Optimal. Leaf size=36 \[ \frac {\cos (x)}{b}-\frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
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Rubi [A] time = 0.05, 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} \[ \frac {\cos (x)}{b}-\frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 3190
Rubi steps
\begin {align*} \int \frac {\sin ^3(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1-x^2}{a+b x^2} \, dx,x,\cos (x)\right )\\ &=\frac {\cos (x)}{b}-\frac {(a+b) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\cos (x)\right )}{b}\\ &=-\frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {\cos (x)}{b}\\ \end {align*}
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Mathematica [B] time = 0.18, size = 90, normalized size = 2.50 \[ \frac {\sqrt {a} \sqrt {b} \cos (x)-\left ((a+b) \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {a+b} \tan \left (\frac {x}{2}\right )}{\sqrt {a}}\right )\right )-(a+b) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan \left (\frac {x}{2}\right )+\sqrt {b}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 95, normalized size = 2.64 \[ \left [\frac {2 \, a b \cos \relax (x) - \sqrt {-a b} {\left (a + b\right )} \log \left (-\frac {b \cos \relax (x)^{2} + 2 \, \sqrt {-a b} \cos \relax (x) - a}{b \cos \relax (x)^{2} + a}\right )}{2 \, a b^{2}}, \frac {a b \cos \relax (x) - \sqrt {a b} {\left (a + b\right )} \arctan \left (\frac {\sqrt {a b} \cos \relax (x)}{a}\right )}{a b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 30, normalized size = 0.83 \[ -\frac {{\left (a + b\right )} \arctan \left (\frac {b \cos \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {\cos \relax (x)}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 46, normalized size = 1.28 \[ \frac {\cos \relax (x )}{b}-\frac {\arctan \left (\frac {\cos \relax (x ) b}{\sqrt {a b}}\right ) a}{b \sqrt {a b}}-\frac {\arctan \left (\frac {\cos \relax (x ) b}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.34, size = 30, normalized size = 0.83 \[ -\frac {{\left (a + b\right )} \arctan \left (\frac {b \cos \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {\cos \relax (x)}{b} \]
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 \[ \frac {\cos \relax (x)}{b}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\cos \relax (x)}{\sqrt {a}}\right )\,\left (a+b\right )}{\sqrt {a}\,b^{3/2}} \]
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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