Optimal. Leaf size=56 \[ \frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(a-b) \sin (x)}{b^2}-\frac {\sin ^3(x)}{3 b} \]
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Rubi [A] time = 0.07, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3186, 390, 208} \[ \frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(a-b) \sin (x)}{b^2}-\frac {\sin ^3(x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 208
Rule 390
Rule 3186
Rubi steps
\begin {align*} \int \frac {\cos ^5(x)}{a+b \cos ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a+b-b x^2} \, dx,x,\sin (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {a-b}{b^2}-\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b-b x^2\right )}\right ) \, dx,x,\sin (x)\right )\\ &=-\frac {(a-b) \sin (x)}{b^2}-\frac {\sin ^3(x)}{3 b}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\sin (x)\right )}{b^2}\\ &=\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(a-b) \sin (x)}{b^2}-\frac {\sin ^3(x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 86, normalized size = 1.54 \[ \frac {\frac {6 a^2 \left (\log \left (\sqrt {a+b}+\sqrt {b} \sin (x)\right )-\log \left (\sqrt {a+b}-\sqrt {b} \sin (x)\right )\right )}{\sqrt {a+b}}+3 \sqrt {b} (3 b-4 a) \sin (x)+b^{3/2} \sin (3 x)}{12 b^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 191, normalized size = 3.41 \[ \left [\frac {3 \, \sqrt {a b + b^{2}} a^{2} \log \left (-\frac {b \cos \relax (x)^{2} - 2 \, \sqrt {a b + b^{2}} \sin \relax (x) - a - 2 \, b}{b \cos \relax (x)^{2} + a}\right ) - 2 \, {\left (3 \, a^{2} b + a b^{2} - 2 \, b^{3} - {\left (a b^{2} + b^{3}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)}{6 \, {\left (a b^{3} + b^{4}\right )}}, -\frac {3 \, \sqrt {-a b - b^{2}} a^{2} \arctan \left (\frac {\sqrt {-a b - b^{2}} \sin \relax (x)}{a + b}\right ) + {\left (3 \, a^{2} b + a b^{2} - 2 \, b^{3} - {\left (a b^{2} + b^{3}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)}{3 \, {\left (a b^{3} + b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 65, normalized size = 1.16 \[ -\frac {a^{2} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} b^{2}} - \frac {b^{2} \sin \relax (x)^{3} + 3 \, a b \sin \relax (x) - 3 \, b^{2} \sin \relax (x)}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 50, normalized size = 0.89 \[ -\frac {\frac {\left (\sin ^{3}\relax (x )\right ) b}{3}+\sin \relax (x ) a -\sin \relax (x ) b}{b^{2}}+\frac {a^{2} \arctanh \left (\frac {\sin \relax (x ) b}{\sqrt {\left (a +b \right ) b}}\right )}{b^{2} \sqrt {\left (a +b \right ) b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 67, normalized size = 1.20 \[ -\frac {a^{2} \log \left (\frac {b \sin \relax (x) - \sqrt {{\left (a + b\right )} b}}{b \sin \relax (x) + \sqrt {{\left (a + b\right )} b}}\right )}{2 \, \sqrt {{\left (a + b\right )} b} b^{2}} - \frac {b \sin \relax (x)^{3} + 3 \, {\left (a - b\right )} \sin \relax (x)}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.31, size = 51, normalized size = 0.91 \[ \frac {a^2\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sin \relax (x)}{\sqrt {a+b}}\right )}{b^{5/2}\,\sqrt {a+b}}-\frac {{\sin \relax (x)}^3}{3\,b}-\sin \relax (x)\,\left (\frac {a+b}{b^2}-\frac {2}{b}\right ) \]
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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