Optimal. Leaf size=267 \[ \frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2}}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2}}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{3/2}}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d e^{3/2}}+\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}} \]
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Rubi [A] time = 0.257626, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3542, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2}}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2}}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{3/2}}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d e^{3/2}}+\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3542
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(a+b \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 b e-\left (a^2-b^2\right ) e \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{e^2}\\ &=\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{-2 a b e^2+\left (a^2-b^2\right ) e x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d e^2}\\ &=\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}+\frac{\left (a^2-2 a b-b^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 (a^2+2 a b-b^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{\left (a^2+2 a b-b^2\right ) \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 )}{2 \sqrt{2} d e^{3/2}}+\frac{\left (a^2+2 a b-b^2\right ) \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 )}{2 \sqrt{2} d e^{3/2}}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 d e}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 d e}\\ &=\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d e^{3/2}}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d e^{3/2}}+\frac{\left (a^2-2 a b-b^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 )}{\sqrt{2} d e^{3/2}}-\frac{\left (a^2-2 a b-b^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 )}{\sqrt{2} d e^{3/2}}\\ &=-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{3/2}}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{3/2}}+\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d e^{3/2}}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d e^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.306595, size = 218, normalized size = 0.82 \[ -\frac{\cot ^{\frac{3}{2}}(c+d x) \left (-\frac{2 \left (a^2-b^2\right ) \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},-\cot ^2(c+d x)\right )}{\sqrt{\cot (c+d x)}}+4 a b \left (\frac{1}{2} \left (\frac{\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2}}-\frac{\log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2}}\right )+\frac{1}{2} \left (\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2}}\right )\right )-\frac{2 b^2}{\sqrt{\cot (c+d x)}}\right )}{d (e \cot (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 538, normalized size = 2. \begin{align*} \text{result too large to display} \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*} \int \frac{\left (a + b \cot{\left (c + d x \right )}\right )^{2}}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \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 (b \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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