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 = 0.07, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 697
Rule 2668
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*}
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Mathematica [A] time = 0.16, 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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fricas [A] time = 1.68, size = 34, normalized size = 0.48 \[ \frac {2}{984375} \, {\left (21875 \, \cos \relax (x)^{4} + 5000 \, \cos \relax (x)^{3} - 77550 \, \cos \relax (x)^{2} - 20680 \, \cos \relax (x) + 188603\right )} \sqrt {-5 \, \cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 75, normalized size = 1.06 \[ \frac {2}{28125} \, {\left (5 \, \cos \relax (x) - 1\right )}^{4} \sqrt {-5 \, \cos \relax (x) + 1} + \frac {8}{21875} \, {\left (5 \, \cos \relax (x) - 1\right )}^{3} \sqrt {-5 \, \cos \relax (x) + 1} - \frac {88}{15625} \, {\left (5 \, \cos \relax (x) - 1\right )}^{2} \sqrt {-5 \, \cos \relax (x) + 1} + \frac {64}{3125} \, {\left (-5 \, \cos \relax (x) + 1\right )}^{\frac {3}{2}} + \frac {1152}{3125} \, \sqrt {-5 \, \cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 49, normalized size = 0.69 \[ \frac {32 \sqrt {10 \left (\sin ^{2}\left (\frac {x}{2}\right )\right )-4}\, \left (21875 \left (\sin ^{8}\left (\frac {x}{2}\right )\right )-46250 \left (\sin ^{6}\left (\frac {x}{2}\right )\right )+17175 \left (\sin ^{4}\left (\frac {x}{2}\right )\right )+9160 \left (\sin ^{2}\left (\frac {x}{2}\right )\right )+7328\right )}{984375} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 51, normalized size = 0.72 \[ \frac {2}{28125} \, {\left (-5 \, \cos \relax (x) + 1\right )}^{\frac {9}{2}} - \frac {8}{21875} \, {\left (-5 \, \cos \relax (x) + 1\right )}^{\frac {7}{2}} - \frac {88}{15625} \, {\left (-5 \, \cos \relax (x) + 1\right )}^{\frac {5}{2}} + \frac {64}{3125} \, {\left (-5 \, \cos \relax (x) + 1\right )}^{\frac {3}{2}} + \frac {1152}{3125} \, \sqrt {-5 \, \cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \relax (x)}^5}{\sqrt {1-5\,\cos \relax (x)}} \,d x \]
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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