Optimal. Leaf size=58 \[ -\frac{\csc ^5(x)}{5 a}+\frac{\csc ^4(x)}{4 a}+\frac{2 \csc ^3(x)}{3 a}-\frac{\csc ^2(x)}{a}-\frac{\csc (x)}{a}-\frac{\log (\sin (x))}{a} \]
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Rubi [A] time = 0.0581809, antiderivative size = 58, normalized size of antiderivative = 1., 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 = {3879, 88} \[ -\frac{\csc ^5(x)}{5 a}+\frac{\csc ^4(x)}{4 a}+\frac{2 \csc ^3(x)}{3 a}-\frac{\csc ^2(x)}{a}-\frac{\csc (x)}{a}-\frac{\log (\sin (x))}{a} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \frac{\cot ^7(x)}{a+a \csc (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^3 (a+a x)^2}{x^6} \, dx,x,\sin (x)\right )}{a^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^5}{x^6}-\frac{a^5}{x^5}-\frac{2 a^5}{x^4}+\frac{2 a^5}{x^3}+\frac{a^5}{x^2}-\frac{a^5}{x}\right ) \, dx,x,\sin (x)\right )}{a^6}\\ &=-\frac{\csc (x)}{a}-\frac{\csc ^2(x)}{a}+\frac{2 \csc ^3(x)}{3 a}+\frac{\csc ^4(x)}{4 a}-\frac{\csc ^5(x)}{5 a}-\frac{\log (\sin (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.049801, size = 39, normalized size = 0.67 \[ -\frac{\frac{\csc ^5(x)}{5}-\frac{\csc ^4(x)}{4}-\frac{2 \csc ^3(x)}{3}+\csc ^2(x)+\csc (x)+\log (\sin (x))}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 55, normalized size = 1. \begin{align*}{\frac{2}{3\,a \left ( \sin \left ( x \right ) \right ) ^{3}}}-{\frac{1}{a \left ( \sin \left ( x \right ) \right ) ^{2}}}+{\frac{1}{4\,a \left ( \sin \left ( x \right ) \right ) ^{4}}}-{\frac{\ln \left ( \sin \left ( x \right ) \right ) }{a}}-{\frac{1}{a\sin \left ( x \right ) }}-{\frac{1}{5\,a \left ( \sin \left ( x \right ) \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960434, size = 57, normalized size = 0.98 \begin{align*} -\frac{\log \left (\sin \left (x\right )\right )}{a} - \frac{60 \, \sin \left (x\right )^{4} + 60 \, \sin \left (x\right )^{3} - 40 \, \sin \left (x\right )^{2} - 15 \, \sin \left (x\right ) + 12}{60 \, a \sin \left (x\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.508639, size = 224, normalized size = 3.86 \begin{align*} -\frac{60 \, \cos \left (x\right )^{4} + 60 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right ) - 80 \, \cos \left (x\right )^{2} - 15 \,{\left (4 \, \cos \left (x\right )^{2} - 3\right )} \sin \left (x\right ) + 32}{60 \,{\left (a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a\right )} \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38837, size = 58, normalized size = 1. \begin{align*} -\frac{\log \left ({\left | \sin \left (x\right ) \right |}\right )}{a} - \frac{60 \, \sin \left (x\right )^{4} + 60 \, \sin \left (x\right )^{3} - 40 \, \sin \left (x\right )^{2} - 15 \, \sin \left (x\right ) + 12}{60 \, a \sin \left (x\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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