Optimal. Leaf size=79 \[ \frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}-\frac {4 a \sin (c+d x)}{3 d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2851, 2850}
\begin {gather*} \frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}-\frac {4 a \sin (c+d x)}{3 d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2850
Rule 2851
Rubi steps
\begin {align*} \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx &=\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}-\frac {2}{3} \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}-\frac {4 a \sin (c+d x)}{3 d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 52, normalized size = 0.66 \begin {gather*} -\frac {2 (-1+2 \cos (c+d x)) \sqrt {a-a \cos (c+d x)} \cot \left (\frac {1}{2} (c+d x)\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 56, normalized size = 0.71
method | result | size |
default | \(\frac {\left (-1+2 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {-2 \left (-1+\cos \left (d x +c \right )\right ) a}\, \sqrt {2}}{3 d \left (-1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {3}{2}}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 174 vs.
\(2 (67) = 134\).
time = 0.55, size = 174, normalized size = 2.20 \begin {gather*} -\frac {2 \, {\left (\sqrt {2} \sqrt {a} - \frac {4 \, \sqrt {2} \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sqrt {2} \sqrt {a} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 52, normalized size = 0.66 \begin {gather*} -\frac {2 \, \sqrt {-a \cos \left (d x + c\right ) + a} {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) - 1\right )}}{3 \, d \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right )} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- a \left (\cos {\left (c + d x \right )} - 1\right )}}{\cos ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 90, normalized size = 1.14 \begin {gather*} \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 )} \sqrt {a} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.39, size = 85, normalized size = 1.08 \begin {gather*} \frac {4\,\sqrt {-a\,\left (\cos \left (c+d\,x\right )-1\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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