Optimal. Leaf size=164 \[ \frac{5 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d}+\frac{e \sqrt{e \cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+a^3\right )}-\frac{e \sqrt{e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2} \]
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Rubi [A] time = 0.662592, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3567, 3649, 3653, 3532, 205, 3634, 63} \[ \frac{5 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d}+\frac{e \sqrt{e \cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+a^3\right )}-\frac{e \sqrt{e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3567
Rule 3649
Rule 3653
Rule 3532
Rule 205
Rule 3634
Rule 63
Rubi steps
\begin{align*} \int \frac{(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^3} \, dx &=-\frac{e \sqrt{e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}-\frac{\int \frac{\frac{a e^2}{2}-2 a e^2 \cot (c+d x)-\frac{3}{2} a e^2 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2} \, dx}{4 a^2}\\ &=-\frac{e \sqrt{e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac{e \sqrt{e \cot (c+d x)}}{8 d \left (a^3+a^3 \cot (c+d x)\right )}+\frac{\int \frac{-\frac{1}{2} a^3 e^3+4 a^3 e^3 \cot (c+d x)-\frac{1}{2} a^3 e^3 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{8 a^5 e}\\ &=-\frac{e \sqrt{e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac{e \sqrt{e \cot (c+d x)}}{8 d \left (a^3+a^3 \cot (c+d x)\right )}+\frac{\int \frac{4 a^4 e^3+4 a^4 e^3 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{16 a^7 e}-\frac{\left (5 e^2\right ) \int \frac{1+\cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{16 a^2}\\ &=-\frac{e \sqrt{e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac{e \sqrt{e \cot (c+d x)}}{8 d \left (a^3+a^3 \cot (c+d x)\right )}-\frac{\left (5 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{16 a^2 d}-\frac{\left (2 a e^5\right ) \operatorname{Subst}\left (\int \frac{1}{-32 a^8 e^6-e x^2} \, dx,x,\frac{4 a^4 e^3-4 a^4 e^3 \cot (c+d x)}{\sqrt{e \cot (c+d x)}}\right )}{d}\\ &=\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d}-\frac{e \sqrt{e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac{e \sqrt{e \cot (c+d x)}}{8 d \left (a^3+a^3 \cot (c+d x)\right )}+\frac{(5 e) \operatorname{Subst}\left (\int \frac{1}{a+\frac{a x^2}{e}} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{8 a^2 d}\\ &=\frac{5 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d}-\frac{e \sqrt{e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac{e \sqrt{e \cot (c+d x)}}{8 d \left (a^3+a^3 \cot (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 1.98, size = 131, normalized size = 0.8 \[ \frac{e \sqrt{e \cot (c+d x)} \left (\frac{2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+5 \tan ^{-1}\left (\sqrt{\cot (c+d x)}\right )}{\sqrt{\cot (c+d x)}}+\frac{\tan (c+d x)-\sec ^2(c+d x)+1}{(\tan (c+d x)+1)^2}\right )}{8 a^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 434, normalized size = 2.7 \begin{align*} -{\frac{e\sqrt{2}}{16\,d{a}^{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{e\sqrt{2}}{8\,d{a}^{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{e\sqrt{2}}{8\,d{a}^{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{{e}^{2}\sqrt{2}}{16\,d{a}^{3}}\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}^{2}\sqrt{2}}{8\,d{a}^{3}}\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}^{2}\sqrt{2}}{8\,d{a}^{3}}\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}^{2}}{8\,d{a}^{3} \left ( e\cot \left ( dx+c \right ) +e \right ) ^{2}} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{3}}{8\,d{a}^{3} \left ( e\cot \left ( dx+c \right ) +e \right ) ^{2}}\sqrt{e\cot \left ( dx+c \right ) }}+{\frac{5}{8\,d{a}^{3}}{e}^{{\frac{3}{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 = 2.0411, size = 1355, normalized size = 8.26 \begin{align*} \left [\frac{2 \,{\left (\sqrt{2} e \sin \left (2 \, d x + 2 \, c\right ) + \sqrt{2} e\right )} \sqrt{-e} \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 ) + 5 \,{\left (e \sin \left (2 \, d x + 2 \, c\right ) + e\right )} \sqrt{-e} \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 ) +{\left (e \cos \left (2 \, d x + 2 \, c\right ) + e \sin \left (2 \, d x + 2 \, c\right ) - e\right )} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{16 \,{\left (a^{3} d \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d\right )}}, \frac{4 \,{\left (\sqrt{2} e \sin \left (2 \, d x + 2 \, c\right ) + \sqrt{2} e\right )} \sqrt{e} \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 ) + 10 \,{\left (e \sin \left (2 \, d x + 2 \, c\right ) + e\right )} \sqrt{e} \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 ) +{\left (e \cos \left (2 \, d x + 2 \, c\right ) + e \sin \left (2 \, d x + 2 \, c\right ) - e\right )} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{16 \,{\left (a^{3} d \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d\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*} \frac{\int \frac{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}{\cot ^{3}{\left (c + d x \right )} + 3 \cot ^{2}{\left (c + d x \right )} + 3 \cot{\left (c + d x \right )} + 1}\, dx}{a^{3}} \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{3}{2}}}{{\left (a \cot \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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