Integrand size = 15, antiderivative size = 36 \[ \int \frac {\cos ^3(x)}{a+b \sin ^2(x)} \, dx=\frac {(a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\sin (x)}{b} \]
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Time = 0.06 (sec) , antiderivative size = 36, 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, 396, 211} \[ \int \frac {\cos ^3(x)}{a+b \sin ^2(x)} \, dx=\frac {(a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\sin (x)}{b} \]
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Rule 211
Rule 396
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1-x^2}{a+b x^2} \, dx,x,\sin (x)\right ) \\ & = -\frac {\sin (x)}{b}+\frac {(a+b) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{b} \\ & = \frac {(a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\sin (x)}{b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^3(x)}{a+b \sin ^2(x)} \, dx=\frac {(a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\sin (x)}{b} \]
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Time = 0.64 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\frac {\sin \left (x \right )}{b}+\frac {\left (a +b \right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(31\) |
default | \(-\frac {\sin \left (x \right )}{b}+\frac {\left (a +b \right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(31\) |
risch | \(\frac {i {\mathrm e}^{i x}}{2 b}-\frac {i {\mathrm e}^{-i x}}{2 b}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right ) a}{2 \sqrt {-a b}\, b}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right ) a}{2 \sqrt {-a b}\, b}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}}\) | \(156\) |
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Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.81 \[ \int \frac {\cos ^3(x)}{a+b \sin ^2(x)} \, dx=\left [-\frac {2 \, a b \sin \left (x\right ) + \sqrt {-a b} {\left (a + b\right )} \log \left (-\frac {b \cos \left (x\right )^{2} + 2 \, \sqrt {-a b} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right )}{2 \, a b^{2}}, -\frac {a b \sin \left (x\right ) - \sqrt {a b} {\left (a + b\right )} \arctan \left (\frac {\sqrt {a b} \sin \left (x\right )}{a}\right )}{a b^{2}}\right ] \]
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Timed out. \[ \int \frac {\cos ^3(x)}{a+b \sin ^2(x)} \, dx=\text {Timed out} \]
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Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^3(x)}{a+b \sin ^2(x)} \, dx=\frac {{\left (a + b\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {\sin \left (x\right )}{b} \]
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Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^3(x)}{a+b \sin ^2(x)} \, dx=\frac {{\left (a + b\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {\sin \left (x\right )}{b} \]
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Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^3(x)}{a+b \sin ^2(x)} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \left (x\right )}{\sqrt {a}}\right )\,\left (a+b\right )}{\sqrt {a}\,b^{3/2}}-\frac {\sin \left (x\right )}{b} \]
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