Optimal. Leaf size=118 \[ -\frac {8 a \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}+\frac {16 a \sin (c+d x)}{15 d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}} \]
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Rubi [A] time = 0.17, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2772, 2771} \[ -\frac {8 a \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}+\frac {16 a \sin (c+d x)}{15 d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2771
Rule 2772
Rubi steps
\begin {align*} \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx &=\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}-\frac {4}{5} \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}-\frac {8 a \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}+\frac {8}{15} \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}-\frac {8 a \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}+\frac {16 a \sin (c+d x)}{15 d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 62, normalized size = 0.53 \[ \frac {2 (-4 \cos (c+d x)+4 \cos (2 (c+d x))+7) \cot \left (\frac {1}{2} (c+d x)\right ) \sqrt {a-a \cos (c+d x)}}{15 d \cos ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 64, normalized size = 0.54 \[ \frac {2 \, {\left (8 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 3\right )} \sqrt {-a \cos \left (d x + c\right ) + a}}{15 \, d \cos \left (d x + c\right )^{\frac {5}{2}} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.30, size = 120, normalized size = 1.02 \[ -\frac {2 \, \sqrt {2} {\left ({\left ({\left ({\left ({\left (7 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 75\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 430\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 430\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 75\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 7\right )} \sqrt {a} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{15 \, {\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1\right )}^{\frac {5}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 66, normalized size = 0.56 \[ -\frac {\left (8 \left (\cos ^{2}\left (d x +c \right )\right )-4 \cos \left (d x +c \right )+3\right ) \sqrt {-2 a \left (-1+\cos \left (d x +c \right )\right )}\, \sin \left (d x +c \right ) \sqrt {2}}{15 d \left (-1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.98, size = 221, normalized size = 1.87 \[ \frac {2 \, {\left (7 \, \sqrt {2} \sqrt {a} - \frac {17 \, \sqrt {2} \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {25 \, \sqrt {2} \sqrt {a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {15 \, \sqrt {2} \sqrt {a} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{15 \, d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (\frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.89, size = 158, normalized size = 1.34 \[ \frac {8\,\sqrt {2\,a\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\,\left (7\,\sin \left (c+d\,x\right )-4\,\sin \left (2\,c+2\,d\,x\right )+9\,\sin \left (3\,c+3\,d\,x\right )-2\,\sin \left (4\,c+4\,d\,x\right )+2\,\sin \left (5\,c+5\,d\,x\right )\right )}{15\,d\,\sqrt {1-2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\,\left (-16\,{\sin \left (c+d\,x\right )}^2-4\,{\sin \left (2\,c+2\,d\,x\right )}^2+20\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+10\,{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2+2\,{\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}^2\right )} \]
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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