Optimal. Leaf size=394 \[ -\frac{b^2 \sqrt{e \cot (c+d x)}}{a d e \left (a^2+b^2\right ) (a+b \cot (c+d x))}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \sqrt{e} \left (a^2+b^2\right )^2}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \sqrt{e} \left (a^2+b^2\right )^2}-\frac{b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{a^{3/2} d \sqrt{e} \left (a^2+b^2\right )^2}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d \sqrt{e} \left (a^2+b^2\right )^2}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d \sqrt{e} \left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.738044, antiderivative size = 394, 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 = {3569, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac{b^2 \sqrt{e \cot (c+d x)}}{a d e \left (a^2+b^2\right ) (a+b \cot (c+d x))}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \sqrt{e} \left (a^2+b^2\right )^2}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \sqrt{e} \left (a^2+b^2\right )^2}-\frac{b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{a^{3/2} d \sqrt{e} \left (a^2+b^2\right )^2}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d \sqrt{e} \left (a^2+b^2\right )^2}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d \sqrt{e} \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3569
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{1}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2} \, dx &=-\frac{b^2 \sqrt{e \cot (c+d x)}}{a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))}-\frac{\int \frac{-\frac{1}{2} \left (2 a^2+b^2\right ) e+a b e \cot (c+d x)-\frac{1}{2} b^2 e \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{a \left (a^2+b^2\right ) e}\\ &=-\frac{b^2 \sqrt{e \cot (c+d x)}}{a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))}+\frac{\left (b^2 \left (5 a^2+b^2\right )\right ) \int \frac{1+\cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{2 a \left (a^2+b^2\right )^2}-\frac{\int \frac{-a \left (a^2-b^2\right ) e+2 a^2 b e \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{a \left (a^2+b^2\right )^2 e}\\ &=-\frac{b^2 \sqrt{e \cot (c+d x)}}{a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))}+\frac{\left (b^2 \left (5 a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{2 a \left (a^2+b^2\right )^2 d}-\frac{2 \operatorname{Subst}\left (\int \frac{a \left (a^2-b^2\right ) e^2-2 a^2 b e x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d e}\\ &=-\frac{b^2 \sqrt{e \cot (c+d x)}}{a \left (a^2+b^2\right ) d e (a+b \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 )}{\left (a^2+b^2\right )^2 d}-\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 )}{\left (a^2+b^2\right )^2 d}-\frac{\left (b^2 \left (5 a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+\frac{b x^2}{e}} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d e}\\ &=-\frac{b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d \sqrt{e}}-\frac{b^2 \sqrt{e \cot (c+d x)}}{a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))}-\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 \left (a^2+b^2\right )^2 d}-\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 \left (a^2+b^2\right )^2 d}+\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} \left (a^2+b^2\right )^2 d \sqrt{e}}+\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} \left (a^2+b^2\right )^2 d \sqrt{e}}\\ &=-\frac{b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d \sqrt{e}}-\frac{b^2 \sqrt{e \cot (c+d x)}}{a \left (a^2+b^2\right ) d e (a+b \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} \left (a^2+b^2\right )^2 d \sqrt{e}}-\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} \left (a^2+b^2\right )^2 d \sqrt{e}}-\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} \left (a^2+b^2\right )^2 d \sqrt{e}}+\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} \left (a^2+b^2\right )^2 d \sqrt{e}}\\ &=-\frac{b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d \sqrt{e}}+\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} \left (a^2+b^2\right )^2 d \sqrt{e}}-\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} \left (a^2+b^2\right )^2 d \sqrt{e}}-\frac{b^2 \sqrt{e \cot (c+d x)}}{a \left (a^2+b^2\right ) d e (a+b \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} \left (a^2+b^2\right )^2 d \sqrt{e}}-\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} \left (a^2+b^2\right )^2 d \sqrt{e}}\\ \end{align*}
Mathematica [C] time = 2.75739, size = 300, normalized size = 0.76 \[ -\frac{\sqrt{\cot (c+d x)} \left (-32 a b \cot ^{\frac{3}{2}}(c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )+\frac{24 b^2 \left (a^2+b^2\right ) \sqrt{\cot (c+d x)} \left (\frac{a}{a+b \cot (c+d x)}+\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{\cot (c+d x)}}\right )}{a^2}+96 \sqrt{a} b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right )-6 \sqrt{2} (a-b) (a+b) \left (\log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-2 \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )\right )}{24 d \left (a^2+b^2\right )^2 \sqrt{e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 765, 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 \frac{1}{\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 \frac{1}{{\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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