Integrand size = 43, antiderivative size = 343 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a^3 (264 A+253 B+210 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d} \]
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Time = 0.79 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4173, 4103, 4082, 3872, 3853, 3856, 2719, 2720} \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {4 a^3 (264 A+253 B+210 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{1155 d}+\frac {4 a^3 (143 A+121 B+105 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{231 d}+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{693 d}+\frac {4 a^3 (21 A+17 B+15 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}-\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (11 B+6 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{99 a d}+\frac {2 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^3}{11 d} \]
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 3872
Rule 4082
Rule 4103
Rule 4173
Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (\frac {1}{2} a (11 A+3 C)+\frac {1}{2} a (11 B+6 C) \sec (c+d x)\right ) \, dx}{11 a} \\ & = \frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {4 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (\frac {3}{4} a^2 (33 A+11 B+15 C)+\frac {1}{4} a^2 (99 A+143 B+105 C) \sec (c+d x)\right ) \, dx}{99 a} \\ & = \frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d}+\frac {8 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x)) \left (\frac {15}{4} a^3 (33 A+22 B+21 C)+\frac {3}{4} a^3 (264 A+253 B+210 C) \sec (c+d x)\right ) \, dx}{693 a} \\ & = \frac {4 a^3 (264 A+253 B+210 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d}+\frac {16 \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {231}{8} a^4 (21 A+17 B+15 C)+\frac {45}{8} a^4 (143 A+121 B+105 C) \sec (c+d x)\right ) \, dx}{3465 a} \\ & = \frac {4 a^3 (264 A+253 B+210 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d}+\frac {1}{15} \left (2 a^3 (21 A+17 B+15 C)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{77} \left (2 a^3 (143 A+121 B+105 C)\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx \\ & = \frac {4 a^3 (21 A+17 B+15 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a^3 (264 A+253 B+210 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d}-\frac {1}{15} \left (2 a^3 (21 A+17 B+15 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{231} \left (2 a^3 (143 A+121 B+105 C)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {4 a^3 (21 A+17 B+15 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a^3 (264 A+253 B+210 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d}-\frac {1}{15} \left (2 a^3 (21 A+17 B+15 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (2 a^3 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a^3 (264 A+253 B+210 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{693 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.58 (sec) , antiderivative size = 1324, normalized size of antiderivative = 3.86 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {7 A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{15 \sqrt {2} d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {17 B e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{45 \sqrt {2} d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{3 \sqrt {2} d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {13 A \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{21 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {11 B \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{21 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {5 C \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{11 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {\sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {(21 A+17 B+15 C) \cos (d x) \csc (c)}{15 d}+\frac {C \sec (c) \sec ^5(c+d x) \sin (d x)}{22 d}+\frac {\sec (c) \sec ^4(c+d x) (9 C \sin (c)+11 B \sin (d x)+33 C \sin (d x))}{198 d}+\frac {\sec (c) \sec ^3(c+d x) (77 B \sin (c)+231 C \sin (c)+99 A \sin (d x)+297 B \sin (d x)+378 C \sin (d x))}{1386 d}+\frac {\sec (c) \sec ^2(c+d x) (495 A \sin (c)+1485 B \sin (c)+1890 C \sin (c)+2079 A \sin (d x)+2618 B \sin (d x)+2310 C \sin (d x))}{6930 d}+\frac {\sec (c) \sec (c+d x) (2079 A \sin (c)+2618 B \sin (c)+2310 C \sin (c)+4290 A \sin (d x)+3630 B \sin (d x)+3150 C \sin (d x))}{6930 d}+\frac {(143 A+121 B+105 C) \tan (c)}{231 d}\right )}{(A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1399\) vs. \(2(359)=718\).
Time = 10.52 (sec) , antiderivative size = 1400, normalized size of antiderivative = 4.08
method | result | size |
default | \(\text {Expression too large to display}\) | \(1400\) |
parts | \(\text {Expression too large to display}\) | \(1739\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.95 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (143 \, A + 121 \, B + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (143 \, A + 121 \, B + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 i \, \sqrt {2} {\left (21 \, A + 17 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (21 \, A + 17 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (462 \, {\left (21 \, A + 17 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 30 \, {\left (143 \, A + 121 \, B + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 77 \, {\left (27 \, A + 34 \, B + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 45 \, {\left (11 \, A + 33 \, B + 42 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 385 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 315 \, C a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{3465 \, d \cos \left (d x + c\right )^{5}} \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
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