Integrand size = 15, antiderivative size = 72 \[ \int \frac {\cos ^5(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=-\frac {(3 a-b) (a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}}+\frac {\sin (x)}{b^2}+\frac {(a+b)^2 \sin (x)}{2 a b^2 \left (a+b \sin ^2(x)\right )} \]
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Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3269, 398, 393, 211} \[ \int \frac {\cos ^5(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=-\frac {(3 a-b) (a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}}+\frac {(a+b)^2 \sin (x)}{2 a b^2 \left (a+b \sin ^2(x)\right )}+\frac {\sin (x)}{b^2} \]
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Rule 211
Rule 393
Rule 398
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a+b x^2\right )^2} \, dx,x,\sin (x)\right ) \\ & = \text {Subst}\left (\int \left (\frac {1}{b^2}-\frac {a^2-b^2+2 b (a+b) x^2}{b^2 \left (a+b x^2\right )^2}\right ) \, dx,x,\sin (x)\right ) \\ & = \frac {\sin (x)}{b^2}-\frac {\text {Subst}\left (\int \frac {a^2-b^2+2 b (a+b) x^2}{\left (a+b x^2\right )^2} \, dx,x,\sin (x)\right )}{b^2} \\ & = \frac {\sin (x)}{b^2}+\frac {(a+b)^2 \sin (x)}{2 a b^2 \left (a+b \sin ^2(x)\right )}-\frac {((3 a-b) (a+b)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{2 a b^2} \\ & = -\frac {(3 a-b) (a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}}+\frac {\sin (x)}{b^2}+\frac {(a+b)^2 \sin (x)}{2 a b^2 \left (a+b \sin ^2(x)\right )} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.64 \[ \int \frac {\cos ^5(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {\frac {\left (3 a^2+2 a b-b^2\right ) \arctan \left (\frac {\sqrt {a} \csc (x)}{\sqrt {b}}\right )}{a^{3/2}}+\frac {\left (-3 a^2-2 a b+b^2\right ) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{a^{3/2}}+4 \sqrt {b} \sin (x)+\frac {4 \sqrt {b} (a+b)^2 \sin (x)}{a (2 a+b-b \cos (2 x))}}{4 b^{5/2}} \]
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Time = 0.98 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(\frac {\sin \left (x \right )}{b^{2}}-\frac {-\frac {\left (a^{2}+2 a b +b^{2}\right ) \sin \left (x \right )}{2 a \left (a +b \left (\sin ^{2}\left (x \right )\right )\right )}+\frac {\left (3 a^{2}+2 a b -b^{2}\right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{b^{2}}\) | \(77\) |
default | \(\frac {\sin \left (x \right )}{b^{2}}-\frac {-\frac {\left (a^{2}+2 a b +b^{2}\right ) \sin \left (x \right )}{2 a \left (a +b \left (\sin ^{2}\left (x \right )\right )\right )}+\frac {\left (3 a^{2}+2 a b -b^{2}\right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{b^{2}}\) | \(77\) |
risch | \(-\frac {i {\mathrm e}^{i x}}{2 b^{2}}+\frac {i {\mathrm e}^{-i x}}{2 b^{2}}-\frac {i \left (a^{2}+2 a b +b^{2}\right ) \left ({\mathrm e}^{3 i x}-{\mathrm e}^{i x}\right )}{a \,b^{2} \left (-b \,{\mathrm e}^{4 i x}+4 a \,{\mathrm e}^{2 i x}+2 b \,{\mathrm e}^{2 i x}-b \right )}-\frac {3 a \ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, b^{2}}-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, b}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, a}+\frac {3 a \ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, b^{2}}+\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, b}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, a}\) | \(293\) |
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Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (60) = 120\).
Time = 0.32 (sec) , antiderivative size = 296, normalized size of antiderivative = 4.11 \[ \int \frac {\cos ^5(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\left [-\frac {{\left (3 \, a^{3} + 5 \, a^{2} b + a b^{2} - b^{3} - {\left (3 \, a^{2} b + 2 \, a b^{2} - b^{3}\right )} \cos \left (x\right )^{2}\right )} \sqrt {-a b} \log \left (-\frac {b \cos \left (x\right )^{2} + 2 \, \sqrt {-a b} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right ) - 2 \, {\left (2 \, a^{2} b^{2} \cos \left (x\right )^{2} - 3 \, a^{3} b - 4 \, a^{2} b^{2} - a b^{3}\right )} \sin \left (x\right )}{4 \, {\left (a^{2} b^{4} \cos \left (x\right )^{2} - a^{3} b^{3} - a^{2} b^{4}\right )}}, \frac {{\left (3 \, a^{3} + 5 \, a^{2} b + a b^{2} - b^{3} - {\left (3 \, a^{2} b + 2 \, a b^{2} - b^{3}\right )} \cos \left (x\right )^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} \sin \left (x\right )}{a}\right ) + {\left (2 \, a^{2} b^{2} \cos \left (x\right )^{2} - 3 \, a^{3} b - 4 \, a^{2} b^{2} - a b^{3}\right )} \sin \left (x\right )}{2 \, {\left (a^{2} b^{4} \cos \left (x\right )^{2} - a^{3} b^{3} - a^{2} b^{4}\right )}}\right ] \]
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Timed out. \[ \int \frac {\cos ^5(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\text {Timed out} \]
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Time = 0.40 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.10 \[ \int \frac {\cos ^5(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \sin \left (x\right )}{2 \, {\left (a b^{3} \sin \left (x\right )^{2} + a^{2} b^{2}\right )}} + \frac {\sin \left (x\right )}{b^{2}} - \frac {{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.14 \[ \int \frac {\cos ^5(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {\sin \left (x\right )}{b^{2}} - \frac {{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{2}} + \frac {a^{2} \sin \left (x\right ) + 2 \, a b \sin \left (x\right ) + b^{2} \sin \left (x\right )}{2 \, {\left (b \sin \left (x\right )^{2} + a\right )} a b^{2}} \]
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Time = 14.74 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^5(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {\sin \left (x\right )}{b^2}+\frac {\sin \left (x\right )\,\left (a^2+2\,a\,b+b^2\right )}{2\,a\,\left (b^3\,{\sin \left (x\right )}^2+a\,b^2\right )}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \left (x\right )\,\left (a+b\right )\,\left (3\,a-b\right )}{\sqrt {a}\,\left (3\,a^2+2\,a\,b-b^2\right )}\right )\,\left (a+b\right )\,\left (3\,a-b\right )}{2\,a^{3/2}\,b^{5/2}} \]
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