Optimal. Leaf size=313 \[ -\frac{2 b \left (a^2+b^2\right ) \sqrt{e \cot (c+d x)}}{d e^2}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2}}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2}}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{3/2}}+\frac{(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d e^{3/2}}+\frac{2 a^2 (a+b \cot (c+d x))}{d e \sqrt{e \cot (c+d x)}} \]
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Rubi [A] time = 0.419893, 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 = {3565, 3630, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{2 b \left (a^2+b^2\right ) \sqrt{e \cot (c+d x)}}{d e^2}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2}}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2}}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{3/2}}+\frac{(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d e^{3/2}}+\frac{2 a^2 (a+b \cot (c+d x))}{d e \sqrt{e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3565
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}{(e \cot (c+d x))^{3/2}} \, dx &=\frac{2 a^2 (a+b \cot (c+d x))}{d e \sqrt{e \cot (c+d x)}}-\frac{2 \int \frac{-2 a^2 b e^2+\frac{1}{2} a \left (a^2-3 b^2\right ) e^2 \cot (c+d x)-\frac{1}{2} b \left (a^2+b^2\right ) e^2 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{e^3}\\ &=-\frac{2 b \left (a^2+b^2\right ) \sqrt{e \cot (c+d x)}}{d e^2}+\frac{2 a^2 (a+b \cot (c+d x))}{d e \sqrt{e \cot (c+d x)}}-\frac{2 \int \frac{-\frac{1}{2} b \left (3 a^2-b^2\right ) e^2+\frac{1}{2} a \left (a^2-3 b^2\right ) e^2 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{e^3}\\ &=-\frac{2 b \left (a^2+b^2\right ) \sqrt{e \cot (c+d x)}}{d e^2}+\frac{2 a^2 (a+b \cot (c+d x))}{d e \sqrt{e \cot (c+d x)}}-\frac{4 \operatorname{Subst}\left (\int \frac{\frac{1}{2} b \left (3 a^2-b^2\right ) e^3-\frac{1}{2} a \left (a^2-3 b^2\right ) e^2 x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d e^3}\\ &=-\frac{2 b \left (a^2+b^2\right ) \sqrt{e \cot (c+d x)}}{d e^2}+\frac{2 a^2 (a+b \cot (c+d x))}{d e \sqrt{e \cot (c+d x)}}+\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 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 e}\\ &=-\frac{2 b \left (a^2+b^2\right ) \sqrt{e \cot (c+d x)}}{d e^2}+\frac{2 a^2 (a+b \cot (c+d x))}{d e \sqrt{e \cot (c+d x)}}+\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 e^{3/2}}+\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 e^{3/2}}+\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 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 e}\\ &=-\frac{2 b \left (a^2+b^2\right ) \sqrt{e \cot (c+d x)}}{d e^2}+\frac{2 a^2 (a+b \cot (c+d x))}{d e \sqrt{e \cot (c+d x)}}+\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 e^{3/2}}-\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 e^{3/2}}+\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 e^{3/2}}-\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 e^{3/2}}\\ &=-\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 e^{3/2}}+\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 e^{3/2}}-\frac{2 b \left (a^2+b^2\right ) \sqrt{e \cot (c+d x)}}{d e^2}+\frac{2 a^2 (a+b \cot (c+d x))}{d e \sqrt{e \cot (c+d x)}}+\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 e^{3/2}}-\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 e^{3/2}}\\ \end{align*}
Mathematica [C] time = 3.31059, size = 193, normalized size = 0.62 \[ \frac{2 \left (a \left (a^2-3 b^2\right ) \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},-\cot ^2(c+d x)\right )-\frac{b \left (b^2-3 a^2\right ) \sqrt{\cot (c+d x)} \left (\log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-2 \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )}{4 \sqrt{2}}+3 a b^2+b^3 (-\cot (c+d x))\right )}{d e \sqrt{e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 742, normalized size = 2.4 \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}}{\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 )}^{3}}{\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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