Optimal. Leaf size=92 \[ -\frac{b^3 \log (a+b \cos (x))}{\left (a^2-b^2\right )^2}+\frac{\csc ^2(x) (b-a \cos (x))}{2 \left (a^2-b^2\right )}+\frac{(a+2 b) \log (1-\cos (x))}{4 (a+b)^2}-\frac{(a-2 b) \log (\cos (x)+1)}{4 (a-b)^2} \]
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Rubi [A] time = 0.154131, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2668, 741, 801} \[ -\frac{b^3 \log (a+b \cos (x))}{\left (a^2-b^2\right )^2}+\frac{\csc ^2(x) (b-a \cos (x))}{2 \left (a^2-b^2\right )}+\frac{(a+2 b) \log (1-\cos (x))}{4 (a+b)^2}-\frac{(a-2 b) \log (\cos (x)+1)}{4 (a-b)^2} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 741
Rule 801
Rubi steps
\begin{align*} \int \frac{\csc ^3(x)}{a+b \cos (x)} \, dx &=-\left (b^3 \operatorname{Subst}\left (\int \frac{1}{(a+x) \left (b^2-x^2\right )^2} \, dx,x,b \cos (x)\right )\right )\\ &=\frac{(b-a \cos (x)) \csc ^2(x)}{2 \left (a^2-b^2\right )}-\frac{b \operatorname{Subst}\left (\int \frac{a^2-2 b^2+a x}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \cos (x)\right )}{2 \left (a^2-b^2\right )}\\ &=\frac{(b-a \cos (x)) \csc ^2(x)}{2 \left (a^2-b^2\right )}-\frac{b \operatorname{Subst}\left (\int \left (\frac{(a-b) (a+2 b)}{2 b (a+b) (b-x)}+\frac{2 b^2}{(a-b) (a+b) (a+x)}+\frac{(a-2 b) (a+b)}{2 (a-b) b (b+x)}\right ) \, dx,x,b \cos (x)\right )}{2 \left (a^2-b^2\right )}\\ &=\frac{(b-a \cos (x)) \csc ^2(x)}{2 \left (a^2-b^2\right )}+\frac{(a+2 b) \log (1-\cos (x))}{4 (a+b)^2}-\frac{(a-2 b) \log (1+\cos (x))}{4 (a-b)^2}-\frac{b^3 \log (a+b \cos (x))}{\left (a^2-b^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.42895, size = 99, normalized size = 1.08 \[ \frac{1}{8} \left (-\frac{8 b^3 \log (a+b \cos (x))}{\left (a^2-b^2\right )^2}-\frac{\csc ^2\left (\frac{x}{2}\right )}{a+b}+\frac{\sec ^2\left (\frac{x}{2}\right )}{a-b}+\frac{4 (a+2 b) \log \left (\sin \left (\frac{x}{2}\right )\right )}{(a+b)^2}-\frac{4 (a-2 b) \log \left (\cos \left (\frac{x}{2}\right )\right )}{(a-b)^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 114, normalized size = 1.2 \begin{align*} -{\frac{{b}^{3}\ln \left ( a+b\cos \left ( x \right ) \right ) }{ \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}}}+{\frac{1}{ \left ( 4\,a+4\,b \right ) \left ( -1+\cos \left ( x \right ) \right ) }}+{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) a}{4\, \left ( a+b \right ) ^{2}}}+{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) b}{2\, \left ( a+b \right ) ^{2}}}+{\frac{1}{ \left ( 4\,a-4\,b \right ) \left ( \cos \left ( x \right ) +1 \right ) }}-{\frac{\ln \left ( \cos \left ( x \right ) +1 \right ) a}{4\, \left ( a-b \right ) ^{2}}}+{\frac{\ln \left ( \cos \left ( x \right ) +1 \right ) b}{2\, \left ( a-b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50236, size = 155, normalized size = 1.68 \begin{align*} -\frac{b^{3} \log \left (b \cos \left (x\right ) + a\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (a - 2 \, b\right )} \log \left (\cos \left (x\right ) + 1\right )}{4 \,{\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac{{\left (a + 2 \, b\right )} \log \left (\cos \left (x\right ) - 1\right )}{4 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac{a \cos \left (x\right ) - b}{2 \,{\left ({\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - a^{2} + b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02705, size = 429, normalized size = 4.66 \begin{align*} \frac{2 \, a^{2} b - 2 \, b^{3} - 2 \,{\left (a^{3} - a b^{2}\right )} \cos \left (x\right ) + 4 \,{\left (b^{3} \cos \left (x\right )^{2} - b^{3}\right )} \log \left (-b \cos \left (x\right ) - a\right ) -{\left (a^{3} - 3 \, a b^{2} - 2 \, b^{3} -{\left (a^{3} - 3 \, a b^{2} - 2 \, b^{3}\right )} \cos \left (x\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) +{\left (a^{3} - 3 \, a b^{2} + 2 \, b^{3} -{\left (a^{3} - 3 \, a b^{2} + 2 \, b^{3}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{4 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} -{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{3}{\left (x \right )}}{a + b \cos{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16905, size = 184, normalized size = 2. \begin{align*} -\frac{b^{4} \log \left ({\left | b \cos \left (x\right ) + a \right |}\right )}{a^{4} b - 2 \, a^{2} b^{3} + b^{5}} - \frac{{\left (a - 2 \, b\right )} \log \left (\cos \left (x\right ) + 1\right )}{4 \,{\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac{{\left (a + 2 \, b\right )} \log \left (-\cos \left (x\right ) + 1\right )}{4 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac{a^{2} b - b^{3} -{\left (a^{3} - a b^{2}\right )} \cos \left (x\right )}{2 \,{\left (a + b\right )}^{2}{\left (a - b\right )}^{2}{\left (\cos \left (x\right ) + 1\right )}{\left (\cos \left (x\right ) - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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