Optimal. Leaf size=288 \[ -\frac{\sqrt{e} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d}+\frac{\sqrt{e} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d}+\frac{\sqrt{e} \left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d}-\frac{\sqrt{e} \left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d}-\frac{4 a b \sqrt{e \cot (c+d x)}}{d}-\frac{2 b^2 (e \cot (c+d x))^{3/2}}{3 d e} \]
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Rubi [A] time = 0.274558, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {3543, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt{e} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d}+\frac{\sqrt{e} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d}+\frac{\sqrt{e} \left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d}-\frac{\sqrt{e} \left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d}-\frac{4 a b \sqrt{e \cot (c+d x)}}{d}-\frac{2 b^2 (e \cot (c+d x))^{3/2}}{3 d e} \]
Antiderivative was successfully verified.
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Rule 3543
Rule 3528
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2 \, dx &=-\frac{2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}+\int \sqrt{e \cot (c+d x)} \left (a^2-b^2+2 a b \cot (c+d x)\right ) \, dx\\ &=-\frac{4 a b \sqrt{e \cot (c+d x)}}{d}-\frac{2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}+\int \frac{-2 a b e+\left (a^2-b^2\right ) e \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx\\ &=-\frac{4 a b \sqrt{e \cot (c+d x)}}{d}-\frac{2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}+\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}\\ &=-\frac{4 a b \sqrt{e \cot (c+d x)}}{d}-\frac{2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}-\frac{\left (\left (a^2-2 a b-b^2\right ) e\right ) \operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}+\frac{\left (\left (a^2+2 a b-b^2\right ) e\right ) \operatorname{Subst}\left (\int \frac{e-x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}\\ &=-\frac{4 a b \sqrt{e \cot (c+d x)}}{d}-\frac{2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}-\frac{\left (\left (a^2+2 a b-b^2\right ) \sqrt{e}\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}-\frac{\left (\left (a^2+2 a b-b^2\right ) \sqrt{e}\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}-\frac{\left (\left (a^2-2 a b-b^2\right ) e\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}-\frac{\left (\left (a^2-2 a b-b^2\right ) e\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}\\ &=-\frac{4 a b \sqrt{e \cot (c+d x)}}{d}-\frac{2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}-\frac{\left (a^2+2 a b-b^2\right ) \sqrt{e} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d}+\frac{\left (a^2+2 a b-b^2\right ) \sqrt{e} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d}-\frac{\left (\left (a^2-2 a b-b^2\right ) \sqrt{e}\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}+\frac{\left (\left (a^2-2 a b-b^2\right ) \sqrt{e}\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}\\ &=\frac{\left (a^2-2 a b-b^2\right ) \sqrt{e} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d}-\frac{\left (a^2-2 a b-b^2\right ) \sqrt{e} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d}-\frac{4 a b \sqrt{e \cot (c+d x)}}{d}-\frac{2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}-\frac{\left (a^2+2 a b-b^2\right ) \sqrt{e} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d}+\frac{\left (a^2+2 a b-b^2\right ) \sqrt{e} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d}\\ \end{align*}
Mathematica [C] time = 0.569856, size = 220, normalized size = 0.76 \[ -\frac{\sqrt{e \cot (c+d x)} \left (4 \left (a^2-b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )+b \left (24 a \sqrt{\cot (c+d x)}+3 \sqrt{2} a \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-3 \sqrt{2} a \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+6 \sqrt{2} a \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-6 \sqrt{2} a \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+4 b \cot ^{\frac{3}{2}}(c+d x)\right )\right )}{6 d \sqrt{\cot (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 534, normalized size = 1.9 \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 \sqrt{e \cot{\left (c + d x \right )}} \left (a + b \cot{\left (c + d x \right )}\right )^{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{\left (b \cot \left (d x + c\right ) + a\right )}^{2} \sqrt{e \cot \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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