Optimal. Leaf size=140 \[ -\frac {47 E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{20 \sqrt {7} d}-\frac {59 F\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{60 \sqrt {7} d}+\frac {59 \sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{105 d}-\frac {3 (3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{70 d}-\frac {(3-4 \cos (c+d x))^{3/2} \cos (c+d x) \sin (c+d x)}{14 d} \]
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Rubi [A]
time = 0.13, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2872, 3102,
2832, 2831, 2741, 2733} \begin {gather*} -\frac {59 F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{60 \sqrt {7} d}-\frac {47 E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{20 \sqrt {7} d}-\frac {\sin (c+d x) \cos (c+d x) (3-4 \cos (c+d x))^{3/2}}{14 d}-\frac {3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{70 d}+\frac {59 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{105 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2733
Rule 2741
Rule 2831
Rule 2832
Rule 2872
Rule 3102
Rubi steps
\begin {align*} \int \sqrt {3-4 \cos (c+d x)} \cos ^3(c+d x) \, dx &=-\frac {(3-4 \cos (c+d x))^{3/2} \cos (c+d x) \sin (c+d x)}{14 d}-\frac {1}{14} \int \sqrt {3-4 \cos (c+d x)} \left (3-10 \cos (c+d x)-6 \cos ^2(c+d x)\right ) \, dx\\ &=-\frac {3 (3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{70 d}-\frac {(3-4 \cos (c+d x))^{3/2} \cos (c+d x) \sin (c+d x)}{14 d}+\frac {1}{140} \int \sqrt {3-4 \cos (c+d x)} (6+118 \cos (c+d x)) \, dx\\ &=\frac {59 \sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{105 d}-\frac {3 (3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{70 d}-\frac {(3-4 \cos (c+d x))^{3/2} \cos (c+d x) \sin (c+d x)}{14 d}+\frac {1}{210} \int \frac {-209+141 \cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=\frac {59 \sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{105 d}-\frac {3 (3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{70 d}-\frac {(3-4 \cos (c+d x))^{3/2} \cos (c+d x) \sin (c+d x)}{14 d}-\frac {47}{280} \int \sqrt {3-4 \cos (c+d x)} \, dx-\frac {59}{120} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=-\frac {47 E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{20 \sqrt {7} d}-\frac {59 F\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{60 \sqrt {7} d}+\frac {59 \sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{105 d}-\frac {3 (3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{70 d}-\frac {(3-4 \cos (c+d x))^{3/2} \cos (c+d x) \sin (c+d x)}{14 d}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 114, normalized size = 0.81 \begin {gather*} \frac {141 \sqrt {-3+4 \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |8\right )-413 \sqrt {-3+4 \cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |8\right )+654 \sin (c+d x)-511 \sin (2 (c+d x))+108 \sin (3 (c+d x))-60 \sin (4 (c+d x))}{420 d \sqrt {3-4 \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 276, normalized size = 1.97
method | result | size |
default | \(\frac {\sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (7680 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8064 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5432 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-568 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+59 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )+141 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )\right )}{420 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d}\) | \(276\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.12, size = 148, normalized size = 1.06 \begin {gather*} \frac {4 \, {\left (60 \, \cos \left (d x + c\right )^{2} - 9 \, \cos \left (d x + c\right ) + 91\right )} \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sin \left (d x + c\right ) + 277 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + 277 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + 282 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right ) + 282 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right )}{840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\cos \left (c+d\,x\right )}^3\,\sqrt {3-4\,\cos \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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