Optimal. Leaf size=111 \[ -\frac {23 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{20 \sqrt {7} d}+\frac {9 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{20 d}+\frac {\sin (c+d x) \cos (c+d x) \sqrt {4 \cos (c+d x)+3}}{10 d}-\frac {\sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{10 d} \]
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Rubi [A] time = 0.15, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2793, 3023, 2752, 2661, 2653} \[ -\frac {23 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{20 \sqrt {7} d}+\frac {9 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{20 d}+\frac {\sin (c+d x) \cos (c+d x) \sqrt {4 \cos (c+d x)+3}}{10 d}-\frac {\sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{10 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2661
Rule 2752
Rule 2793
Rule 3023
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx &=\frac {\cos (c+d x) \sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{10 d}+\frac {1}{10} \int \frac {3+6 \cos (c+d x)-6 \cos ^2(c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx\\ &=-\frac {\sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{10 d}+\frac {\cos (c+d x) \sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{10 d}+\frac {1}{60} \int \frac {6+54 \cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx\\ &=-\frac {\sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{10 d}+\frac {\cos (c+d x) \sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{10 d}+\frac {9}{40} \int \sqrt {3+4 \cos (c+d x)} \, dx-\frac {23}{40} \int \frac {1}{\sqrt {3+4 \cos (c+d x)}} \, dx\\ &=\frac {9 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{20 d}-\frac {23 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{20 \sqrt {7} d}-\frac {\sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{10 d}+\frac {\cos (c+d x) \sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 81, normalized size = 0.73 \[ \frac {-23 \sqrt {7} F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )+63 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )+7 (\sin (2 (c+d x))-2 \sin (c+d x)) \sqrt {4 \cos (c+d x)+3}}{140 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (d x + c\right )^{3}}{\sqrt {4 \, \cos \left (d x + c\right ) + 3}}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{3}}{\sqrt {4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.76, size = 231, normalized size = 2.08 \[ -\frac {\sqrt {\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-64 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-23 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )-9 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )\right )}{20 \sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{3}}{\sqrt {4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3}{\sqrt {4\,\cos \left (c+d\,x\right )+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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