Optimal. Leaf size=50 \[ -\frac {8 F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d}-\frac {6 \Pi \left (2;\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d} \]
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Rubi [A] time = 0.09, antiderivative size = 50, 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 = {2803, 2662, 2806} \[ -\frac {8 F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d}-\frac {6 \Pi \left (2;\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d} \]
Antiderivative was successfully verified.
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Rule 2662
Rule 2803
Rule 2806
Rubi steps
\begin {align*} \int \sqrt {3-4 \cos (c+d x)} \sec (c+d x) \, dx &=3 \int \frac {\sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx-4 \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=-\frac {8 F\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{\sqrt {7} d}-\frac {6 \Pi \left (2;\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{\sqrt {7} d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 61, normalized size = 1.22 \[ \frac {2 \sqrt {4 \cos (c+d x)-3} \left (3 \Pi \left (2;\left .\frac {1}{2} (c+d x)\right |8\right )-4 F\left (\left .\frac {1}{2} (c+d x)\right |8\right )\right )}{d \sqrt {3-4 \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right ), x\right ) \]
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 \[ \int \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.80, size = 159, normalized size = 3.18 \[ \frac {2 \sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \left (4 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )+3 \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, \frac {2 \sqrt {14}}{7}\right )\right )}{7 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d} \]
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 \[ \int \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {3-4\,\cos \left (c+d\,x\right )}}{\cos \left (c+d\,x\right )} \,d x \]
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 \[ \int \sqrt {3 - 4 \cos {\left (c + d x \right )}} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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