Optimal. Leaf size=313 \[ \frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \sqrt{e}}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \sqrt{e}}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d \sqrt{e}}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d \sqrt{e}}-\frac{2 b^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac{16 a b^2 \sqrt{e \cot (c+d x)}}{3 d e} \]
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Rubi [A] time = 0.425366, antiderivative size = 313, 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 = {3566, 3630, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \sqrt{e}}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \sqrt{e}}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d \sqrt{e}}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d \sqrt{e}}-\frac{2 b^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac{16 a b^2 \sqrt{e \cot (c+d x)}}{3 d e} \]
Antiderivative was successfully verified.
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Rule 3566
Rule 3630
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))^3}{\sqrt{e \cot (c+d x)}} \, dx &=-\frac{2 b^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac{2 \int \frac{-\frac{1}{2} a \left (3 a^2-b^2\right ) e-\frac{3}{2} b \left (3 a^2-b^2\right ) e \cot (c+d x)-4 a b^2 e \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{3 e}\\ &=-\frac{16 a b^2 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 b^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac{2 \int \frac{-\frac{3}{2} a \left (a^2-3 b^2\right ) e-\frac{3}{2} b \left (3 a^2-b^2\right ) e \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{3 e}\\ &=-\frac{16 a b^2 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 b^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac{4 \operatorname{Subst}\left (\int \frac{\frac{3}{2} a \left (a^2-3 b^2\right ) e^2+\frac{3}{2} b \left (3 a^2-b^2\right ) e x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{3 d e}\\ &=-\frac{16 a b^2 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 b^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\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 ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}\\ &=-\frac{16 a b^2 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 b^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\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 ((a-b) \left (a^2+4 a b+b^2\right )\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 ((a+b) \left (a^2-4 a b+b^2\right )\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 \sqrt{e}}+\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\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 \sqrt{e}}\\ &=-\frac{16 a b^2 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 b^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}+\frac{(a+b) \left (a^2-4 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 \sqrt{e}}-\frac{(a+b) \left (a^2-4 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 \sqrt{e}}-\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\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 \sqrt{e}}+\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\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 \sqrt{e}}\\ &=\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d \sqrt{e}}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d \sqrt{e}}-\frac{16 a b^2 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 b^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}+\frac{(a+b) \left (a^2-4 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 \sqrt{e}}-\frac{(a+b) \left (a^2-4 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 \sqrt{e}}\\ \end{align*}
Mathematica [C] time = 1.01618, size = 216, normalized size = 0.69 \[ -\frac{2 \sqrt{\cot (c+d x)} \left (-b \left (b^2-3 a^2\right ) \cot ^{\frac{3}{2}}(c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )-\frac{3 a \left (a^2-3 b^2\right ) \left (\log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-2 \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )}{4 \sqrt{2}}+9 a b^2 \sqrt{\cot (c+d x)}+b^3 \cot ^{\frac{3}{2}}(c+d x)\right )}{3 d \sqrt{e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.031, size = 725, normalized size = 2.3 \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 )^{3}}{\sqrt{e \cot{\left (c + d x \right )}}}\, 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 )}^{3}}{\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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