Integrand size = 16, antiderivative size = 92 \[ \int \frac {\sin ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {\left (a^2-2 b^2\right ) \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac {a \left (3 a b \cos (x)+\left (a^2+4 b^2\right ) \sin (x)\right )}{2 \left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))^2} \]
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Leaf count is larger than twice the leaf count of optimal. \(300\) vs. \(2(92)=184\).
Time = 0.83 (sec) , antiderivative size = 300, normalized size of antiderivative = 3.26, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4486, 1674, 12, 632, 212, 3234, 3153} \[ \int \frac {\sin ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {a^2 \left (2 a^2-b^2\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right )^{5/2}}+\frac {2 a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right )^{3/2}}-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}+\frac {2 \left (\left (a^2+2 b^2\right ) \tan \left (\frac {x}{2}\right )+a b\right )}{a \left (a^2+b^2\right ) \left (-a \tan ^2\left (\frac {x}{2}\right )+a+2 b \tan \left (\frac {x}{2}\right )\right )^2}+\frac {2 a}{b \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}-\frac {4 a^4+a b \left (5 a^2+2 b^2\right ) \tan \left (\frac {x}{2}\right )+3 a^2 b^2+2 b^4}{a b \left (a^2+b^2\right )^2 \left (-a \tan ^2\left (\frac {x}{2}\right )+a+2 b \tan \left (\frac {x}{2}\right )\right )} \]
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Rule 12
Rule 212
Rule 632
Rule 1674
Rule 3153
Rule 3234
Rule 4486
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 \cos ^2(x)}{b^2 (a \cos (x)+b \sin (x))^3}-\frac {2 a \cos (x)}{b^2 (a \cos (x)+b \sin (x))^2}+\frac {1}{b^2 (a \cos (x)+b \sin (x))}\right ) \, dx \\ & = \frac {\int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{b^2}-\frac {(2 a) \int \frac {\cos (x)}{(a \cos (x)+b \sin (x))^2} \, dx}{b^2}+\frac {a^2 \int \frac {\cos ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx}{b^2} \\ & = \frac {2 a}{b \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}-\frac {\text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{b^2}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a+2 b x-a x^2\right )^3} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^2}-\frac {\left (2 a^2\right ) \int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{b^2 \left (a^2+b^2\right )} \\ & = -\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}+\frac {2 a}{b \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}+\frac {2 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac {x}{2}\right )\right )}{a \left (a^2+b^2\right ) \left (a+2 b \tan \left (\frac {x}{2}\right )-a \tan ^2\left (\frac {x}{2}\right )\right )^2}-\frac {a^2 \text {Subst}\left (\int \frac {-\frac {8 \left (a^4+2 b^4\right )}{a^3}+16 b \left (1+\frac {b^2}{a^2}\right ) x+8 \left (a+\frac {b^2}{a}\right ) x^2}{\left (a+2 b x-a x^2\right )^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{4 b^2 \left (a^2+b^2\right )}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{b^2 \left (a^2+b^2\right )} \\ & = \frac {2 a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right )^{3/2}}-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}+\frac {2 a}{b \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}+\frac {2 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac {x}{2}\right )\right )}{a \left (a^2+b^2\right ) \left (a+2 b \tan \left (\frac {x}{2}\right )-a \tan ^2\left (\frac {x}{2}\right )\right )^2}-\frac {4 a^4+3 a^2 b^2+2 b^4+a b \left (5 a^2+2 b^2\right ) \tan \left (\frac {x}{2}\right )}{a b \left (a^2+b^2\right )^2 \left (a+2 b \tan \left (\frac {x}{2}\right )-a \tan ^2\left (\frac {x}{2}\right )\right )}+\frac {a^2 \text {Subst}\left (\int \frac {16 \left (2 a^2-b^2\right )}{a+2 b x-a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{16 b^2 \left (a^2+b^2\right )^2} \\ & = \frac {2 a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right )^{3/2}}-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}+\frac {2 a}{b \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}+\frac {2 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac {x}{2}\right )\right )}{a \left (a^2+b^2\right ) \left (a+2 b \tan \left (\frac {x}{2}\right )-a \tan ^2\left (\frac {x}{2}\right )\right )^2}-\frac {4 a^4+3 a^2 b^2+2 b^4+a b \left (5 a^2+2 b^2\right ) \tan \left (\frac {x}{2}\right )}{a b \left (a^2+b^2\right )^2 \left (a+2 b \tan \left (\frac {x}{2}\right )-a \tan ^2\left (\frac {x}{2}\right )\right )}+\frac {\left (a^2 \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^2 \left (a^2+b^2\right )^2} \\ & = \frac {2 a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right )^{3/2}}-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}+\frac {2 a}{b \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}+\frac {2 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac {x}{2}\right )\right )}{a \left (a^2+b^2\right ) \left (a+2 b \tan \left (\frac {x}{2}\right )-a \tan ^2\left (\frac {x}{2}\right )\right )^2}-\frac {4 a^4+3 a^2 b^2+2 b^4+a b \left (5 a^2+2 b^2\right ) \tan \left (\frac {x}{2}\right )}{a b \left (a^2+b^2\right )^2 \left (a+2 b \tan \left (\frac {x}{2}\right )-a \tan ^2\left (\frac {x}{2}\right )\right )}-\frac {\left (2 a^2 \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tan \left (\frac {x}{2}\right )\right )}{b^2 \left (a^2+b^2\right )^2} \\ & = \frac {2 a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right )^{3/2}}-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}-\frac {a^2 \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 )}{b^2 \left (a^2+b^2\right )^{5/2}}+\frac {2 a}{b \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}+\frac {2 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac {x}{2}\right )\right )}{a \left (a^2+b^2\right ) \left (a+2 b \tan \left (\frac {x}{2}\right )-a \tan ^2\left (\frac {x}{2}\right )\right )^2}-\frac {4 a^4+3 a^2 b^2+2 b^4+a b \left (5 a^2+2 b^2\right ) \tan \left (\frac {x}{2}\right )}{a b \left (a^2+b^2\right )^2 \left (a+2 b \tan \left (\frac {x}{2}\right )-a \tan ^2\left (\frac {x}{2}\right )\right )} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {\left (a^2-2 b^2\right ) \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac {a \left (3 a b \cos (x)+\left (a^2+4 b^2\right ) \sin (x)\right )}{2 \left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(211\) vs. \(2(84)=168\).
Time = 0.60 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.30
method | result | size |
default | \(-\frac {8 \left (-\frac {a \left (a^{2}-2 b^{2}\right ) \tan \left (\frac {x}{2}\right )^{3}}{8 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 b \left (a^{2}-2 b^{2}\right ) \tan \left (\frac {x}{2}\right )^{2}}{8 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (a^{2}+10 b^{2}\right ) a \tan \left (\frac {x}{2}\right )}{8 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {3 a^{2} b}{8 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\right )}{\left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )^{2}}-\frac {\left (a^{2}-2 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}+b^{2}}}\) | \(212\) |
risch | \(-\frac {i a \,{\mathrm e}^{i x} \left (3 i a b \,{\mathrm e}^{2 i x}+a^{2} {\mathrm e}^{2 i x}+4 b^{2} {\mathrm e}^{2 i x}+3 i b a -a^{2}-4 b^{2}\right )}{\left (-i b \,{\mathrm e}^{2 i x}+a \,{\mathrm e}^{2 i x}+i b +a \right )^{2} \left (i b +a \right )^{2} \left (-i b +a \right )^{2}}-\frac {\ln \left ({\mathrm e}^{i x}+\frac {i a^{5}+2 i a^{3} b^{2}+i a \,b^{4}-a^{4} b -2 a^{2} b^{3}-b^{5}}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) a^{2}}{2 \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left ({\mathrm e}^{i x}+\frac {i a^{5}+2 i a^{3} b^{2}+i a \,b^{4}-a^{4} b -2 a^{2} b^{3}-b^{5}}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) b^{2}}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left ({\mathrm e}^{i x}-\frac {i a^{5}+2 i a^{3} b^{2}+i a \,b^{4}-a^{4} b -2 a^{2} b^{3}-b^{5}}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) a^{2}}{2 \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left ({\mathrm e}^{i x}-\frac {i a^{5}+2 i a^{3} b^{2}+i a \,b^{4}-a^{4} b -2 a^{2} b^{3}-b^{5}}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) b^{2}}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\) | \(399\) |
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Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (84) = 168\).
Time = 0.25 (sec) , antiderivative size = 282, normalized size of antiderivative = 3.07 \[ \int \frac {\sin ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {{\left (a^{2} b^{2} - 2 \, b^{4} + {\left (a^{4} - 3 \, a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{3} b - 2 \, a b^{3}\right )} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) - 6 \, {\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (x\right ) - 2 \, {\left (a^{5} + 5 \, a^{3} b^{2} + 4 \, a b^{4}\right )} \sin \left (x\right )}{4 \, {\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8} + {\left (a^{8} + 2 \, a^{6} b^{2} - 2 \, a^{2} b^{6} - b^{8}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (x\right ) \sin \left (x\right )\right )}} \]
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Timed out. \[ \int \frac {\sin ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (84) = 168\).
Time = 0.31 (sec) , antiderivative size = 299, normalized size of antiderivative = 3.25 \[ \int \frac {\sin ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left (\frac {b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {3 \, a^{2} b + \frac {{\left (a^{3} + 10 \, a b^{2}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {3 \, {\left (a^{2} b - 2 \, b^{3}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {{\left (a^{3} - 2 \, a b^{2}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} + \frac {4 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {2 \, {\left (a^{6} - 3 \, a^{2} b^{4} - 2 \, b^{6}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {4 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (84) = 168\).
Time = 0.31 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.14 \[ \int \frac {\sin ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {a^{3} \tan \left (\frac {1}{2} \, x\right )^{3} - 2 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} - 3 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )^{2} + 6 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{2} + a^{3} \tan \left (\frac {1}{2} \, x\right ) + 10 \, a b^{2} \tan \left (\frac {1}{2} \, x\right ) + 3 \, a^{2} b}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, x\right ) - a\right )}^{2}} \]
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Time = 21.14 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.86 \[ \int \frac {\sin ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {\frac {3\,a^2\,b}{a^4+2\,a^2\,b^2+b^4}+\frac {a\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2+10\,b^2\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (a^2-2\,b^2\right )}{a^4+2\,a^2\,b^2+b^4}-\frac {3\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (a^2-2\,b^2\right )}{a^4+2\,a^2\,b^2+b^4}}{a^2-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (2\,a^2-4\,b^2\right )+a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+4\,a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )-4\,a\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}+\frac {\mathrm {atanh}\left (\frac {2\,a^4\,b+4\,a^2\,b^3+2\,b^5}{2\,{\left (a^2+b^2\right )}^{5/2}}-\frac {a\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}{{\left (a^2+b^2\right )}^{5/2}}\right )\,\left (a^2-2\,b^2\right )}{{\left (a^2+b^2\right )}^{5/2}} \]
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