Optimal. Leaf size=75 \[ -\frac {3 \cot (c+d x) \Pi \left (\frac {5}{2};\left .\text {ArcSin}\left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d} \]
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Rubi [A]
time = 0.04, antiderivative size = 75, 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 = {2887}
\begin {gather*} -\frac {3 \cot (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \Pi \left (\frac {5}{2};\left .\text {ArcSin}\left (\frac {\sqrt {-2 \cos (c+d x)-3}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2887
Rubi steps
\begin {align*} \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx &=-\frac {3 \cot (c+d x) \Pi \left (\frac {5}{2};\left .\sin ^{-1}\left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 115, normalized size = 1.53 \begin {gather*} \frac {2 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (F\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|-\frac {1}{5}\right )-2 \Pi \left (-1;\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|-\frac {1}{5}\right )\right ) \sqrt {\cos (c+d x) (3+2 \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right )}}{\sqrt {5} d \sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 163 vs. \(2 (68 ) = 136\).
time = 0.21, size = 164, normalized size = 2.19
method | result | size |
default | \(\frac {\sqrt {10}\, \sqrt {2}\, \left (\EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right )-2 \EllipticPi \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, -1, \frac {i \sqrt {5}}{5}\right )\right ) \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {-3-2 \cos \left (d x +c \right )}\, \sqrt {-\cos \left (d x +c \right )}}{5 d \left (2 \left (\cos ^{2}\left (d x +c \right )\right )+\cos \left (d x +c \right )-3\right ) \cos \left (d x +c \right )}\) | \(164\) |
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 {\sqrt {- \cos {\left (c + d x \right )}}}{\sqrt {- 2 \cos {\left (c + d x \right )} - 3}}\, 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 = -0.01 \begin {gather*} \int \frac {\sqrt {-\cos \left (c+d\,x\right )}}{\sqrt {-2\,\cos \left (c+d\,x\right )-3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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