Optimal. Leaf size=72 \[ \frac {2 \cot (c+d x) E\left (\left .\text {ArcSin}\left (\frac {\sqrt {3+2 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {3073}
\begin {gather*} \frac {2 \cot (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {2 \cos (c+d x)+3}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |-5\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3073
Rubi steps
\begin {align*} \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx &=\frac {2 \cot (c+d x) E\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 {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 d}\\ \end {align*}
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Mathematica [F]
time = 48.75, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx \end {gather*}
Verification is not applicable to the result.
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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. 664 vs. \(2 (65 ) = 130\).
time = 2.00, size = 665, normalized size = 9.24
method | result | size |
default | \(-\frac {3 \sqrt {2}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \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 )+6 \sqrt {2}\, \sin \left (d x +c \right ) \cos \left (d x +c \right ) \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 )-5 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+3 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \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 ) \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+3 \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 ) \sin \left (d x +c \right )-5 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \sin \left (d x +c \right ) \cos \left (d x +c \right )+3 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \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 ) \sin \left (d x +c \right ) \cos \left (d x +c \right )+20 \left (\cos ^{3}\left (d x +c \right )\right )+10 \left (\cos ^{2}\left (d x +c \right )\right )-30 \cos \left (d x +c \right )}{15 d \sqrt {3+2 \cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{\frac {3}{2}} \sin \left (d x +c \right )}\) | \(665\) |
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 {\cos {\left (c + d x \right )} + 1}{\sqrt {2 \cos {\left (c + d x \right )} + 3} \cos ^{\frac {3}{2}}{\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 {\cos \left (c+d\,x\right )+1}{{\cos \left (c+d\,x\right )}^{3/2}\,\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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