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 = 0.04, 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 = {3269, 396, 211}
\begin {gather*} \frac {(a+b) \text {ArcTan}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\sin (x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 396
Rule 3269
Rubi steps
\begin {align*} \int \frac {\cos ^3(x)}{a+b \sin ^2(x)} \, dx &=\text {Subst}\left (\int \frac {1-x^2}{a+b x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {\sin (x)}{b}+\frac {(a+b) \text {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 \begin {gather*} \frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\sin (x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 31, normalized size = 0.86
method | result | size |
default | \(-\frac {\sin \left (x \right )}{b}+\frac {\left (a +b \right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(31\) |
risch | \(\frac {i {\mathrm e}^{i x}}{2 b}-\frac {i {\mathrm e}^{-i x}}{2 b}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right ) a}{2 \sqrt {-a b}\, b}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right ) a}{2 \sqrt {-a b}\, b}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}}\) | \(156\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 30, normalized size = 0.83 \begin {gather*} \frac {{\left (a + b\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {\sin \left (x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 101, normalized size = 2.81 \begin {gather*} \left [-\frac {2 \, a b \sin \left (x\right ) + \sqrt {-a b} {\left (a + b\right )} \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 \, a b^{2}}, -\frac {a b \sin \left (x\right ) - \sqrt {a b} {\left (a + b\right )} \arctan \left (\frac {\sqrt {a b} \sin \left (x\right )}{a}\right )}{a b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 30, normalized size = 0.83 \begin {gather*} \frac {{\left (a + b\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {\sin \left (x\right )}{b} \end {gather*}
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 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \left (x\right )}{\sqrt {a}}\right )\,\left (a+b\right )}{\sqrt {a}\,b^{3/2}}-\frac {\sin \left (x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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