Optimal. Leaf size=61 \[ \frac {b \sin ^2(x)}{2 a^2}-\frac {b \left (a^2-b^2\right ) \log (a \sin (x)+b)}{a^4}+\frac {\left (a^2-b^2\right ) \sin (x)}{a^3}-\frac {\sin ^3(x)}{3 a} \]
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Rubi [A] time = 0.13, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3872, 2837, 12, 772} \[ \frac {\left (a^2-b^2\right ) \sin (x)}{a^3}-\frac {b \left (a^2-b^2\right ) \log (a \sin (x)+b)}{a^4}+\frac {b \sin ^2(x)}{2 a^2}-\frac {\sin ^3(x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 12
Rule 772
Rule 2837
Rule 3872
Rubi steps
\begin {align*} \int \frac {\cos ^3(x)}{a+b \csc (x)} \, dx &=\int \frac {\cos ^3(x) \sin (x)}{b+a \sin (x)} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x \left (a^2-x^2\right )}{a (b+x)} \, dx,x,a \sin (x)\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x \left (a^2-x^2\right )}{b+x} \, dx,x,a \sin (x)\right )}{a^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 \left (1-\frac {b^2}{a^2}\right )+b x-x^2+\frac {-a^2 b+b^3}{b+x}\right ) \, dx,x,a \sin (x)\right )}{a^4}\\ &=-\frac {b \left (a^2-b^2\right ) \log (b+a \sin (x))}{a^4}+\frac {\left (a^2-b^2\right ) \sin (x)}{a^3}+\frac {b \sin ^2(x)}{2 a^2}-\frac {\sin ^3(x)}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 60, normalized size = 0.98 \[ \frac {-2 a^3 \sin ^3(x)+6 a \left (a^2-b^2\right ) \sin (x)+6 b \left (b^2-a^2\right ) \log (a \sin (x)+b)+3 a^2 b \sin ^2(x)}{6 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 60, normalized size = 0.98 \[ -\frac {3 \, a^{2} b \cos \relax (x)^{2} + 6 \, {\left (a^{2} b - b^{3}\right )} \log \left (a \sin \relax (x) + b\right ) - 2 \, {\left (a^{3} \cos \relax (x)^{2} + 2 \, a^{3} - 3 \, a b^{2}\right )} \sin \relax (x)}{6 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 62, normalized size = 1.02 \[ -\frac {2 \, a^{2} \sin \relax (x)^{3} - 3 \, a b \sin \relax (x)^{2} - 6 \, a^{2} \sin \relax (x) + 6 \, b^{2} \sin \relax (x)}{6 \, a^{3}} - \frac {{\left (a^{2} b - b^{3}\right )} \log \left ({\left | a \sin \relax (x) + b \right |}\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 64, normalized size = 1.05 \[ -\frac {\sin ^{3}\relax (x )}{3 a}+\frac {b \left (\sin ^{2}\relax (x )\right )}{2 a^{2}}+\frac {\sin \relax (x )}{a}-\frac {b^{2} \sin \relax (x )}{a^{3}}-\frac {b \ln \left (b +a \sin \relax (x )\right )}{a^{2}}+\frac {b^{3} \ln \left (b +a \sin \relax (x )\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 60, normalized size = 0.98 \[ -\frac {2 \, a^{2} \sin \relax (x)^{3} - 3 \, a b \sin \relax (x)^{2} - 6 \, {\left (a^{2} - b^{2}\right )} \sin \relax (x)}{6 \, a^{3}} - \frac {{\left (a^{2} b - b^{3}\right )} \log \left (a \sin \relax (x) + b\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 58, normalized size = 0.95 \[ \sin \relax (x)\,\left (\frac {1}{a}-\frac {b^2}{a^3}\right )-\frac {{\sin \relax (x)}^3}{3\,a}+\frac {b\,{\sin \relax (x)}^2}{2\,a^2}-\frac {\ln \left (b+a\,\sin \relax (x)\right )\,\left (a^2\,b-b^3\right )}{a^4} \]
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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