Optimal. Leaf size=36 \[ \frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\sin (x)}{b} \]
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Rubi [A] time = 0.06, antiderivative size = 36, 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, 388, 205} \[ \frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\sin (x)}{b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cos ^3(x)}{a+b \sin ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1-x^2}{a+b x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {\sin (x)}{b}+\frac {(a+b) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{b}\\ &=\frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\sin (x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 1.00 \[ \frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\sin (x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 101, normalized size = 2.81 \[ \left [-\frac {2 \, a b \sin \relax (x) + \sqrt {-a b} {\left (a + b\right )} \log \left (-\frac {b \cos \relax (x)^{2} + 2 \, \sqrt {-a b} \sin \relax (x) + a - b}{b \cos \relax (x)^{2} - a - b}\right )}{2 \, a b^{2}}, -\frac {a b \sin \relax (x) - \sqrt {a b} {\left (a + b\right )} \arctan \left (\frac {\sqrt {a b} \sin \relax (x)}{a}\right )}{a b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 30, normalized size = 0.83 \[ \frac {{\left (a + b\right )} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {\sin \relax (x)}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 45, normalized size = 1.25 \[ -\frac {\sin \relax (x )}{b}+\frac {\arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right ) a}{b \sqrt {a b}}+\frac {\arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 30, normalized size = 0.83 \[ \frac {{\left (a + b\right )} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {\sin \relax (x)}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 28, normalized size = 0.78 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \relax (x)}{\sqrt {a}}\right )\,\left (a+b\right )}{\sqrt {a}\,b^{3/2}}-\frac {\sin \relax (x)}{b} \]
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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