Integrand size = 23, antiderivative size = 135 \[ \int \sqrt {3+4 \cos (c+d x)} \sec ^3(c+d x) \, dx=-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{3 d}+\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}+\frac {5 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {8}{7}\right )}{3 \sqrt {7} d}+\frac {\sqrt {3+4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3+4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.42 (sec) , antiderivative size = 135, 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, 2732, 3081, 2740, 2884} \[ \int \sqrt {3+4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{3 d}+\frac {5 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {8}{7}\right )}{3 \sqrt {7} d}+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rule 2732
Rule 2740
Rule 2875
Rule 2884
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+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+d x)|\frac {8}{7}\right )}{3 d}+\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}+\frac {5 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+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 = 0.90 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.44 \[ \int \sqrt {3+4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\frac {\frac {12 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7}}+\frac {6 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7}}+\frac {2 i \left (21 E\left (i \text {arcsinh}\left (\sqrt {3+4 \cos (c+d x)}\right )|-\frac {1}{7}\right )-12 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {3+4 \cos (c+d x)}\right ),-\frac {1}{7}\right )-8 \operatorname {EllipticPi}\left (-\frac {1}{3},i \text {arcsinh}\left (\sqrt {3+4 \cos (c+d x)}\right ),-\frac {1}{7}\right )\right ) \sin (c+d x)}{3 \sqrt {7} \sqrt {\sin ^2(c+d x)}}+(3+2 \cos (c+d x)) \sqrt {3+4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{6 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(195)=390\).
Time = 4.08 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.02
method | result | size |
default | \(-\frac {\sqrt {-\left (1-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\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 )}}{{\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 {3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, F\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 {1-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, E\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 {5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \Pi \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}\) | \(408\) |
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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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\[ \int \sqrt {3+4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int \sqrt {4 \cos {\left (c + d x \right )} + 3} \sec ^{3}{\left (c + d x \right )}\, dx \]
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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. \[ \int \sqrt {3+4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int \frac {\sqrt {4\,\cos \left (c+d\,x\right )+3}}{{\cos \left (c+d\,x\right )}^3} \,d x \]
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