Integrand size = 33, antiderivative size = 277 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=-\frac {4 a^3 (21 A+17 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (13 A+11 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (21 A+17 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (13 A+11 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {4 a^3 (24 A+23 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d} \]
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Time = 0.53 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4103, 4082, 3872, 3853, 3856, 2719, 2720} \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {4 a^3 (24 A+23 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{105 d}+\frac {4 a^3 (13 A+11 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 (9 A+13 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{63 d}+\frac {4 a^3 (21 A+17 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (13 A+11 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {4 a^3 (21 A+17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^2}{9 d} \]
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 3872
Rule 4082
Rule 4103
Rubi steps \begin{align*} \text {integral}& = \frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2}{9} \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (\frac {3}{2} a (3 A+B)+\frac {1}{2} a (9 A+13 B) \sec (c+d x)\right ) \, dx \\ & = \frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {4}{63} \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x)) \left (\frac {15}{2} a^2 (3 A+2 B)+\frac {3}{2} a^2 (24 A+23 B) \sec (c+d x)\right ) \, dx \\ & = \frac {4 a^3 (24 A+23 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {8}{315} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {21}{4} a^3 (21 A+17 B)+\frac {45}{4} a^3 (13 A+11 B) \sec (c+d x)\right ) \, dx \\ & = \frac {4 a^3 (24 A+23 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {1}{7} \left (2 a^3 (13 A+11 B)\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{15} \left (2 a^3 (21 A+17 B)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {4 a^3 (21 A+17 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (13 A+11 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {4 a^3 (24 A+23 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {1}{21} \left (2 a^3 (13 A+11 B)\right ) \int \sqrt {\sec (c+d x)} \, dx-\frac {1}{15} \left (2 a^3 (21 A+17 B)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {4 a^3 (21 A+17 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (13 A+11 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {4 a^3 (24 A+23 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {1}{21} \left (2 a^3 (13 A+11 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{15} \left (2 a^3 (21 A+17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {4 a^3 (21 A+17 B) \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 (13 A+11 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (21 A+17 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (13 A+11 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {4 a^3 (24 A+23 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.72 (sec) , antiderivative size = 793, normalized size of antiderivative = 2.86 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, 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 ^4(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 (A+B \sec (c+d x))}{30 \sqrt {2} d (B+A \cos (c+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 ^4(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 (A+B \sec (c+d x))}{90 \sqrt {2} d (B+A \cos (c+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 (A+B \sec (c+d x))}{42 d (B+A \cos (c+d x)) \sec ^{\frac {7}{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 (A+B \sec (c+d x))}{42 d (B+A \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)}+\frac {\sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \left (\frac {(21 A+17 B) \cos (d x) \csc (c)}{30 d}+\frac {B \sec (c) \sec ^4(c+d x) \sin (d x)}{36 d}+\frac {\sec (c) \sec ^3(c+d x) (7 B \sin (c)+9 A \sin (d x)+27 B \sin (d x))}{252 d}+\frac {\sec (c) \sec ^2(c+d x) (45 A \sin (c)+135 B \sin (c)+189 A \sin (d x)+238 B \sin (d x))}{1260 d}+\frac {\sec (c) \sec (c+d x) (189 A \sin (c)+238 B \sin (c)+390 A \sin (d x)+330 B \sin (d x))}{1260 d}+\frac {(13 A+11 B) \tan (c)}{42 d}\right )}{(B+A \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1152\) vs. \(2(297)=594\).
Time = 58.79 (sec) , antiderivative size = 1153, normalized size of antiderivative = 4.16
method | result | size |
default | \(\text {Expression too large to display}\) | \(1153\) |
parts | \(\text {Expression too large to display}\) | \(1421\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.02 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (13 \, A + 11 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (13 \, A + 11 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (21 \, A + 17 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (21 \, A + 17 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (42 \, {\left (21 \, A + 17 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} + 30 \, {\left (13 \, A + 11 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 7 \, {\left (27 \, A + 34 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 45 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right ) + 35 \, B a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{315 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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\[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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