Optimal. Leaf size=387 \[ -\frac{e^{3/2} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}+\frac{e^{3/2} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\sqrt{a} e^{3/2} \left (a^2-3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{b} d \left (a^2+b^2\right )^2}-\frac{e^{3/2} \left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}+\frac{e^{3/2} \left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{a e \sqrt{e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))} \]
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Rubi [A] time = 0.676897, antiderivative size = 387, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {3567, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac{e^{3/2} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}+\frac{e^{3/2} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\sqrt{a} e^{3/2} \left (a^2-3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{b} d \left (a^2+b^2\right )^2}-\frac{e^{3/2} \left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}+\frac{e^{3/2} \left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{a e \sqrt{e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))} \]
Antiderivative was successfully verified.
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Rule 3567
Rule 3653
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 3634
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{(e \cot (c+d x))^{3/2}}{(a+b \cot (c+d x))^2} \, dx &=-\frac{a e \sqrt{e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac{\int \frac{\frac{a e^2}{2}-b e^2 \cot (c+d x)-\frac{1}{2} a e^2 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{a^2+b^2}\\ &=-\frac{a e \sqrt{e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac{\int \frac{\left (a^2-b^2\right ) e^2-2 a b e^2 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}+\frac{\left (a \left (a^2-3 b^2\right ) e^2\right ) \int \frac{1+\cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{2 \left (a^2+b^2\right )^2}\\ &=-\frac{a e \sqrt{e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac{2 \operatorname{Subst}\left (\int \frac{-\left (a^2-b^2\right ) e^3+2 a b e^2 x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac{\left (a \left (a^2-3 b^2\right ) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=-\frac{a e \sqrt{e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac{\left (a \left (a^2-3 b^2\right ) e\right ) \operatorname{Subst}\left (\int \frac{1}{a+\frac{b x^2}{e}} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac{\left (\left (a^2-2 a b-b^2\right ) e^2\right ) \operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac{\left (\left (a^2+2 a b-b^2\right ) e^2\right ) \operatorname{Subst}\left (\int \frac{e-x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac{\sqrt{a} \left (a^2-3 b^2\right ) e^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{b} \left (a^2+b^2\right )^2 d}-\frac{a e \sqrt{e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac{\left (\left (a^2+2 a b-b^2\right ) e^{3/2}\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} \left (a^2+b^2\right )^2 d}-\frac{\left (\left (a^2+2 a b-b^2\right ) e^{3/2}\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} \left (a^2+b^2\right )^2 d}+\frac{\left (\left (a^2-2 a b-b^2\right ) e^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 \left (a^2+b^2\right )^2 d}+\frac{\left (\left (a^2-2 a b-b^2\right ) e^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 \left (a^2+b^2\right )^2 d}\\ &=-\frac{\sqrt{a} \left (a^2-3 b^2\right ) e^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{b} \left (a^2+b^2\right )^2 d}-\frac{a e \sqrt{e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac{\left (a^2+2 a b-b^2\right ) e^{3/2} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}+\frac{\left (a^2+2 a b-b^2\right ) e^{3/2} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}+\frac{\left (\left (a^2-2 a b-b^2\right ) e^{3/2}\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} \left (a^2+b^2\right )^2 d}-\frac{\left (\left (a^2-2 a b-b^2\right ) e^{3/2}\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} \left (a^2+b^2\right )^2 d}\\ &=-\frac{\sqrt{a} \left (a^2-3 b^2\right ) e^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{b} \left (a^2+b^2\right )^2 d}-\frac{\left (a^2-2 a b-b^2\right ) e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}+\frac{\left (a^2-2 a b-b^2\right ) e^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{a e \sqrt{e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac{\left (a^2+2 a b-b^2\right ) e^{3/2} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}+\frac{\left (a^2+2 a b-b^2\right ) e^{3/2} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}\\ \end{align*}
Mathematica [C] time = 3.18089, size = 322, normalized size = 0.83 \[ -\frac{(e \cot (c+d x))^{3/2} \left (\frac{24 b^2 \left (a^2+b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \text{Hypergeometric2F1}\left (2,\frac{5}{2},\frac{7}{2},-\frac{b \cot (c+d x)}{a}\right )}{a^2}+80 a b \cot ^{\frac{3}{2}}(c+d x) \left (\text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )-1\right )-240 a^2 \left (\sqrt{\cot (c+d x)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right )}{\sqrt{b}}\right )+80 a b \cot ^{\frac{3}{2}}(c+d x)+15 (a-b) (a+b) \left (8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\sqrt{2} \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )\right )}{60 d \left (a^2+b^2\right )^2 \cot ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 768, 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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \cot \left (d x + c\right )\right )^{\frac{3}{2}}}{{\left (b \cot \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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