Optimal. Leaf size=72 \[ \frac{a x \left (a^2+3 b^2\right )}{2 \left (a^2+b^2\right )^2}-\frac{\sin ^2(x) (a \cot (x)+b)}{2 \left (a^2+b^2\right )}-\frac{b^3 \log (a \sin (x)+b \cos (x))}{\left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.129704, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {3506, 741, 801, 635, 203, 260} \[ \frac{a x \left (a^2+3 b^2\right )}{2 \left (a^2+b^2\right )^2}-\frac{\sin ^2(x) (a \cot (x)+b)}{2 \left (a^2+b^2\right )}-\frac{b^3 \log (a \sin (x)+b \cos (x))}{\left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 741
Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{\sin ^2(x)}{a+b \cot (x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x) \left (1+\frac{x^2}{b^2}\right )^2} \, dx,x,b \cot (x)\right )}{b}\\ &=-\frac{(b+a \cot (x)) \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac{b \operatorname{Subst}\left (\int \frac{-2-\frac{a^2}{b^2}-\frac{a x}{b^2}}{(a+x) \left (1+\frac{x^2}{b^2}\right )} \, dx,x,b \cot (x)\right )}{2 \left (a^2+b^2\right )}\\ &=-\frac{(b+a \cot (x)) \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac{b \operatorname{Subst}\left (\int \left (-\frac{2 b^2}{\left (a^2+b^2\right ) (a+x)}+\frac{-a^3-3 a b^2+2 b^2 x}{\left (a^2+b^2\right ) \left (b^2+x^2\right )}\right ) \, dx,x,b \cot (x)\right )}{2 \left (a^2+b^2\right )}\\ &=-\frac{b^3 \log (a+b \cot (x))}{\left (a^2+b^2\right )^2}-\frac{(b+a \cot (x)) \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac{b \operatorname{Subst}\left (\int \frac{-a^3-3 a b^2+2 b^2 x}{b^2+x^2} \, dx,x,b \cot (x)\right )}{2 \left (a^2+b^2\right )^2}\\ &=-\frac{b^3 \log (a+b \cot (x))}{\left (a^2+b^2\right )^2}-\frac{(b+a \cot (x)) \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac{b^3 \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \cot (x)\right )}{\left (a^2+b^2\right )^2}-\frac{\left (a b \left (a^2+3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \cot (x)\right )}{2 \left (a^2+b^2\right )^2}\\ &=\frac{a \left (a^2+3 b^2\right ) x}{2 \left (a^2+b^2\right )^2}-\frac{b^3 \log (a+b \cot (x))}{\left (a^2+b^2\right )^2}-\frac{b^3 \log (\sin (x))}{\left (a^2+b^2\right )^2}-\frac{(b+a \cot (x)) \sin ^2(x)}{2 \left (a^2+b^2\right )}\\ \end{align*}
Mathematica [C] time = 0.172764, size = 94, normalized size = 1.31 \[ \frac{b \left (a^2+b^2\right ) \cos (2 x)+2 a^3 x-a^3 \sin (2 x)+6 a b^2 x-a b^2 \sin (2 x)-2 b^3 \log \left ((a \sin (x)+b \cos (x))^2\right )-4 i b^3 x+4 i b^3 \tan ^{-1}(\tan (x))}{4 \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.12, size = 173, normalized size = 2.4 \begin{align*} -{\frac{\tan \left ( x \right ){a}^{3}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) }}-{\frac{a\tan \left ( x \right ){b}^{2}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) }}+{\frac{b{a}^{2}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) }}+{\frac{{b}^{3}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) }}+{\frac{{b}^{3}\ln \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) }{2\, \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{3\,\arctan \left ( \tan \left ( x \right ) \right ) a{b}^{2}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{\arctan \left ( \tan \left ( x \right ) \right ){a}^{3}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{{b}^{3}\ln \left ( a\tan \left ( x \right ) +b \right ) }{ \left ({a}^{2}+{b}^{2} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.83733, size = 162, normalized size = 2.25 \begin{align*} -\frac{b^{3} \log \left (a \tan \left (x\right ) + b\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{b^{3} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac{{\left (a^{3} + 3 \, a b^{2}\right )} x}{2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac{a \tan \left (x\right ) - b}{2 \,{\left ({\left (a^{2} + b^{2}\right )} \tan \left (x\right )^{2} + a^{2} + b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77594, size = 223, normalized size = 3.1 \begin{align*} -\frac{b^{3} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}\right ) -{\left (a^{2} b + b^{3}\right )} \cos \left (x\right )^{2} +{\left (a^{3} + a b^{2}\right )} \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{3} + 3 \, a b^{2}\right )} x}{2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \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{\sin ^{2}{\left (x \right )}}{a + b \cot{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.32006, size = 200, normalized size = 2.78 \begin{align*} -\frac{a b^{3} \log \left ({\left | a \tan \left (x\right ) + b \right |}\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4}} + \frac{b^{3} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac{{\left (a^{3} + 3 \, a b^{2}\right )} x}{2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac{b^{3} \tan \left (x\right )^{2} + a^{3} \tan \left (x\right ) + a b^{2} \tan \left (x\right ) - a^{2} b}{2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}{\left (\tan \left (x\right )^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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