Optimal. Leaf size=75 \[ \frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}-\frac {4 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.08, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2772, 2771} \[ \frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}-\frac {4 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2771
Rule 2772
Rubi steps
\begin {align*} \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx &=\frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}-\frac {2}{3} \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}-\frac {4 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 51, normalized size = 0.68 \[ -\frac {2 \sqrt {1-\cos (c+d x)} (2 \cos (c+d x)-1) \cot \left (\frac {1}{2} (c+d x)\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.05, size = 51, normalized size = 0.68 \[ -\frac {2 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) - 1\right )} \sqrt {-\cos \left (d x + c\right ) + 1}}{3 \, d \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.41, size = 87, normalized size = 1.16 \[ \frac {2 \, \sqrt {2} {\left ({\left ({\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 15\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 15\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 1\right )} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{3 \, {\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1\right )}^{\frac {3}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 55, normalized size = 0.73 \[ \frac {\left (-1+2 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2-2 \cos \left (d x +c \right )}\, \sqrt {2}}{3 d \left (-1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.89, size = 164, normalized size = 2.19 \[ -\frac {2 \, {\left (\sqrt {2} - \frac {4 \, \sqrt {2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sqrt {2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{3 \, d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (\frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.55, size = 84, normalized size = 1.12 \[ \frac {4\,\sqrt {1-\cos \left (c+d\,x\right )}\,\left (\sin \left (c+d\,x\right )-\sin \left (2\,c+2\,d\,x\right )+\sin \left (3\,c+3\,d\,x\right )\right )}{3\,d\,\sqrt {\cos \left (c+d\,x\right )}\,\left (3\,\cos \left (c+d\,x\right )-2\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (3\,c+3\,d\,x\right )-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1 - \cos {\left (c + d x \right )}}}{\cos ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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