Optimal. Leaf size=73 \[ \frac {a x \left (a^2-b^2\right )}{2 \left (a^2+b^2\right )^2}+\frac {\sin ^2(x) (a \cot (x)+b)}{2 \left (a^2+b^2\right )}-\frac {a^2 b \log (a \sin (x)+b \cos (x))}{\left (a^2+b^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 73, 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 = {3516, 1647, 801, 635, 203, 260} \[ \frac {a x \left (a^2-b^2\right )}{2 \left (a^2+b^2\right )^2}+\frac {\sin ^2(x) (a \cot (x)+b)}{2 \left (a^2+b^2\right )}-\frac {a^2 b \log (a \sin (x)+b \cos (x))}{\left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 260
Rule 635
Rule 801
Rule 1647
Rule 3516
Rubi steps
\begin {align*} \int \frac {\cos ^2(x)}{a+b \cot (x)} \, dx &=-\left (b \operatorname {Subst}\left (\int \frac {x^2}{(a+x) \left (b^2+x^2\right )^2} \, dx,x,b \cot (x)\right )\right )\\ &=\frac {(b+a \cot (x)) \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {a^2 b^2}{a^2+b^2}+\frac {a b^2 x}{a^2+b^2}}{(a+x) \left (b^2+x^2\right )} \, dx,x,b \cot (x)\right )}{2 b}\\ &=\frac {(b+a \cot (x)) \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {\operatorname {Subst}\left (\int \left (-\frac {2 a^2 b^2}{\left (a^2+b^2\right )^2 (a+x)}-\frac {a b^2 \left (a^2-b^2-2 a x\right )}{\left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \cot (x)\right )}{2 b}\\ &=-\frac {a^2 b \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 {(a b) \operatorname {Subst}\left (\int \frac {a^2-b^2-2 a x}{b^2+x^2} \, dx,x,b \cot (x)\right )}{2 \left (a^2+b^2\right )^2}\\ &=-\frac {a^2 b \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 {\left (a^2 b\right ) \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-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-b^2\right ) x}{2 \left (a^2+b^2\right )^2}-\frac {a^2 b \log (a+b \cot (x))}{\left (a^2+b^2\right )^2}-\frac {a^2 b \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 = 0.32, size = 82, normalized size = 1.12 \[ \frac {-b \left (a^2+b^2\right ) \cos (2 x)+a \left (\left (a^2+b^2\right ) \sin (2 x)+2 x (a-i b)^2-2 a b \log \left ((a \sin (x)+b \cos (x))^2\right )\right )+4 i a^2 b \tan ^{-1}(\tan (x))}{4 \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.64, size = 95, normalized size = 1.30 \[ -\frac {a^{2} b \log \left (2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + a^{2}\right ) + {\left (a^{2} b + b^{3}\right )} \cos \relax (x)^{2} - {\left (a^{3} + a b^{2}\right )} \cos \relax (x) \sin \relax (x) - {\left (a^{3} - a b^{2}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.38, size = 155, normalized size = 2.12 \[ -\frac {a^{3} b \log \left ({\left | a \tan \relax (x) + b \right |}\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4}} + \frac {a^{2} b \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {{\left (a^{3} - a b^{2}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {a^{2} b \tan \relax (x)^{2} - a^{3} \tan \relax (x) - a b^{2} \tan \relax (x) + 2 \, a^{2} b + b^{3}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.27, size = 175, normalized size = 2.40 \[ -\frac {a^{2} b \ln \left (a \tan \relax (x )+b \right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\tan \relax (x ) a^{3}}{2 \left (a^{2}+b^{2}\right )^{2} \left (1+\tan ^{2}\relax (x )\right )}+\frac {\tan \relax (x ) b^{2} a}{2 \left (a^{2}+b^{2}\right )^{2} \left (1+\tan ^{2}\relax (x )\right )}-\frac {a^{2} b}{2 \left (a^{2}+b^{2}\right )^{2} \left (1+\tan ^{2}\relax (x )\right )}-\frac {b^{3}}{2 \left (a^{2}+b^{2}\right )^{2} \left (1+\tan ^{2}\relax (x )\right )}+\frac {\arctan \left (\tan \relax (x )\right ) a^{3}}{2 \left (a^{2}+b^{2}\right )^{2}}-\frac {\arctan \left (\tan \relax (x )\right ) b^{2} a}{2 \left (a^{2}+b^{2}\right )^{2}}+\frac {\ln \left (1+\tan ^{2}\relax (x )\right ) a^{2} b}{2 \left (a^{2}+b^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.72, size = 122, normalized size = 1.67 \[ -\frac {a^{2} b \log \left (a \tan \relax (x) + b\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {a^{2} b \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {{\left (a^{3} - a b^{2}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {a \tan \relax (x) - b}{2 \, {\left ({\left (a^{2} + b^{2}\right )} \tan \relax (x)^{2} + a^{2} + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.37, size = 118, normalized size = 1.62 \[ -{\cos \relax (x)}^2\,\left (\frac {b}{2\,\left (a^2+b^2\right )}-\frac {a\,\mathrm {tan}\relax (x)}{2\,\left (a^2+b^2\right )}\right )+\frac {a\,\ln \left (\mathrm {tan}\relax (x)+1{}\mathrm {i}\right )}{4\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {a^2\,b\,\ln \left (b+a\,\mathrm {tan}\relax (x)\right )}{{\left (a^2+b^2\right )}^2}+\frac {a\,\ln \left (\mathrm {tan}\relax (x)-\mathrm {i}\right )\,1{}\mathrm {i}}{4\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}{\relax (x )}}{a + b \cot {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________