Optimal. Leaf size=58 \[ \frac {2 \cot (c+d x) F\left (\left .\text {ArcSin}\left (\frac {\sqrt {3+2 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |-5\right ) \sqrt {-\tan ^2(c+d x)}}{d} \]
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Rubi [A]
time = 0.04, antiderivative size = 58, 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 = {2894}
\begin {gather*} \frac {2 \sqrt {-\tan ^2(c+d x)} \cot (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
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx &=\frac {2 \cot (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 [B] Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(58)=116\).
time = 1.09, size = 140, normalized size = 2.41 \begin {gather*} \frac {4 \sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)} \sqrt {-\cot ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) F\left (\left .\text {ArcSin}\left (\frac {\sqrt {(3+2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}}{\sqrt {6}}\right )\right |6\right )}{d \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 )}} \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. 115 vs. \(2 (55 ) = 110\).
time = 0.60, size = 116, normalized size = 2.00
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{5 d \sqrt {3+2 \cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{\frac {3}{2}} \left (-1+\cos \left (d x +c \right )\right )^{2}}\) | \(116\) |
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 = -0.02 \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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