Optimal. Leaf size=34 \[ -\frac {2 F\left (\text {ArcSin}\left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right )|\frac {1}{5}\right )}{\sqrt {5} d} \]
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Rubi [A]
time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2892}
\begin {gather*} -\frac {2 F\left (\text {ArcSin}\left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right )|\frac {1}{5}\right )}{\sqrt {5} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2892
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx &=-\frac {2 F\left (\sin ^{-1}\left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right )|\frac {1}{5}\right )}{\sqrt {5} d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(145\) vs. \(2(34)=68\).
time = 0.63, size = 145, normalized size = 4.26 \begin {gather*} -\frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(2-3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) F\left (\left .\text {ArcSin}\left (\frac {1}{2} \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right )\right |-4\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{d \sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(120\) vs.
\(2(33)=66\).
time = 0.31, size = 121, normalized size = 3.56
method | result | size |
default | \(-\frac {2 \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2-3 \cos \left (d x +c \right )}\, \left (\sin ^{2}\left (d x +c \right )\right )}{d \left (3 \left (\cos ^{2}\left (d x +c \right )\right )-5 \cos \left (d x +c \right )+2\right ) \sqrt {-\cos \left (d x +c \right )}}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {1}{\sqrt {- \cos {\left (c + d x \right )}} \sqrt {2 - 3 \cos {\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {-\cos \left (c+d\,x\right )}\,\sqrt {2-3\,\cos \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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