Optimal. Leaf size=40 \[ \frac {\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3}-\frac {a \cos (x)}{b^2}+\frac {\cos ^2(x)}{2 b} \]
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Rubi [A] time = 0.06, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2668, 697} \[ \frac {\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3}-\frac {a \cos (x)}{b^2}+\frac {\cos ^2(x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rubi steps
\begin {align*} \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {b^2-x^2}{a+x} \, dx,x,b \cos (x)\right )}{b^3}\\ &=-\frac {\operatorname {Subst}\left (\int \left (a-x+\frac {-a^2+b^2}{a+x}\right ) \, dx,x,b \cos (x)\right )}{b^3}\\ &=-\frac {a \cos (x)}{b^2}+\frac {\cos ^2(x)}{2 b}+\frac {\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 40, normalized size = 1.00 \[ \frac {\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3}-\frac {a \cos (x)}{b^2}+\frac {\cos (2 x)}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 41, normalized size = 1.02 \[ \frac {b^{2} \cos \relax (x)^{2} - 2 \, a b \cos \relax (x) + 2 \, {\left (a^{2} - b^{2}\right )} \log \left (-b \cos \relax (x) - a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 39, normalized size = 0.98 \[ \frac {b \cos \relax (x)^{2} - 2 \, a \cos \relax (x)}{2 \, b^{2}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left | b \cos \relax (x) + a \right |}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 45, normalized size = 1.12 \[ \frac {\cos ^{2}\relax (x )}{2 b}-\frac {a \cos \relax (x )}{b^{2}}+\frac {\ln \left (a +b \cos \relax (x )\right ) a^{2}}{b^{3}}-\frac {\ln \left (a +b \cos \relax (x )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.29, size = 38, normalized size = 0.95 \[ \frac {b \cos \relax (x)^{2} - 2 \, a \cos \relax (x)}{2 \, b^{2}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (b \cos \relax (x) + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 38, normalized size = 0.95 \[ \frac {{\cos \relax (x)}^2}{2\,b}+\frac {\ln \left (a+b\,\cos \relax (x)\right )\,\left (a^2-b^2\right )}{b^3}-\frac {a\,\cos \relax (x)}{b^2} \]
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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