Optimal. Leaf size=54 \[ \frac {(a+b)^2 \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}-\frac {(a+2 b) \sin (x)}{b^2}+\frac {\sin ^3(x)}{3 b} \]
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Rubi [A] time = 0.07, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3190, 390, 205} \[ -\frac {(a+2 b) \sin (x)}{b^2}+\frac {(a+b)^2 \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}+\frac {\sin ^3(x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 390
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cos ^5(x)}{a+b \sin ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a+b x^2} \, dx,x,\sin (x)\right )\\ &=\operatorname {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 \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{b^2}\\ &=\frac {(a+b)^2 \tan ^{-1}\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*}
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Mathematica [A] time = 0.17, size = 84, normalized size = 1.56 \[ \frac {6 (a+b)^2 \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )-2 \sqrt {a} \sqrt {b} \sin (x) (6 a+b \cos (2 x)+11 b)-6 (a+b)^2 \tan ^{-1}\left (\frac {\sqrt {a} \csc (x)}{\sqrt {b}}\right )}{12 \sqrt {a} b^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 159, normalized size = 2.94 \[ \left [-\frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-a b} \log \left (-\frac {b \cos \relax (x)^{2} + 2 \, \sqrt {-a b} \sin \relax (x) + a - b}{b \cos \relax (x)^{2} - a - b}\right ) + 2 \, {\left (a b^{2} \cos \relax (x)^{2} + 3 \, a^{2} b + 5 \, a b^{2}\right )} \sin \relax (x)}{6 \, a b^{3}}, \frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} \sin \relax (x)}{a}\right ) - {\left (a b^{2} \cos \relax (x)^{2} + 3 \, a^{2} b + 5 \, a b^{2}\right )} \sin \relax (x)}{3 \, a b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 58, normalized size = 1.07 \[ \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b^{2} \sin \relax (x)^{3} - 3 \, a b \sin \relax (x) - 6 \, b^{2} \sin \relax (x)}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 85, normalized size = 1.57 \[ \frac {\sin ^{3}\relax (x )}{3 b}-\frac {a \sin \relax (x )}{b^{2}}-\frac {2 \sin \relax (x )}{b}+\frac {\arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right ) a^{2}}{b^{2} \sqrt {a b}}+\frac {2 \arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right ) a}{b \sqrt {a b}}+\frac {\arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 52, normalized size = 0.96 \[ \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b \sin \relax (x)^{3} - 3 \, {\left (a + 2 \, b\right )} \sin \relax (x)}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.30, size = 65, normalized size = 1.20 \[ \frac {{\sin \relax (x)}^3}{3\,b}-\sin \relax (x)\,\left (\frac {a}{b^2}+\frac {2}{b}\right )+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \relax (x)\,{\left (a+b\right )}^2}{\sqrt {a}\,\left (a^2+2\,a\,b+b^2\right )}\right )\,{\left (a+b\right )}^2}{\sqrt {a}\,b^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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