Integrand size = 23, antiderivative size = 112 \[ \int \frac {\sqrt {\cos (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\frac {4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d}-\frac {5 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d}-\frac {5 \sin (c+d x)}{3 a^2 d \sqrt {\cos (c+d x)} (1+\sec (c+d x))}-\frac {\sin (c+d x)}{3 d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2} \]
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Time = 0.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4349, 3902, 4105, 3872, 3856, 2719, 2720} \[ \int \frac {\sqrt {\cos (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=-\frac {5 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d}+\frac {4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d}-\frac {5 \sin (c+d x)}{3 a^2 d \sqrt {\cos (c+d x)} (\sec (c+d x)+1)}-\frac {\sin (c+d x)}{3 d \sqrt {\cos (c+d x)} (a \sec (c+d x)+a)^2} \]
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Rule 2719
Rule 2720
Rule 3856
Rule 3872
Rule 3902
Rule 4105
Rule 4349
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2} \, dx \\ & = -\frac {\sin (c+d x)}{3 d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {7 a}{2}+\frac {3}{2} a \sec (c+d x)}{\sqrt {\sec (c+d x)} (a+a \sec (c+d x))} \, dx}{3 a^2} \\ & = -\frac {5 \sin (c+d x)}{3 a^2 d \sqrt {\cos (c+d x)} (1+\sec (c+d x))}-\frac {\sin (c+d x)}{3 d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-6 a^2+\frac {5}{2} a^2 \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{3 a^4} \\ & = -\frac {5 \sin (c+d x)}{3 a^2 d \sqrt {\cos (c+d x)} (1+\sec (c+d x))}-\frac {\sin (c+d x)}{3 d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}-\frac {\left (5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{6 a^2}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{a^2} \\ & = -\frac {5 \sin (c+d x)}{3 a^2 d \sqrt {\cos (c+d x)} (1+\sec (c+d x))}-\frac {\sin (c+d x)}{3 d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}-\frac {5 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{6 a^2}+\frac {2 \int \sqrt {\cos (c+d x)} \, dx}{a^2} \\ & = \frac {4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d}-\frac {5 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d}-\frac {5 \sin (c+d x)}{3 a^2 d \sqrt {\cos (c+d x)} (1+\sec (c+d x))}-\frac {\sin (c+d x)}{3 d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.53 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {\cos (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\frac {\sqrt {\cos (c+d x)} \csc ^3(c+d x) \left (-6-46 \cos (c+d x)+14 \cos (2 (c+d x))+6 \cos (3 (c+d x))+20 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\cos ^2(c+d x)\right ) \sin ^2(c+d x)^{3/2}+64 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},\cos ^2(c+d x)\right ) \sin ^2(c+d x)^{3/2}\right )}{12 a^2 d} \]
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Time = 7.61 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.29
method | result | size |
default | \(\frac {\sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (24 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+10 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+24 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-38 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}{6 a^{2} \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(257\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.39 \[ \int \frac {\sqrt {\cos (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=-\frac {2 \, {\left (6 \, \cos \left (d x + c\right ) + 5\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 5 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 12 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 12 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
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\[ \int \frac {\sqrt {\cos (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\sqrt {\cos {\left (c + d x \right )}}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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\[ \int \frac {\sqrt {\cos (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\sqrt {\cos \left (d x + c\right )}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\sqrt {\cos (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\sqrt {\cos \left (d x + c\right )}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {\cos (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {\sqrt {\cos \left (c+d\,x\right )}}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2} \,d x \]
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