Optimal. Leaf size=101 \[ -\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \Pi \left (\frac {5}{3};\left .\text {ArcSin}\left (\frac {\sqrt {-2-3 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{3 d \sqrt {-\cos (c+d x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 101, 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 = {2889, 2888}
\begin {gather*} -\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} \Pi \left (\frac {5}{3};\left .\text {ArcSin}\left (\frac {\sqrt {-3 \cos (c+d x)-2}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right )}{3 d \sqrt {-\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2888
Rule 2889
Rubi steps
\begin {align*} \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx}{\sqrt {-\cos (c+d x)}}\\ &=-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \Pi \left (\frac {5}{3};\left .\sin ^{-1}\left (\frac {\sqrt {-2-3 \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)}}{3 d \sqrt {-\cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 2.14, size = 155, normalized size = 1.53 \begin {gather*} -\frac {4 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {-\frac {(2+3 \cos (c+d x))^2}{(1+\cos (c+d x))^2}} \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 {5} d \sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)} \sqrt {-\frac {2+3 \cos (c+d x)}{1+\cos (c+d x)}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 161, normalized size = 1.59
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {10}\, \left (\EllipticF \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, \sqrt {5}\right )-2 \EllipticPi \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, -5, \sqrt {5}\right )\right ) \sqrt {-2-3 \cos \left (d x +c \right )}\, \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {5}}{5 d \left (3 \left (\cos ^{2}\left (d x +c \right )\right )-\cos \left (d x +c \right )-2\right ) \sqrt {\cos \left (d x +c \right )}}\) | \(161\) |
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 {- 3 \cos {\left (c + d x \right )} - 2}}\, 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 {-3\,\cos \left (c+d\,x\right )-2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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