Integrand size = 23, antiderivative size = 68 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^2 \, dx=\frac {2 \left (a^2-b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {4 a b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {2 b^2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
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Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4349, 3873, 3856, 2720, 4131, 2719} \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^2 \, dx=\frac {2 \left (a^2-b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {4 a b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {2 b^2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
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Rule 2719
Rule 2720
Rule 3856
Rule 3873
Rule 4131
Rule 4349
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \sec (c+d x))^2}{\sqrt {\sec (c+d x)}} \, dx \\ & = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {a^2+b^2 \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx+\left (2 a b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {2 b^2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+(2 a b) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\left (\left (a^2-b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {4 a b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {2 b^2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\left (a^2-b^2\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {2 \left (a^2-b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {4 a b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {2 b^2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^2 \, dx=\frac {2 \left (\left (a^2-b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+b \left (2 a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {b \sin (c+d x)}{\sqrt {\cos (c+d x)}}\right )\right )}{d} \]
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Time = 8.26 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.97
method | result | size |
default | \(\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b^{2}-4 \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, a b +2 \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, a^{2}-2 \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, b^{2}}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(202\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.62 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^2 \, dx=\frac {-2 i \, \sqrt {2} a b \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 2 i \, \sqrt {2} a b \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, b^{2} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (i \, a^{2} - i \, b^{2}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + \sqrt {2} {\left (-i \, a^{2} + i \, b^{2}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{d \cos \left (d x + c\right )} \]
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\[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^2 \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{2} \sqrt {\cos {\left (c + d x \right )}}\, dx \]
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\[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^2 \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sqrt {\cos \left (d x + c\right )} \,d x } \]
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\[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^2 \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sqrt {\cos \left (d x + c\right )} \,d x } \]
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Time = 14.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.19 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^2 \, dx=\frac {2\,a^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {4\,a\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,b^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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