Optimal. Leaf size=59 \[ -\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}+\frac {(a+b) \sin (x)}{2 a b \left (a+b \sin ^2(x)\right )} \]
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Rubi [A]
time = 0.04, 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 = {3269, 393, 211}
\begin {gather*} \frac {(a+b) \sin (x)}{2 a b \left (a+b \sin ^2(x)\right )}-\frac {(a-b) \text {ArcTan}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 393
Rule 3269
Rubi steps
\begin {align*} \int \frac {\cos ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx &=\text {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) \text {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 = 0.05, size = 59, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 53, normalized size = 0.90
method | result | size |
default | \(\frac {\left (a +b \right ) \sin \left (x \right )}{2 a b \left (a +b \left (\sin ^{2}\left (x \right )\right )\right )}-\frac {\left (a -b \right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{2 a b \sqrt {a b}}\) | \(53\) |
risch | \(\frac {i \left (a +b \right ) \left ({\mathrm e}^{3 i x}-{\mathrm e}^{i x}\right )}{a b \left (b \,{\mathrm e}^{4 i x}-4 a \,{\mathrm e}^{2 i x}-2 b \,{\mathrm e}^{2 i x}+b \right )}-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, b}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, a}+\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, b}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, a}\) | \(190\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 53, normalized size = 0.90 \begin {gather*} \frac {{\left (a + b\right )} \sin \left (x\right )}{2 \, {\left (a b^{2} \sin \left (x\right )^{2} + a^{2} b\right )}} - \frac {{\left (a - b\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 206, normalized size = 3.49 \begin {gather*} \left [\frac {{\left ({\left (a b - b^{2}\right )} \cos \left (x\right )^{2} - a^{2} + b^{2}\right )} \sqrt {-a b} \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 \, {\left (a^{2} b + a b^{2}\right )} \sin \left (x\right )}{4 \, {\left (a^{2} b^{3} \cos \left (x\right )^{2} - a^{3} b^{2} - a^{2} b^{3}\right )}}, -\frac {{\left ({\left (a b - b^{2}\right )} \cos \left (x\right )^{2} - a^{2} + b^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} \sin \left (x\right )}{a}\right ) + {\left (a^{2} b + a b^{2}\right )} \sin \left (x\right )}{2 \, {\left (a^{2} b^{3} \cos \left (x\right )^{2} - a^{3} b^{2} - a^{2} b^{3}\right )}}\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 = 0.51, size = 56, normalized size = 0.95 \begin {gather*} -\frac {{\left (a - b\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} + \frac {a \sin \left (x\right ) + b \sin \left (x\right )}{2 \, {\left (b \sin \left (x\right )^{2} + a\right )} a b} \end {gather*}
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 \begin {gather*} \frac {\sin \left (x\right )\,\left (a+b\right )}{2\,a\,b\,\left (b\,{\sin \left (x\right )}^2+a\right )}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \left (x\right )}{\sqrt {a}}\right )\,\left (a-b\right )}{2\,a^{3/2}\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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