Optimal. Leaf size=84 \[ -\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {-3+2 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right )|-\frac {1}{5}\right ) \sqrt {-\tan ^2(c+d x)}}{\sqrt {5} d} \]
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Rubi [A]
time = 0.08, antiderivative size = 84, 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 = {2896, 2894}
\begin {gather*} -\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \sqrt {-\tan ^2(c+d x)} \csc (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {2 \cos (c+d x)-3}}{\sqrt {-\cos (c+d x)}}\right )|-\frac {1}{5}\right )}{\sqrt {5} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2894
Rule 2896
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx &=\frac {\sqrt {-\cos (c+d x)} \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx}{\sqrt {\cos (c+d x)}}\\ &=-\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {-3+2 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right )|-\frac {1}{5}\right ) \sqrt {-\tan ^2(c+d x)}}{\sqrt {5} d}\\ \end {align*}
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Mathematica [A]
time = 1.30, size = 144, normalized size = 1.71 \begin {gather*} \frac {2 \sqrt {\cos (c+d x)} \sqrt {\frac {-3+2 \cos (c+d x)}{-1+\cos (c+d x)}} \sqrt {-\cot ^2\left (\frac {1}{2} (c+d x)\right )} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {-3+2 \cos (c+d x)}{-1+\cos (c+d x)}}}{\sqrt {3}}\right )|\frac {6}{5}\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {5} d \sqrt {\frac {\cos (c+d x)}{-1+\cos (c+d x)}} \sqrt {-3+2 \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.02, size = 123, normalized size = 1.46
method | result | size |
default | \(\frac {i \sqrt {2}\, \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \left (\sin ^{4}\left (d x +c \right )\right ) \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \sqrt {5}}{5 d \sqrt {-3+2 \cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{\frac {3}{2}} \left (-1+\cos \left (d x +c \right )\right )^{2}}\) | \(123\) |
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 {2 \cos {\left (c + d x \right )} - 3} \sqrt {\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 = -0.01 \begin {gather*} \int \frac {1}{\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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