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 = 0.13, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 203
Rule 260
Rule 635
Rule 741
Rule 801
Rule 3506
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*}
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Mathematica [C] time = 0.19, size = 94, normalized size = 1.31 \[ \frac {2 a^3 x-a^3 \sin (2 x)+b \left (a^2+b^2\right ) \cos (2 x)-2 b^3 \log \left ((a \sin (x)+b \cos (x))^2\right )+6 a b^2 x-a b^2 \sin (2 x)-4 i b^3 x+4 i b^3 \tan ^{-1}(\tan (x))}{4 \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 94, normalized size = 1.31 \[ -\frac {b^{3} \log \left (2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + a^{2}\right ) - {\left (a^{2} b + b^{3}\right )} \cos \relax (x)^{2} + {\left (a^{3} + a b^{2}\right )} \cos \relax (x) \sin \relax (x) - {\left (a^{3} + 3 \, a b^{2}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.80, size = 148, normalized size = 2.06 \[ -\frac {a b^{3} \log \left ({\left | a \tan \relax (x) + b \right |}\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4}} + \frac {b^{3} \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {{\left (a^{3} + 3 \, a b^{2}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {b^{3} \tan \relax (x)^{2} + a^{3} \tan \relax (x) + a b^{2} \tan \relax (x) - a^{2} b}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 173, normalized size = 2.40 \[ -\frac {b^{3} \ln \left (a \tan \relax (x )+b \right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {\tan \relax (x ) a^{3}}{2 \left (a^{2}+b^{2}\right )^{2} \left (1+\tan ^{2}\relax (x )\right )}-\frac {\tan \relax (x ) b^{2} a}{2 \left (a^{2}+b^{2}\right )^{2} \left (1+\tan ^{2}\relax (x )\right )}+\frac {a^{2} b}{2 \left (a^{2}+b^{2}\right )^{2} \left (1+\tan ^{2}\relax (x )\right )}+\frac {b^{3}}{2 \left (a^{2}+b^{2}\right )^{2} \left (1+\tan ^{2}\relax (x )\right )}+\frac {b^{3} \ln \left (1+\tan ^{2}\relax (x )\right )}{2 \left (a^{2}+b^{2}\right )^{2}}+\frac {3 \arctan \left (\tan \relax (x )\right ) b^{2} a}{2 \left (a^{2}+b^{2}\right )^{2}}+\frac {\arctan \left (\tan \relax (x )\right ) a^{3}}{2 \left (a^{2}+b^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 120, normalized size = 1.67 \[ -\frac {b^{3} \log \left (a \tan \relax (x) + b\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {b^{3} \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {{\left (a^{3} + 3 \, a b^{2}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {a \tan \relax (x) - b}{2 \, {\left ({\left (a^{2} + b^{2}\right )} \tan \relax (x)^{2} + a^{2} + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 126, normalized size = 1.75 \[ {\cos \relax (x)}^2\,\left (\frac {b}{2\,\left (a^2+b^2\right )}-\frac {a\,\mathrm {tan}\relax (x)}{2\,\left (a^2+b^2\right )}\right )-\frac {b^3\,\ln \left (b+a\,\mathrm {tan}\relax (x)\right )}{{\left (a^2+b^2\right )}^2}+\frac {\ln \left (\mathrm {tan}\relax (x)-\mathrm {i}\right )\,\left (2\,b+a\,1{}\mathrm {i}\right )}{4\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )}+\frac {\ln \left (\mathrm {tan}\relax (x)+1{}\mathrm {i}\right )\,\left (a+b\,2{}\mathrm {i}\right )}{4\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{2}{\relax (x )}}{a + b \cot {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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