Optimal. Leaf size=137 \[ -\frac {F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{3 \sqrt {7} d}+\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{3 d}+\frac {\sqrt {7} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{3 d}-\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{6 d} \]
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Rubi [A] time = 0.37, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2802, 3055, 3059, 2653, 3002, 2661, 2805} \[ -\frac {F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{3 \sqrt {7} d}+\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{3 d}+\frac {\sqrt {7} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{3 d}-\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2661
Rule 2802
Rule 2805
Rule 3002
Rule 3055
Rule 3059
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx &=\frac {\sqrt {3+4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{6 d}+\frac {1}{6} \int \frac {\left (-6+3 \cos (c+d x)+2 \cos ^2(c+d x)\right ) \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 {\sqrt {3+4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{6 d}+\frac {1}{18} \int \frac {\left (21+6 \cos (c+d x)+12 \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 {\sqrt {3+4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{6 d}-\frac {1}{72} \int \frac {(-84+12 \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+d x)|\frac {8}{7}\right )}{3 d}-\frac {\sqrt {3+4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3+4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{6 d}-\frac {1}{6} \int \frac {1}{\sqrt {3+4 \cos (c+d x)}} \, dx+\frac {7}{6} \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 )}{3 d}-\frac {F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{3 \sqrt {7} d}+\frac {\sqrt {7} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{3 d}-\frac {\sqrt {3+4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3+4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{6 d}\\ \end {align*}
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Mathematica [C] time = 1.32, size = 195, normalized size = 1.42 \[ \frac {\frac {4 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7}}+\frac {18 \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7}}-(2 \cos (c+d x)-1) \sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)-\frac {2 i \sin (c+d x) \left (-12 F\left (i \sinh ^{-1}\left (\sqrt {4 \cos (c+d x)+3}\right )|-\frac {1}{7}\right )+21 E\left (i \sinh ^{-1}\left (\sqrt {4 \cos (c+d x)+3}\right )|-\frac {1}{7}\right )-8 \Pi \left (-\frac {1}{3};i \sinh ^{-1}\left (\sqrt {4 \cos (c+d x)+3}\right )|-\frac {1}{7}\right )\right )}{3 \sqrt {7} \sqrt {\sin ^2(c+d x)}}}{6 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.28, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (d x + c\right )^{3}}{\sqrt {4 \, \cos \left (d x + c\right ) + 3}}, x\right ) \]
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 \[ \int \frac {\sec \left (d x + c\right )^{3}}{\sqrt {4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.84, size = 408, normalized size = 2.98 \[ -\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 {\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 )}}{3 \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )^{2}}+\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 )}}{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 {-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 )}{3 \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 )}{3 \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 {7 \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 )}{3 \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 \[ \int \frac {\sec \left (d x + c\right )^{3}}{\sqrt {4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^3\,\sqrt {4\,\cos \left (c+d\,x\right )+3}} \,d x \]
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 \[ \int \frac {\sec ^{3}{\left (c + d x \right )}}{\sqrt {4 \cos {\left (c + d x \right )} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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