Optimal. Leaf size=82 \[ -\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) F\left (\left .\text {ArcSin}\left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right ) \sqrt {-\tan ^2(c+d x)}}{d} \]
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Rubi [A]
time = 0.07, antiderivative size = 82, 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 (\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 2894
Rule 2896
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx &=\frac {\sqrt {-\cos (c+d x)} \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\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 (\left .\sin ^{-1}\left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right ) \sqrt {-\tan ^2(c+d x)}}{d}\\ \end {align*}
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Mathematica [A]
time = 1.11, size = 153, normalized size = 1.87 \begin {gather*} \frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(3+2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) F\left (\text {ArcSin}\left (\sqrt {\frac {5}{3}} \sqrt {\frac {\cos (c+d x)}{-1+\cos (c+d x)}}\right )|\frac {6}{5}\right ) \sin ^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 [A]
time = 0.64, size = 137, normalized size = 1.67
method | result | size |
default | \(-\frac {i \sqrt {2}\, \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {-3-2 \cos \left (d x +c \right )}\, \left (\sin ^{4}\left (d x +c \right )\right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \sqrt {5}}{5 d \cos \left (d x +c \right )^{\frac {3}{2}} \left (3+2 \cos \left (d x +c \right )\right ) \left (-1+\cos \left (d x +c \right )\right )^{2}}\) | \(137\) |
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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