Integrand size = 23, antiderivative size = 138 \[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\frac {\sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{3 d}-\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{\sqrt {7} d}-\frac {5 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{3 \sqrt {7} d}-\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3-4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.44 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2875, 3134, 3138, 2733, 3081, 2741, 2885} \[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=-\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{\sqrt {7} d}+\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{3 d}-\frac {5 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{3 \sqrt {7} d}-\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rule 2733
Rule 2741
Rule 2875
Rule 2885
Rule 3081
Rule 3134
Rule 3138
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {3-4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int \frac {\left (-2+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)}{2 d}+\frac {1}{6} \int \frac {\left (5-6 \cos (c+d x)-4 \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)}{2 d}+\frac {1}{24} \int \frac {(20-36 \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 {\sqrt {3-4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d}+\frac {5}{6} \int \frac {\sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx-\frac {3}{2} \int \frac {1}{\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 {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{\sqrt {7} d}-\frac {5 \operatorname {EllipticPi}\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}+\frac {\sqrt {3-4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.42 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.72 \[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\frac {-\frac {12 \sqrt {-3+4 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),8\right )}{\sqrt {3-4 \cos (c+d x)}}+\frac {6 \sqrt {-3+4 \cos (c+d x)} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),8\right )}{\sqrt {3-4 \cos (c+d x)}}+\frac {2 i \left (21 E\left (i \text {arcsinh}\left (\sqrt {3-4 \cos (c+d x)}\right )|-\frac {1}{7}\right )-12 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {3-4 \cos (c+d x)}\right ),-\frac {1}{7}\right )-8 \operatorname {EllipticPi}\left (-\frac {1}{3},i \text {arcsinh}\left (\sqrt {3-4 \cos (c+d x)}\right ),-\frac {1}{7}\right )\right ) \sin (c+d x)}{3 \sqrt {7} \sqrt {\sin ^2(c+d x)}}-\sqrt {3-4 \cos (c+d x)} (-3+2 \cos (c+d x)) \sec (c+d x) \tan (c+d x)}{6 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(195)=390\).
Time = 5.86 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.96
method | result | size |
default | \(-\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 {\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 )}}{{\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 )-\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 {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}\, F\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}\, E\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 {5 \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}\, \Pi \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}\) | \(408\) |
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\[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{3} \,d x } \]
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\[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int \sqrt {3 - 4 \cos {\left (c + d x \right )}} \sec ^{3}{\left (c + d x \right )}\, dx \]
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\[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{3} \,d x } \]
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\[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int \frac {\sqrt {3-4\,\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^3} \,d x \]
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