Optimal. Leaf size=59 \[ \frac {(a+b) \sin (x)}{2 a b \left (a+b \sin ^2(x)\right )}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3190, 385, 205} \[ \frac {(a+b) \sin (x)}{2 a b \left (a+b \sin ^2(x)\right )}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 385
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cos ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx &=\operatorname {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) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{2 a b}\\ &=-\frac {(a-b) \tan ^{-1}\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*}
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Mathematica [A] time = 0.07, size = 59, normalized size = 1.00 \[ \frac {(a+b) \sin (x)}{2 a b \left (a+b \sin ^2(x)\right )}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 206, normalized size = 3.49 \[ \left [\frac {{\left ({\left (a b - b^{2}\right )} \cos \relax (x)^{2} - a^{2} + b^{2}\right )} \sqrt {-a b} \log \left (-\frac {b \cos \relax (x)^{2} + 2 \, \sqrt {-a b} \sin \relax (x) + a - b}{b \cos \relax (x)^{2} - a - b}\right ) - 2 \, {\left (a^{2} b + a b^{2}\right )} \sin \relax (x)}{4 \, {\left (a^{2} b^{3} \cos \relax (x)^{2} - a^{3} b^{2} - a^{2} b^{3}\right )}}, -\frac {{\left ({\left (a b - b^{2}\right )} \cos \relax (x)^{2} - a^{2} + b^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} \sin \relax (x)}{a}\right ) + {\left (a^{2} b + a b^{2}\right )} \sin \relax (x)}{2 \, {\left (a^{2} b^{3} \cos \relax (x)^{2} - a^{3} b^{2} - a^{2} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 56, normalized size = 0.95 \[ -\frac {{\left (a - b\right )} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} + \frac {a \sin \relax (x) + b \sin \relax (x)}{2 \, {\left (b \sin \relax (x)^{2} + a\right )} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 65, normalized size = 1.10 \[ \frac {\left (a +b \right ) \sin \relax (x )}{2 a b \left (a +b \left (\sin ^{2}\relax (x )\right )\right )}-\frac {\arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right )}{2 b \sqrt {a b}}+\frac {\arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right )}{2 a \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.83, size = 53, normalized size = 0.90 \[ \frac {{\left (a + b\right )} \sin \relax (x)}{2 \, {\left (a b^{2} \sin \relax (x)^{2} + a^{2} b\right )}} - \frac {{\left (a - b\right )} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 47, normalized size = 0.80 \[ \frac {\sin \relax (x)\,\left (a+b\right )}{2\,a\,b\,\left (b\,{\sin \relax (x)}^2+a\right )}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \relax (x)}{\sqrt {a}}\right )\,\left (a-b\right )}{2\,a^{3/2}\,b^{3/2}} \]
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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