Integrand size = 23, antiderivative size = 162 \[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx=\frac {7 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d}+\frac {10 \sin (c+d x)}{3 a^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {7 \sin (c+d x)}{a^2 d \sqrt {\cos (c+d x)}}-\frac {7 \sin (c+d x)}{3 a^2 d \cos ^{\frac {5}{2}}(c+d x) (1+\sec (c+d x))}-\frac {\sin (c+d x)}{3 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^2} \]
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Time = 0.38 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4349, 3901, 4104, 3872, 3853, 3856, 2719, 2720} \[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx=\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d}+\frac {7 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac {10 \sin (c+d x)}{3 a^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {7 \sin (c+d x)}{a^2 d \sqrt {\cos (c+d x)}}-\frac {7 \sin (c+d x)}{3 a^2 d \cos ^{\frac {5}{2}}(c+d x) (\sec (c+d x)+1)}-\frac {\sin (c+d x)}{3 d \cos ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^2} \]
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 3872
Rule 3901
Rule 4104
Rule 4349
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx \\ & = -\frac {\sin (c+d x)}{3 d \cos ^{\frac {7}{2}}(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 {\sec ^{\frac {5}{2}}(c+d x) \left (\frac {5 a}{2}-\frac {9}{2} a \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{3 a^2} \\ & = -\frac {7 \sin (c+d x)}{3 a^2 d \cos ^{\frac {5}{2}}(c+d x) (1+\sec (c+d x))}-\frac {\sin (c+d x)}{3 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {21 a^2}{2}-15 a^2 \sec (c+d x)\right ) \, dx}{3 a^4} \\ & = -\frac {7 \sin (c+d x)}{3 a^2 d \cos ^{\frac {5}{2}}(c+d x) (1+\sec (c+d x))}-\frac {\sin (c+d x)}{3 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {\left (7 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx}{2 a^2}+\frac {\left (5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx}{a^2} \\ & = \frac {10 \sin (c+d x)}{3 a^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {7 \sin (c+d x)}{a^2 d \sqrt {\cos (c+d x)}}-\frac {7 \sin (c+d x)}{3 a^2 d \cos ^{\frac {5}{2}}(c+d x) (1+\sec (c+d x))}-\frac {\sin (c+d x)}{3 d \cos ^{\frac {7}{2}}(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}{3 a^2}+\frac {\left (7 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{2 a^2} \\ & = \frac {10 \sin (c+d x)}{3 a^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {7 \sin (c+d x)}{a^2 d \sqrt {\cos (c+d x)}}-\frac {7 \sin (c+d x)}{3 a^2 d \cos ^{\frac {5}{2}}(c+d x) (1+\sec (c+d x))}-\frac {\sin (c+d x)}{3 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^2}+\frac {5 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 a^2}+\frac {7 \int \sqrt {\cos (c+d x)} \, dx}{2 a^2} \\ & = \frac {7 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d}+\frac {10 \sin (c+d x)}{3 a^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {7 \sin (c+d x)}{a^2 d \sqrt {\cos (c+d x)}}-\frac {7 \sin (c+d x)}{3 a^2 d \cos ^{\frac {5}{2}}(c+d x) (1+\sec (c+d x))}-\frac {\sin (c+d x)}{3 d \cos ^{\frac {7}{2}}(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 = 2.46 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.30 \[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx=\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (-\frac {\left (82 \cos \left (\frac {1}{2} (c-d x)\right )+65 \cos \left (\frac {1}{2} (3 c+d x)\right )+68 \cos \left (\frac {1}{2} (c+3 d x)\right )+37 \cos \left (\frac {1}{2} (5 c+3 d x)\right )+53 \cos \left (\frac {1}{2} (3 c+5 d x)\right )+10 \cos \left (\frac {1}{2} (7 c+5 d x)\right )+21 \cos \left (\frac {1}{2} (5 c+7 d x)\right )\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right )}{8 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 i \sqrt {2} e^{-i (c+d x)} \left (21 \left (1+e^{2 i (c+d x)}\right )+21 \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 i (c+d x)}\right )-10 e^{i (c+d x)} \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^2(c+d x)}{d \left (-1+e^{2 i c}\right ) \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{3 a^2 (1+\sec (c+d x))^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(412\) vs. \(2(198)=396\).
Time = 11.27 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.55
method | result | size |
default | \(-\frac {\sqrt {-\left (-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (\frac {\sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {6 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {22 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}+\frac {14 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \left (\operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-\operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{\sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}+\frac {16 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {-\left (-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}-\frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{3 \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {1}{2}\right )^{2}}\right )}{2 a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(413\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.09 \[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx=-\frac {2 \, {\left (21 \, \cos \left (d x + c\right )^{3} + 32 \, \cos \left (d x + c\right )^{2} + 8 \, \cos \left (d x + c\right ) - 2\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 10 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{4} + 2 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + i \, \sqrt {2} \cos \left (d x + c\right )^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 10 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{4} - 2 i \, \sqrt {2} \cos \left (d x + c\right )^{3} - i \, \sqrt {2} \cos \left (d x + c\right )^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{4} - 2 i \, \sqrt {2} \cos \left (d x + c\right )^{3} - i \, \sqrt {2} \cos \left (d x + c\right )^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{4} + 2 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + i \, \sqrt {2} \cos \left (d x + c\right )^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} + a^{2} d \cos \left (d x + c\right )^{2}\right )}} \]
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Timed out. \[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^{9/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2} \,d x \]
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