Integrand size = 13, antiderivative size = 40 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=-\frac {a \cos (x)}{b^2}+\frac {\cos ^2(x)}{2 b}+\frac {\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3} \]
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Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2747, 711} \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=\frac {\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3}-\frac {a \cos (x)}{b^2}+\frac {\cos ^2(x)}{2 b} \]
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Rule 711
Rule 2747
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {b^2-x^2}{a+x} \, dx,x,b \cos (x)\right )}{b^3} \\ & = -\frac {\text {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*}
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=-\frac {a \cos (x)}{b^2}+\frac {\cos (2 x)}{4 b}+\frac {\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3} \]
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Time = 0.80 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(-\frac {-\frac {b \left (\cos ^{2}\left (x \right )\right )}{2}+\cos \left (x \right ) a}{b^{2}}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (a +\cos \left (x \right ) b \right )}{b^{3}}\) | \(39\) |
default | \(-\frac {-\frac {b \left (\cos ^{2}\left (x \right )\right )}{2}+\cos \left (x \right ) a}{b^{2}}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (a +\cos \left (x \right ) b \right )}{b^{3}}\) | \(39\) |
parallelrisch | \(\frac {\left (a^{2}-b^{2}\right ) \ln \left (\frac {a +\cos \left (x \right ) b}{\cos \left (x \right )+1}\right )+\left (-a^{2}+b^{2}\right ) \ln \left (\frac {1}{\cos \left (x \right )+1}\right )-b \left (\cos \left (x \right ) a -\frac {b \cos \left (2 x \right )}{4}+a +\frac {b}{4}\right )}{b^{3}}\) | \(66\) |
norman | \(\frac {\frac {2 a \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{b^{2}}-\frac {2 a -2 b}{3 b^{2}}+\frac {\left (4 a +2 b \right ) \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3 b^{2}}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3}}+\frac {\left (a -b \right ) \left (a +b \right ) \ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+a +b \right )}{b^{3}}-\frac {\left (a -b \right ) \left (a +b \right ) \ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{b^{3}}\) | \(111\) |
risch | \(-\frac {i a^{2} x}{b^{3}}+\frac {i x}{b}+\frac {{\mathrm e}^{2 i x}}{8 b}-\frac {a \,{\mathrm e}^{i x}}{2 b^{2}}-\frac {a \,{\mathrm e}^{-i x}}{2 b^{2}}+\frac {{\mathrm e}^{-2 i x}}{8 b}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 a \,{\mathrm e}^{i x}}{b}+1\right ) a^{2}}{b^{3}}-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 a \,{\mathrm e}^{i x}}{b}+1\right )}{b}\) | \(111\) |
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Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=\frac {b^{2} \cos \left (x\right )^{2} - 2 \, a b \cos \left (x\right ) + 2 \, {\left (a^{2} - b^{2}\right )} \log \left (-b \cos \left (x\right ) - a\right )}{2 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1421 vs. \(2 (34) = 68\).
Time = 173.47 (sec) , antiderivative size = 1421, normalized size of antiderivative = 35.52 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=\text {Too large to display} \]
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Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=\frac {b \cos \left (x\right )^{2} - 2 \, a \cos \left (x\right )}{2 \, b^{2}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (b \cos \left (x\right ) + a\right )}{b^{3}} \]
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Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=\frac {b \cos \left (x\right )^{2} - 2 \, a \cos \left (x\right )}{2 \, b^{2}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left | b \cos \left (x\right ) + a \right |}\right )}{b^{3}} \]
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Time = 0.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=\frac {{\cos \left (x\right )}^2}{2\,b}+\frac {\ln \left (a+b\,\cos \left (x\right )\right )\,\left (a^2-b^2\right )}{b^3}-\frac {a\,\cos \left (x\right )}{b^2} \]
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