Optimal. Leaf size=73 \[ -\frac {3 \cot (c+d x) \Pi \left (\frac {5}{2};\left .\text {ArcSin}\left (\frac {\sqrt {3+2 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d} \]
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Rubi [A]
time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, 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 = 1.46, size = 115, normalized size = 1.58 \begin {gather*} -\frac {2 \sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)} \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 ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{\sqrt {5} d \sqrt {(1+3 \cos (c+d x)+\cos (2 (c+d x))) \sec ^4\left (\frac {1}{2} (c+d x)\right )}} \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. 143 vs. \(2 (66 ) = 132\).
time = 0.21, size = 144, normalized size = 1.97
method | result | size |
default | \(-\frac {\sqrt {10}\, \sqrt {2}\, \left (\EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right )-2 \EllipticPi \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, -1, \frac {i \sqrt {5}}{5}\right )\right ) \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\sin ^{2}\left (d x +c \right )\right )}{5 d \sqrt {3+2 \cos \left (d x +c \right )}\, \left (-1+\cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}}\) | \(144\) |
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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