Integrand size = 23, antiderivative size = 207 \[ \int \frac {1}{\cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {11 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{2 a^3 d}+\frac {11 \sin (c+d x)}{2 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {119 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )} \]
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Time = 0.49 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ \int \frac {1}{\cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\frac {11 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{2 a^3 d}+\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {11 \sin (c+d x)}{2 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {119 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {9}{2}}(c+d x) (a \sec (c+d x)+a)^3} \]
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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 {11}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx \\ & = -\frac {\sin (c+d x)}{5 d \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (\frac {7 a}{2}-\frac {13}{2} a \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {\sin (c+d x)}{5 d \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 \sin (c+d x)}{3 a 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 (25 a^2-\frac {69}{2} a^2 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4} \\ & = -\frac {\sin (c+d x)}{5 d \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {357 a^3}{4}-\frac {495}{4} a^3 \sec (c+d x)\right ) \, dx}{15 a^6} \\ & = -\frac {\sin (c+d x)}{5 d \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (119 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx}{20 a^3}+\frac {\left (33 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx}{4 a^3} \\ & = \frac {11 \sin (c+d x)}{2 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {119 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\left (11 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{4 a^3}+\frac {\left (119 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3} \\ & = \frac {11 \sin (c+d x)}{2 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {119 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}+\frac {11 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{4 a^3}+\frac {119 \int \sqrt {\cos (c+d x)} \, dx}{20 a^3} \\ & = \frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {11 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{2 a^3 d}+\frac {11 \sin (c+d x)}{2 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {119 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 2.92 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.94 \[ \int \frac {1}{\cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (-\frac {\left (5134 \cos \left (\frac {1}{2} (c-d x)\right )+4148 \cos \left (\frac {1}{2} (3 c+d x)\right )+4664 \cos \left (\frac {1}{2} (c+3 d x)\right )+2476 \cos \left (\frac {1}{2} (5 c+3 d x)\right )+3340 \cos \left (\frac {1}{2} (3 c+5 d x)\right )+944 \cos \left (\frac {1}{2} (7 c+5 d x)\right )+1620 \cos \left (\frac {1}{2} (5 c+7 d x)\right )+165 \cos \left (\frac {1}{2} (9 c+7 d x)\right )+357 \cos \left (\frac {1}{2} (7 c+9 d x)\right )\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )}{96 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {4 i \sqrt {2} e^{-i (c+d x)} \left (119 \left (1+e^{2 i (c+d x)}\right )+119 \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 i (c+d x)}\right )-55 e^{i (c+d x)} \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^3(c+d x)}{d \left (-1+e^{2 i c}\right ) \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{5 a^3 (1+\sec (c+d x))^3} \]
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Time = 13.14 (sec) , antiderivative size = 453, normalized size of antiderivative = 2.19
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}}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {32 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {118 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {128 \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 )}{5 \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 {238 \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 )}{5 \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 {4 \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}}+\frac {48 \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}}}\right )}{4 a^{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}\) | \(453\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=-\frac {2 \, {\left (357 \, \cos \left (d x + c\right )^{4} + 906 \, \cos \left (d x + c\right )^{3} + 695 \, \cos \left (d x + c\right )^{2} + 120 \, \cos \left (d x + c\right ) - 20\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 165 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{5} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{4} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + i \, \sqrt {2} \cos \left (d x + c\right )^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 165 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{5} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{4} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} - i \, \sqrt {2} \cos \left (d x + c\right )^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 357 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{5} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{4} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} - i \, \sqrt {2} \cos \left (d x + c\right )^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 357 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{5} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{4} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + i \, \sqrt {2} \cos \left (d x + c\right )^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2}\right )}} \]
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Timed out. \[ \int \frac {1}{\cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^{11/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]
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