Optimal. Leaf size=53 \[ \frac {3 F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 \sqrt {7} d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2752, 2662, 2654} \[ \frac {3 F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 \sqrt {7} d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2654
Rule 2662
Rule 2752
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx &=-\left (\frac {1}{4} \int \sqrt {3-4 \cos (c+d x)} \, dx\right )+\frac {3}{4} \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 {3 F\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{2 \sqrt {7} d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 60, normalized size = 1.13 \[ \frac {\sqrt {4 \cos (c+d x)-3} \left (3 F\left (\left .\frac {1}{2} (c+d x)\right |8\right )+E\left (\left .\frac {1}{2} (c+d x)\right |8\right )\right )}{2 d \sqrt {3-4 \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )}{4 \, \cos \left (d x + c\right ) - 3}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.73, size = 158, normalized size = 2.98 \[ -\frac {\sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \left (3 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )-7 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )\right )}{14 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d} \]
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 \[ \int \frac {\cos \left (d x + c\right )}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 52, normalized size = 0.98 \[ \frac {\sqrt {4\,\cos \left (c+d\,x\right )-3}\,\left (\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |8\right )+3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |8\right )\right )}{2\,d\,\sqrt {3-4\,\cos \left (c+d\,x\right )}} \]
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 \[ \int \frac {\cos {\left (c + d x \right )}}{\sqrt {3 - 4 \cos {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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