Integrand size = 15, antiderivative size = 59 \[ \int \frac {\cos ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=-\frac {(a-b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}+\frac {(a+b) \sin (x)}{2 a b \left (a+b \sin ^2(x)\right )} \]
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Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3269, 393, 211} \[ \int \frac {\cos ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {(a+b) \sin (x)}{2 a b \left (a+b \sin ^2(x)\right )}-\frac {(a-b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}} \]
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Rule 211
Rule 393
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1-x^2}{\left (a+b x^2\right )^2} \, dx,x,\sin (x)\right ) \\ & = \frac {(a+b) \sin (x)}{2 a b \left (a+b \sin ^2(x)\right )}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{2 a b} \\ & = -\frac {(a-b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}+\frac {(a+b) \sin (x)}{2 a b \left (a+b \sin ^2(x)\right )} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=-\frac {(a-b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}+\frac {(a+b) \sin (x)}{2 a b \left (a+b \sin ^2(x)\right )} \]
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Time = 0.59 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {\left (a +b \right ) \sin \left (x \right )}{2 a b \left (a +b \left (\sin ^{2}\left (x \right )\right )\right )}-\frac {\left (a -b \right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{2 a b \sqrt {a b}}\) | \(53\) |
default | \(\frac {\left (a +b \right ) \sin \left (x \right )}{2 a b \left (a +b \left (\sin ^{2}\left (x \right )\right )\right )}-\frac {\left (a -b \right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{2 a b \sqrt {a b}}\) | \(53\) |
risch | \(\frac {i \left (a +b \right ) \left ({\mathrm e}^{3 i x}-{\mathrm e}^{i x}\right )}{a b \left (b \,{\mathrm e}^{4 i x}-4 a \,{\mathrm e}^{2 i x}-2 b \,{\mathrm e}^{2 i x}+b \right )}-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, b}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, a}+\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, b}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, a}\) | \(190\) |
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Time = 0.31 (sec) , antiderivative size = 206, normalized size of antiderivative = 3.49 \[ \int \frac {\cos ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\left [\frac {{\left ({\left (a b - b^{2}\right )} \cos \left (x\right )^{2} - a^{2} + b^{2}\right )} \sqrt {-a b} \log \left (-\frac {b \cos \left (x\right )^{2} + 2 \, \sqrt {-a b} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right ) - 2 \, {\left (a^{2} b + a b^{2}\right )} \sin \left (x\right )}{4 \, {\left (a^{2} b^{3} \cos \left (x\right )^{2} - a^{3} b^{2} - a^{2} b^{3}\right )}}, -\frac {{\left ({\left (a b - b^{2}\right )} \cos \left (x\right )^{2} - a^{2} + b^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} \sin \left (x\right )}{a}\right ) + {\left (a^{2} b + a b^{2}\right )} \sin \left (x\right )}{2 \, {\left (a^{2} b^{3} \cos \left (x\right )^{2} - a^{3} b^{2} - a^{2} b^{3}\right )}}\right ] \]
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Timed out. \[ \int \frac {\cos ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\text {Timed out} \]
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Time = 0.43 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {{\left (a + b\right )} \sin \left (x\right )}{2 \, {\left (a b^{2} \sin \left (x\right )^{2} + a^{2} b\right )}} - \frac {{\left (a - b\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} \]
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Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=-\frac {{\left (a - b\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} + \frac {a \sin \left (x\right ) + b \sin \left (x\right )}{2 \, {\left (b \sin \left (x\right )^{2} + a\right )} a b} \]
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Time = 14.58 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {\sin \left (x\right )\,\left (a+b\right )}{2\,a\,b\,\left (b\,{\sin \left (x\right )}^2+a\right )}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \left (x\right )}{\sqrt {a}}\right )\,\left (a-b\right )}{2\,a^{3/2}\,b^{3/2}} \]
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