Optimal. Leaf size=104 \[ \frac {F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{3 d}-\frac {4 \Pi \left (2;\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{3 \sqrt {7} d}+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d} \]
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Rubi [A] time = 0.25, antiderivative size = 104, 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 = {2802, 3060, 2654, 3002, 2662, 2806} \[ \frac {F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{3 d}-\frac {4 \Pi \left (2;\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{3 \sqrt {7} d}+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2654
Rule 2662
Rule 2802
Rule 2806
Rule 3002
Rule 3060
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx &=\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {1}{3} \int \frac {\left (2+2 \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {1}{12} \int \frac {(8+6 \cos (c+d x)) \sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx-\frac {1}{6} \int \sqrt {3-4 \cos (c+d x)} \, dx\\ &=-\frac {\sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{3 d}+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {1}{2} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx+\frac {2}{3} \int \frac {\sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=-\frac {\sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{3 d}+\frac {F\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{\sqrt {7} d}-\frac {4 \Pi \left (2;\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{3 \sqrt {7} d}+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\\ \end {align*}
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Mathematica [C] time = 1.52, size = 179, normalized size = 1.72 \[ \frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)+\frac {6 \sqrt {4 \cos (c+d x)-3} \Pi \left (2;\left .\frac {1}{2} (c+d x)\right |8\right )}{\sqrt {3-4 \cos (c+d x)}}-\frac {i \sin (c+d x) \left (-12 F\left (i \sinh ^{-1}\left (\sqrt {3-4 \cos (c+d x)}\right )|-\frac {1}{7}\right )+21 E\left (i \sinh ^{-1}\left (\sqrt {3-4 \cos (c+d x)}\right )|-\frac {1}{7}\right )-8 \Pi \left (-\frac {1}{3};i \sinh ^{-1}\left (\sqrt {3-4 \cos (c+d x)}\right )|-\frac {1}{7}\right )\right )}{3 \sqrt {7} \sqrt {\sin ^2(c+d x)}}}{3 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{2}}{4 \, \cos \left (d x + c\right ) - 3}, x\right ) \]
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 \[ \int \frac {\sec \left (d x + c\right )^{2}}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.83, size = 351, normalized size = 3.38 \[ -\frac {\sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-\frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{3 \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}+\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )}{7 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}-\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )}{3 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}-\frac {4 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, \frac {2 \sqrt {14}}{7}\right )}{21 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d} \]
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 \[ \int \frac {\sec \left (d x + c\right )^{2}}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^2\,\sqrt {3-4\,\cos \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\sec ^{2}{\left (c + d x \right )}}{\sqrt {3 - 4 \cos {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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