Integrand size = 18, antiderivative size = 109 \[ \int \frac {\cos ^2(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {b \left (-2 a^2+b^2\right ) \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {\left (a^2-b^2\right ) \cos (x)}{\left (a^2+b^2\right )^2}+\frac {2 a b \sin (x)}{\left (a^2+b^2\right )^2}+\frac {a b^2}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))} \]
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Time = 0.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.39, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3190, 3179, 2717, 3153, 212, 3188, 2718, 3234} \[ \int \frac {\cos ^2(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {2 a^2 b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {b^3 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac {2 a b \sin (x)}{\left (a^2+b^2\right )^2}+\frac {b^2 \cos (x)}{\left (a^2+b^2\right )^2}-\frac {a^2 \cos (x)}{\left (a^2+b^2\right )^2}+\frac {a b^2}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))} \]
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Rule 212
Rule 2717
Rule 2718
Rule 3153
Rule 3179
Rule 3188
Rule 3190
Rule 3234
Rubi steps \begin{align*} \text {integral}& = \frac {a \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}+\frac {b \int \frac {\cos ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}-\frac {(a b) \int \frac {\cos (x)}{(a \cos (x)+b \sin (x))^2} \, dx}{a^2+b^2} \\ & = \frac {b^2 \cos (x)}{\left (a^2+b^2\right )^2}+\frac {a b^2}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))}+\frac {a^2 \int \sin (x) \, dx}{\left (a^2+b^2\right )^2}+2 \frac {(a b) \int \cos (x) \, dx}{\left (a^2+b^2\right )^2}-2 \frac {\left (a^2 b\right ) \int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {b^3 \int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2} \\ & = -\frac {a^2 \cos (x)}{\left (a^2+b^2\right )^2}+\frac {b^2 \cos (x)}{\left (a^2+b^2\right )^2}+\frac {2 a b \sin (x)}{\left (a^2+b^2\right )^2}+\frac {a b^2}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))}+2 \frac {\left (a^2 b\right ) \text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{\left (a^2+b^2\right )^2}-\frac {b^3 \text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{\left (a^2+b^2\right )^2} \\ & = \frac {2 a^2 b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {b^3 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {a^2 \cos (x)}{\left (a^2+b^2\right )^2}+\frac {b^2 \cos (x)}{\left (a^2+b^2\right )^2}+\frac {2 a b \sin (x)}{\left (a^2+b^2\right )^2}+\frac {a b^2}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))} \\ \end{align*}
Time = 0.81 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^2(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {2 b \left (-2 a^2+b^2\right ) \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {a^3-5 a b^2+a \left (a^2+b^2\right ) \cos (2 x)-b \left (a^2+b^2\right ) \sin (2 x)}{2 \left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))} \]
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Time = 0.64 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.31
method | result | size |
default | \(\frac {4 b \left (\frac {-\frac {b^{2} \tan \left (\frac {x}{2}\right )}{2}-\frac {a b}{2}}{\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a}-\frac {\left (2 a^{2}-b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {4 \tan \left (\frac {x}{2}\right ) a b -2 a^{2}+2 b^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}\) | \(143\) |
risch | \(-\frac {{\mathrm e}^{i x}}{2 \left (-2 i b a +a^{2}-b^{2}\right )}-\frac {{\mathrm e}^{-i x}}{2 \left (2 i b a +a^{2}-b^{2}\right )}+\frac {2 i a \,b^{2} {\mathrm e}^{i x}}{\left (-i a +b \right )^{2} \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i x}+i a \,{\mathrm e}^{2 i x}-b +i a \right )}+\frac {2 i b \ln \left ({\mathrm e}^{i x}+\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right ) a^{2}}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )^{2}}-\frac {i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )^{2}}-\frac {2 i b \ln \left ({\mathrm e}^{i x}-\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right ) a^{2}}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )^{2}}+\frac {i b^{3} \ln \left ({\mathrm e}^{i x}-\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )^{2}}\) | \(326\) |
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Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (106) = 212\).
Time = 0.27 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.31 \[ \int \frac {\cos ^2(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {6 \, a^{3} b^{2} + 6 \, a b^{4} - 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right ) \sin \left (x\right ) - \sqrt {a^{2} + b^{2}} {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \cos \left (x\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sin \left (x\right )\right )} \log \left (-\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right )}{2 \, {\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right ) + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \left (x\right )\right )}} \]
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Timed out. \[ \int \frac {\cos ^2(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (106) = 212\).
Time = 0.32 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.42 \[ \int \frac {\cos ^2(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {{\left (2 \, a^{2} b - b^{3}\right )} \log \left (\frac {b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (a^{3} - 2 \, a b^{2} - \frac {3 \, b^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {{\left (a^{3} + 4 \, a b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {{\left (2 \, a^{2} b - b^{3}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )}}{a^{5} + 2 \, a^{3} b^{2} + a b^{4} + \frac {2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}} \]
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Time = 0.31 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.87 \[ \int \frac {\cos ^2(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {{\left (2 \, a^{2} b - b^{3}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (2 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )^{3} - b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} - a^{3} \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 3 \, b^{3} \tan \left (\frac {1}{2} \, x\right ) + a^{3} - 2 \, a b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, b \tan \left (\frac {1}{2} \, x\right )^{3} - 2 \, b \tan \left (\frac {1}{2} \, x\right ) - a\right )} {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]
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Time = 24.45 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.32 \[ \int \frac {\cos ^2(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {\frac {2\,\left (2\,a\,b^2-a^3\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {6\,b^3\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (a^3+4\,a\,b^2\right )}{a^4+2\,a^2\,b^2+b^4}-\frac {2\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (2\,a^2-b^2\right )}{a^4+2\,a^2\,b^2+b^4}}{-a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+2\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a}+\frac {b\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^5-a^4\,b\,1{}\mathrm {i}+2{}\mathrm {i}\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^3\,b^2-a^2\,b^3\,2{}\mathrm {i}+1{}\mathrm {i}\,\mathrm {tan}\left (\frac {x}{2}\right )\,a\,b^4-b^5\,1{}\mathrm {i}}{{\left (a^2+b^2\right )}^{5/2}}\right )\,\left (2\,a^2-b^2\right )\,2{}\mathrm {i}}{{\left (a^2+b^2\right )}^{5/2}} \]
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