Optimal. Leaf size=80 \[ -\frac {\sqrt {7} F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{6 d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d} \]
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Rubi [A] time = 0.08, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2753, 2752, 2662, 2654} \[ -\frac {\sqrt {7} F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{6 d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2654
Rule 2662
Rule 2752
Rule 2753
Rubi steps
\begin {align*} \int \sqrt {3-4 \cos (c+d x)} \cos (c+d x) \, dx &=\frac {2 \sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2}{3} \int \frac {-2+\frac {3}{2} \cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=\frac {2 \sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{3 d}-\frac {1}{4} \int \sqrt {3-4 \cos (c+d x)} \, dx-\frac {7}{12} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=-\frac {\sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{2 d}-\frac {\sqrt {7} F\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{6 d}+\frac {2 \sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 94, normalized size = 1.18 \[ \frac {12 \sin (c+d x)-8 \sin (2 (c+d x))-7 \sqrt {4 \cos (c+d x)-3} F\left (\left .\frac {1}{2} (c+d x)\right |8\right )+3 \sqrt {4 \cos (c+d x)-3} E\left (\left .\frac {1}{2} (c+d x)\right |8\right )}{6 d \sqrt {3-4 \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right ), x\right ) \]
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 \[ \int \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.74, size = 231, normalized size = 2.89 \[ \frac {\sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (64 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )-8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d} \]
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 \[ \int \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )\,{d x} \]
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 \[ \int \cos \left (c+d\,x\right )\,\sqrt {3-4\,\cos \left (c+d\,x\right )} \,d x \]
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 \[ \int \sqrt {3 - 4 \cos {\left (c + d x \right )}} \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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