Optimal. Leaf size=222 \[ \frac{a^2 \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{\sqrt{2} d e^{3/2}}-\frac{a^2 \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{\sqrt{2} d e^{3/2}}+\frac{\sqrt{2} a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}-\frac{\sqrt{2} a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{d e^{3/2}}+\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}} \]
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Rubi [A] time = 0.208459, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3542, 12, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{a^2 \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{\sqrt{2} d e^{3/2}}-\frac{a^2 \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{\sqrt{2} d e^{3/2}}+\frac{\sqrt{2} a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}-\frac{\sqrt{2} a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{d e^{3/2}}+\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3542
Rule 12
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx &=\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}+\frac{\int \frac{2 a^2 e}{\sqrt{e \cot (c+d x)}} \, dx}{e^2}\\ &=\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}+\frac{\left (2 a^2\right ) \int \frac{1}{\sqrt{e \cot (c+d x)}} \, dx}{e}\\ &=\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (e^2+x^2\right )} \, dx,x,e \cot (c+d x)\right )}{d}\\ &=\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}-\frac{\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}\\ &=\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{e-x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d e}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d e}\\ &=\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}+\frac{a^2 \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 e^{3/2}}+\frac{a^2 \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 e^{3/2}}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d e}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d e}\\ &=\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}+\frac{a^2 \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d e^{3/2}}-\frac{a^2 \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d e^{3/2}}-\frac{\left (\sqrt{2} a^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 e^{3/2}}+\frac{\left (\sqrt{2} a^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 e^{3/2}}\\ &=\frac{\sqrt{2} a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}-\frac{\sqrt{2} a^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}+\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}+\frac{a^2 \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d e^{3/2}}-\frac{a^2 \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d e^{3/2}}\\ \end{align*}
Mathematica [C] time = 1.78299, size = 236, normalized size = 1.06 \[ \frac{a^2 (\cot (c+d x)+1)^2 \left (3 \sin (c+d x) \left (4 \cos (c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},-\cot ^2(c+d x)\right )+\sqrt{2} \sin (c+d x) \cot ^{\frac{3}{2}}(c+d x) \left (\log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-2 \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )\right )-4 \cos ^2(c+d x) \cot (c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )\right )}{6 d (e \cot (c+d x))^{3/2} (\sin (c+d x)+\cos (c+d x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 195, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}\sqrt{2}}{2\,d{e}^{2}}\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}^{2}\sqrt{2}}{d{e}^{2}}\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}^{2}\sqrt{2}}{d{e}^{2}}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+2\,{\frac{{a}^{2}}{de\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 [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 \frac{1}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{2 \cot{\left (c + d x \right )}}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{\cot ^{2}{\left (c + d x \right )}}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{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 )}^{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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