Optimal. Leaf size=71 \[ \frac{2 (1-5 \cos (x))^{9/2}}{28125}-\frac{8 (1-5 \cos (x))^{7/2}}{21875}-\frac{88 (1-5 \cos (x))^{5/2}}{15625}+\frac{64 (1-5 \cos (x))^{3/2}}{3125}+\frac{1152 \sqrt{1-5 \cos (x)}}{3125} \]
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Rubi [A] time = 0.0658991, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2668, 697} \[ \frac{2 (1-5 \cos (x))^{9/2}}{28125}-\frac{8 (1-5 \cos (x))^{7/2}}{21875}-\frac{88 (1-5 \cos (x))^{5/2}}{15625}+\frac{64 (1-5 \cos (x))^{3/2}}{3125}+\frac{1152 \sqrt{1-5 \cos (x)}}{3125} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{\sin ^5(x)}{\sqrt{1-5 \cos (x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (25-x^2\right )^2}{\sqrt{1+x}} \, dx,x,-5 \cos (x)\right )}{3125}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{576}{\sqrt{1+x}}+96 \sqrt{1+x}-44 (1+x)^{3/2}-4 (1+x)^{5/2}+(1+x)^{7/2}\right ) \, dx,x,-5 \cos (x)\right )}{3125}\\ &=\frac{1152 \sqrt{1-5 \cos (x)}}{3125}+\frac{64 (1-5 \cos (x))^{3/2}}{3125}-\frac{88 (1-5 \cos (x))^{5/2}}{15625}-\frac{8 (1-5 \cos (x))^{7/2}}{21875}+\frac{2 (1-5 \cos (x))^{9/2}}{28125}\\ \end{align*}
Mathematica [A] time = 0.15787, size = 59, normalized size = 0.83 \[ \frac{180607 \left (\sqrt{1-5 \cos (x)}-1\right )}{562500}+\sqrt{1-5 \cos (x)} \left (-\frac{6772 \cos (x)}{196875}-\frac{2227 \cos (2 x)}{39375}+\frac{4 \cos (3 x)}{1575}+\frac{1}{180} \cos (4 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 1.081, size = 49, normalized size = 0.7 \begin{align*}{\frac{32}{984375}\sqrt{10\, \left ( \sin \left ( x/2 \right ) \right ) ^{2}-4} \left ( 21875\, \left ( \sin \left ( x/2 \right ) \right ) ^{8}-46250\, \left ( \sin \left ( x/2 \right ) \right ) ^{6}+17175\, \left ( \sin \left ( x/2 \right ) \right ) ^{4}+9160\, \left ( \sin \left ( x/2 \right ) \right ) ^{2}+7328 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985314, size = 69, normalized size = 0.97 \begin{align*} \frac{2}{28125} \,{\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac{9}{2}} - \frac{8}{21875} \,{\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac{7}{2}} - \frac{88}{15625} \,{\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac{5}{2}} + \frac{64}{3125} \,{\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac{3}{2}} + \frac{1152}{3125} \, \sqrt{-5 \, \cos \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1865, size = 140, normalized size = 1.97 \begin{align*} \frac{2}{984375} \,{\left (21875 \, \cos \left (x\right )^{4} + 5000 \, \cos \left (x\right )^{3} - 77550 \, \cos \left (x\right )^{2} - 20680 \, \cos \left (x\right ) + 188603\right )} \sqrt{-5 \, \cos \left (x\right ) + 1} \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 = 1.11028, size = 101, normalized size = 1.42 \begin{align*} \frac{2}{28125} \,{\left (5 \, \cos \left (x\right ) - 1\right )}^{4} \sqrt{-5 \, \cos \left (x\right ) + 1} + \frac{8}{21875} \,{\left (5 \, \cos \left (x\right ) - 1\right )}^{3} \sqrt{-5 \, \cos \left (x\right ) + 1} - \frac{88}{15625} \,{\left (5 \, \cos \left (x\right ) - 1\right )}^{2} \sqrt{-5 \, \cos \left (x\right ) + 1} + \frac{64}{3125} \,{\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac{3}{2}} + \frac{1152}{3125} \, \sqrt{-5 \, \cos \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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