot \left ( dx+c \right ) }}{ad}}+{\frac{{e}^{2}\sqrt{2}}{8\,ad}\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{{e}^{2}\sqrt{2}}{4\,ad}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{{e}^{2}\sqrt{2}}{4\,ad}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{{e}^{3}\sqrt{2}}{8\,ad}\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{{e}^{3}\sqrt{2}}{4\,ad}\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{{e}^{3}\sqrt{2}}{4\,ad}\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{1}{ad}{e}^{{\frac{5}{2}}}\arctan \left ({\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt{e}}}} \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 = 1.40146, size = 1019, normalized size = 9.18 \begin{align*} \left [\frac{\sqrt{2} \sqrt{-e} e^{2} \log \left ({\left (\sqrt{2} \cos \left (2 \, d x + 2 \, c\right ) + \sqrt{2} \sin \left (2 \, d x + 2 \, c\right ) - \sqrt{2}\right )} \sqrt{-e} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) + 2 \, \sqrt{-e} e^{2} \log \left (\frac{e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) + 2 \, \sqrt{-e} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e}{\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}\right ) - 8 \, e^{2} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{4 \, a d}, -\frac{\sqrt{2} e^{\frac{5}{2}} \arctan \left (-\frac{{\left (\sqrt{2} \cos \left (2 \, d x + 2 \, c\right ) - \sqrt{2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt{2}\right )} \sqrt{e} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{2 \,{\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) - 2 \, e^{\frac{5}{2}} \arctan \left (\frac{\sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt{e}}\right ) + 4 \, e^{2} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{2 \, a d}\right ] \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 \frac{\left (e \cot \left (d x + c\right )\right )^{\frac{5}{2}}}{a \cot \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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