Optimal. Leaf size=98 \[ \frac {3 F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{d}+\frac {4 \Pi \left (2;\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d}+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{d} \]
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Rubi [A] time = 0.25, antiderivative size = 98, 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 = {2796, 3060, 2654, 3002, 2662, 2806} \[ \frac {3 F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{d}+\frac {4 \Pi \left (2;\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d}+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2654
Rule 2662
Rule 2796
Rule 2806
Rule 3002
Rule 3060
Rubi steps
\begin {align*} \int \sqrt {3-4 \cos (c+d x)} \sec ^2(c+d x) \, dx &=\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{d}+\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)}{d}+\frac {1}{4} \int \frac {(-8+6 \cos (c+d x)) \sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx-\frac {1}{2} \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 )}{d}+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{d}+\frac {3}{2} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx-2 \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 )}{d}+\frac {3 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 )}{\sqrt {7} d}+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [C] time = 1.47, size = 178, normalized size = 1.82 \[ \frac {21 \sqrt {3-4 \cos (c+d x)} \tan (c+d x)-\frac {42 \sqrt {4 \cos (c+d x)-3} \Pi \left (2;\left .\frac {1}{2} (c+d x)\right |8\right )}{\sqrt {3-4 \cos (c+d x)}}-\frac {i \sqrt {7} \sin (c+d x) \left (-12 F\left (i \sinh ^{-1}\left (\sqrt {3-4 \cos (c+d x)}\right )|-\frac {1}{7}\right )+21 E\left (i \sinh ^{-1}\left (\sqrt {3-4 \cos (c+d x)}\right )|-\frac {1}{7}\right )-8 \Pi \left (-\frac {1}{3};i \sinh ^{-1}\left (\sqrt {3-4 \cos (c+d x)}\right )|-\frac {1}{7}\right )\right )}{\sqrt {\sin ^2(c+d x)}}}{21 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{2}, x\right ) \]
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 \[ \int \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.88, size = 351, normalized size = 3.58 \[ -\frac {\sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-\frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}+\frac {3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )}{7 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}-\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )}{\sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}+\frac {4 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, \frac {2 \sqrt {14}}{7}\right )}{7 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d} \]
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 \[ \int \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int \frac {\sqrt {3-4\,\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^2} \,d x \]
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 \[ \int \sqrt {3 - 4 \cos {\left (c + d x \right )}} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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