Integrand size = 45, antiderivative size = 455 \[ \int \frac {(a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \left (a^2-b^2\right ) \left (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{315 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 (A b+3 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
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Time = 1.98 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4179, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {(a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{315 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 3941
Rule 3943
Rule 4120
Rule 4179
Rule 4189
Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2}{9} \int \frac {\sqrt {a+b \sec (c+d x)} \left (\frac {3}{2} (A b+3 a B)+\frac {1}{2} (7 a A+9 b B+9 a C) \sec (c+d x)+\frac {1}{2} b (4 A+9 C) \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 (A b+3 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4}{63} \int \frac {\frac {1}{4} \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right )+\frac {1}{4} \left (92 a A b+45 a^2 B+63 b^2 B+126 a b C\right ) \sec (c+d x)+\frac {1}{4} b (40 A b+36 a B+63 b C) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {2 (A b+3 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {8 \int \frac {\frac {3}{8} \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right )-\frac {1}{8} a \left (396 a b B+21 a^2 (7 A+9 C)+b^2 (209 A+315 C)\right ) \sec (c+d x)-\frac {1}{4} b \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{315 a} \\ & = \frac {2 (A b+3 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {16 \int \frac {\frac {3}{16} \left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right )+\frac {3}{16} a \left (2 A b^3+75 a^3 B+153 a b^2 B+6 a^2 b (31 A+42 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{945 a^2} \\ & = \frac {2 (A b+3 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{315 a^3}+\frac {\left (\left (a^2-b^2\right ) \left (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b C)\right )\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 a^3} \\ & = \frac {2 (A b+3 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (\left (a^2-b^2\right ) \left (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b C)\right ) \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{315 a^3 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{315 a^3 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ & = \frac {2 (A b+3 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (\left (a^2-b^2\right ) \left (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{315 a^3 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{315 a^3 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \\ & = \frac {2 \left (a^2-b^2\right ) \left (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{315 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 (A b+3 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 13.97 (sec) , antiderivative size = 5997, normalized size of antiderivative = 13.18 \[ \int \frac {(a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(8873\) vs. \(2(473)=946\).
Time = 23.37 (sec) , antiderivative size = 8874, normalized size of antiderivative = 19.50
method | result | size |
parts | \(\text {Expression too large to display}\) | \(8874\) |
default | \(\text {Expression too large to display}\) | \(8932\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {\sqrt {2} {\left (-225 i \, B a^{5} - 6 i \, {\left (44 \, A + 63 \, C\right )} a^{4} b + 33 i \, B a^{3} b^{2} + 6 i \, {\left (10 \, A + 21 \, C\right )} a^{2} b^{3} - 36 i \, B a b^{4} + 16 i \, A b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (225 i \, B a^{5} + 6 i \, {\left (44 \, A + 63 \, C\right )} a^{4} b - 33 i \, B a^{3} b^{2} - 6 i \, {\left (10 \, A + 21 \, C\right )} a^{2} b^{3} + 36 i \, B a b^{4} - 16 i \, A b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (-21 i \, {\left (7 \, A + 9 \, C\right )} a^{5} - 246 i \, B a^{4} b - 3 i \, {\left (11 \, A + 21 \, C\right )} a^{3} b^{2} + 18 i \, B a^{2} b^{3} - 8 i \, A a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (21 i \, {\left (7 \, A + 9 \, C\right )} a^{5} + 246 i \, B a^{4} b + 3 i \, {\left (11 \, A + 21 \, C\right )} a^{3} b^{2} - 18 i \, B a^{2} b^{3} + 8 i \, A a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) + \frac {6 \, {\left (35 \, A a^{5} \cos \left (d x + c\right )^{4} + 5 \, {\left (9 \, B a^{5} + 10 \, A a^{4} b\right )} \cos \left (d x + c\right )^{3} + {\left (7 \, {\left (7 \, A + 9 \, C\right )} a^{5} + 72 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (75 \, B a^{5} + 2 \, {\left (44 \, A + 63 \, C\right )} a^{4} b + 9 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{945 \, a^{4} d} \]
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Timed out. \[ \int \frac {(a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {(a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]
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