Optimal. Leaf size=325 \[ \frac{(a-b) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2} \left (a^2+b^2\right )}-\frac{(a-b) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2} \left (a^2+b^2\right )}+\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{a^{3/2} d e^{3/2} \left (a^2+b^2\right )}-\frac{(a+b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{3/2} \left (a^2+b^2\right )}+\frac{(a+b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d e^{3/2} \left (a^2+b^2\right )}+\frac{2}{a d e \sqrt{e \cot (c+d x)}} \]
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Rubi [A] time = 0.659698, antiderivative size = 325, 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} \[ \frac{(a-b) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2} \left (a^2+b^2\right )}-\frac{(a-b) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2} \left (a^2+b^2\right )}+\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{a^{3/2} d e^{3/2} \left (a^2+b^2\right )}-\frac{(a+b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{3/2} \left (a^2+b^2\right )}+\frac{(a+b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d e^{3/2} \left (a^2+b^2\right )}+\frac{2}{a d e \sqrt{e \cot (c+d x)}} \]
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}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx &=\frac{2}{a d e \sqrt{e \cot (c+d x)}}+\frac{2 \int \frac{-\frac{b e^2}{2}-\frac{1}{2} a e^2 \cot (c+d x)-\frac{1}{2} b e^2 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{a e^3}\\ &=\frac{2}{a d e \sqrt{e \cot (c+d x)}}+\frac{2 \int \frac{-\frac{1}{2} a b e^2-\frac{1}{2} a^2 e^2 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{a \left (a^2+b^2\right ) e^3}-\frac{b^3 \int \frac{1+\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{2}{a d e \sqrt{e \cot (c+d x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{\frac{1}{2} a b e^3+\frac{1}{2} a^2 e^2 x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{a \left (a^2+b^2\right ) d e^3}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{a \left (a^2+b^2\right ) d e}\\ &=\frac{2}{a d e \sqrt{e \cot (c+d x)}}+\frac{\left (2 b^3\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 ) d e^2}-\frac{(a-b) \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 ) d e}+\frac{(a+b) \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 ) d e}\\ &=\frac{2 b^{5/2} \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 ) d e^{3/2}}+\frac{2}{a d e \sqrt{e \cot (c+d x)}}+\frac{(a-b) \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 ) d e^{3/2}}+\frac{(a-b) \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 ) d e^{3/2}}+\frac{(a+b) \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 ) d e}+\frac{(a+b) \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 ) d e}\\ &=\frac{2 b^{5/2} \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 ) d e^{3/2}}+\frac{2}{a d e \sqrt{e \cot (c+d x)}}+\frac{(a-b) \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 ) d e^{3/2}}-\frac{(a-b) \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 ) d e^{3/2}}+\frac{(a+b) \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 ) d e^{3/2}}-\frac{(a+b) \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 ) d e^{3/2}}\\ &=\frac{2 b^{5/2} \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 ) d e^{3/2}}-\frac{(a+b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2+b^2\right ) d e^{3/2}}+\frac{(a+b) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2+b^2\right ) d e^{3/2}}+\frac{2}{a d e \sqrt{e \cot (c+d x)}}+\frac{(a-b) \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 ) d e^{3/2}}-\frac{(a-b) \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 ) d e^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.416671, size = 198, normalized size = 0.61 \[ \frac{8 b^2 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\frac{b \cot (c+d x)}{a}\right )+a \left (8 a \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},-\cot ^2(c+d x)\right )+\sqrt{2} b \sqrt{\cot (c+d x)} \left (-\log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )+2 \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )\right )}{4 a d e \left (a^2+b^2\right ) \sqrt{e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 459, normalized size = 1.4 \begin{align*}{\frac{b\sqrt{2}}{4\,d{e}^{2} \left ({a}^{2}+{b}^{2} \right ) }\sqrt [4]{{e}^{2}}\ln \left ({ \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ) }+{\frac{b\sqrt{2}}{2\,d{e}^{2} \left ({a}^{2}+{b}^{2} \right ) }\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{b\sqrt{2}}{2\,d{e}^{2} \left ({a}^{2}+{b}^{2} \right ) }\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{a\sqrt{2}}{4\,de \left ({a}^{2}+{b}^{2} \right ) }\ln \left ({ \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}+{\frac{a\sqrt{2}}{2\,de \left ({a}^{2}+{b}^{2} \right ) }\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}-{\frac{a\sqrt{2}}{2\,de \left ({a}^{2}+{b}^{2} \right ) }\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}+2\,{\frac{{b}^{3}}{dea \left ({a}^{2}+{b}^{2} \right ) \sqrt{aeb}}\arctan \left ({\frac{\sqrt{e\cot \left ( dx+c \right ) }b}{\sqrt{aeb}}} \right ) }+2\,{\frac{1}{dea\sqrt{e\cot \left ( dx+c \right ) }}} \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}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}} \left (a + b \cot{\left (c + d x \right )}\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{1}{{\left (b \cot \left (d x + c\right ) + a\right )} \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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