Integrand size = 15, antiderivative size = 54 \[ \int \frac {\cos ^5(x)}{a+b \sin ^2(x)} \, dx=\frac {(a+b)^2 \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}-\frac {(a+2 b) \sin (x)}{b^2}+\frac {\sin ^3(x)}{3 b} \]
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Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3269, 398, 211} \[ \int \frac {\cos ^5(x)}{a+b \sin ^2(x)} \, dx=\frac {(a+b)^2 \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}-\frac {(a+2 b) \sin (x)}{b^2}+\frac {\sin ^3(x)}{3 b} \]
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Rule 211
Rule 398
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a+b x^2} \, dx,x,\sin (x)\right ) \\ & = \text {Subst}\left (\int \left (-\frac {a+2 b}{b^2}+\frac {x^2}{b}+\frac {a^2+2 a b+b^2}{b^2 \left (a+b x^2\right )}\right ) \, dx,x,\sin (x)\right ) \\ & = -\frac {(a+2 b) \sin (x)}{b^2}+\frac {\sin ^3(x)}{3 b}+\frac {(a+b)^2 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{b^2} \\ & = \frac {(a+b)^2 \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}-\frac {(a+2 b) \sin (x)}{b^2}+\frac {\sin ^3(x)}{3 b} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.56 \[ \int \frac {\cos ^5(x)}{a+b \sin ^2(x)} \, dx=\frac {-6 (a+b)^2 \arctan \left (\frac {\sqrt {a} \csc (x)}{\sqrt {b}}\right )+6 (a+b)^2 \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )-2 \sqrt {a} \sqrt {b} (6 a+11 b+b \cos (2 x)) \sin (x)}{12 \sqrt {a} b^{5/2}} \]
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Time = 0.80 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(-\frac {-\frac {b \left (\sin ^{3}\left (x \right )\right )}{3}+\sin \left (x \right ) a +2 \sin \left (x \right ) b}{b^{2}}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(54\) |
default | \(-\frac {-\frac {b \left (\sin ^{3}\left (x \right )\right )}{3}+\sin \left (x \right ) a +2 \sin \left (x \right ) b}{b^{2}}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(54\) |
risch | \(\frac {i {\mathrm e}^{i x} a}{2 b^{2}}+\frac {7 i {\mathrm e}^{i x}}{8 b}-\frac {i {\mathrm e}^{-i x} a}{2 b^{2}}-\frac {7 i {\mathrm e}^{-i x}}{8 b}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right ) a^{2}}{2 \sqrt {-a b}\, b^{2}}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right ) a}{\sqrt {-a b}\, b}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right ) a^{2}}{2 \sqrt {-a b}\, b^{2}}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right ) a}{\sqrt {-a b}\, b}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}}-\frac {\sin \left (3 x \right )}{12 b}\) | \(262\) |
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Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.94 \[ \int \frac {\cos ^5(x)}{a+b \sin ^2(x)} \, dx=\left [-\frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-a b} \log \left (-\frac {b \cos \left (x\right )^{2} + 2 \, \sqrt {-a b} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right ) + 2 \, {\left (a b^{2} \cos \left (x\right )^{2} + 3 \, a^{2} b + 5 \, a b^{2}\right )} \sin \left (x\right )}{6 \, a b^{3}}, \frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} \sin \left (x\right )}{a}\right ) - {\left (a b^{2} \cos \left (x\right )^{2} + 3 \, a^{2} b + 5 \, a b^{2}\right )} \sin \left (x\right )}{3 \, a b^{3}}\right ] \]
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Timed out. \[ \int \frac {\cos ^5(x)}{a+b \sin ^2(x)} \, dx=\text {Timed out} \]
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Time = 0.37 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^5(x)}{a+b \sin ^2(x)} \, dx=\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b \sin \left (x\right )^{3} - 3 \, {\left (a + 2 \, b\right )} \sin \left (x\right )}{3 \, b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^5(x)}{a+b \sin ^2(x)} \, dx=\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b^{2} \sin \left (x\right )^{3} - 3 \, a b \sin \left (x\right ) - 6 \, b^{2} \sin \left (x\right )}{3 \, b^{3}} \]
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Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.20 \[ \int \frac {\cos ^5(x)}{a+b \sin ^2(x)} \, dx=\frac {{\sin \left (x\right )}^3}{3\,b}-\sin \left (x\right )\,\left (\frac {a}{b^2}+\frac {2}{b}\right )+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \left (x\right )\,{\left (a+b\right )}^2}{\sqrt {a}\,\left (a^2+2\,a\,b+b^2\right )}\right )\,{\left (a+b\right )}^2}{\sqrt {a}\,b^{5/2}} \]
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