Optimal. Leaf size=95 \[ \frac {3 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d}+\frac {4 \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d}+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{d} \]
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Rubi [A] time = 0.25, antiderivative size = 95, 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, 2653, 3002, 2661, 2805} \[ \frac {3 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d}+\frac {4 \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d}+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2661
Rule 2796
Rule 2805
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+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+d x)|\frac {8}{7}\right )}{d}+\frac {3 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d}+\frac {4 \Pi \left (2;\frac {1}{2} (c+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 = 1.12, size = 157, normalized size = 1.65 \[ \frac {6 \sqrt {7} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )+21 \sqrt {4 \cos (c+d x)+3} \tan (c+d x)+\frac {i \sqrt {7} \sin (c+d x) \left (-12 F\left (i \sinh ^{-1}\left (\sqrt {4 \cos (c+d x)+3}\right )|-\frac {1}{7}\right )+21 E\left (i \sinh ^{-1}\left (\sqrt {4 \cos (c+d x)+3}\right )|-\frac {1}{7}\right )-8 \Pi \left (-\frac {1}{3};i \sinh ^{-1}\left (\sqrt {4 \cos (c+d x)+3}\right )|-\frac {1}{7}\right )\right )}{\sqrt {\sin ^2(c+d x)}}}{21 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.59, 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 = 0.80, size = 350, normalized size = 3.68 \[ -\frac {\sqrt {-\left (-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-\frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}+\frac {3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )}{\sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}+\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )}{\sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}-\frac {4 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, 2 \sqrt {2}\right )}{\sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int \frac {\sqrt {4\,\cos \left (c+d\,x\right )+3}}{{\cos \left (c+d\,x\right )}^2} \,d x \]
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 \[ \int \sqrt {4 \cos {\left (c + d x \right )} + 3} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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