Integrand size = 33, antiderivative size = 263 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=-\frac {2 \left (5 a^2 A+3 A b^2+6 a b B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (5 b^2 B+7 a (2 A b+a B)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (5 a^2 A+3 A b^2+6 a b B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 \left (5 b^2 B+7 a (2 A b+a B)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 b (7 A b+9 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b B \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d} \]
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Time = 0.40 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4111, 4132, 3853, 3856, 2720, 4131, 2719} \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {2 \left (5 a^2 A+6 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}-\frac {2 \left (5 a^2 A+6 a b B+3 A b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (7 a (a B+2 A b)+5 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 \left (7 a (a B+2 A b)+5 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 b (9 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))}{7 d} \]
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 4111
Rule 4131
Rule 4132
Rubi steps \begin{align*} \text {integral}& = \frac {2 b B \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {2}{7} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} a (7 a A+3 b B)+\frac {1}{2} \left (5 b^2 B+7 a (2 A b+a B)\right ) \sec (c+d x)+\frac {1}{2} b (7 A b+9 a B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 b B \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {2}{7} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} a (7 a A+3 b B)+\frac {1}{2} b (7 A b+9 a B) \sec ^2(c+d x)\right ) \, dx+\frac {1}{7} \left (5 b^2 B+7 a (2 A b+a B)\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx \\ & = \frac {2 \left (5 b^2 B+7 a (2 A b+a B)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 b (7 A b+9 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b B \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {1}{5} \left (5 a^2 A+3 A b^2+6 a b B\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{21} \left (5 b^2 B+7 a (2 A b+a B)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {2 \left (5 a^2 A+3 A b^2+6 a b B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 \left (5 b^2 B+7 a (2 A b+a B)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 b (7 A b+9 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b B \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {1}{5} \left (-5 a^2 A-3 A b^2-6 a b B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (\left (5 b^2 B+7 a (2 A b+a B)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (5 b^2 B+7 a (2 A b+a B)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (5 a^2 A+3 A b^2+6 a b B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 \left (5 b^2 B+7 a (2 A b+a B)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 b (7 A b+9 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b B \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {1}{5} \left (\left (-5 a^2 A-3 A b^2-6 a b B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {2 \left (5 a^2 A+3 A b^2+6 a b B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (5 b^2 B+7 a (2 A b+a B)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (5 a^2 A+3 A b^2+6 a b B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 \left (5 b^2 B+7 a (2 A b+a B)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 b (7 A b+9 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b B \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d} \\ \end{align*}
Time = 6.25 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.84 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {\sec ^{\frac {7}{2}}(c+d x) \left (-168 \left (5 a^2 A+3 A b^2+6 a b B\right ) \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+40 \left (14 a A b+7 a^2 B+5 b^2 B\right ) \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (140 a A b+70 a^2 B+110 b^2 B+21 \left (15 a^2 A+13 A b^2+26 a b B\right ) \cos (c+d x)+10 \left (14 a A b+7 a^2 B+5 b^2 B\right ) \cos (2 (c+d x))+105 a^2 A \cos (3 (c+d x))+63 A b^2 \cos (3 (c+d x))+126 a b B \cos (3 (c+d x))\right ) \sin (c+d x)\right )}{420 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(831\) vs. \(2(287)=574\).
Time = 47.66 (sec) , antiderivative size = 832, normalized size of antiderivative = 3.16
method | result | size |
default | \(\text {Expression too large to display}\) | \(832\) |
parts | \(\text {Expression too large to display}\) | \(1024\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.19 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=-\frac {5 \, \sqrt {2} {\left (7 i \, B a^{2} + 14 i \, A a b + 5 i \, B b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-7 i \, B a^{2} - 14 i \, A a b - 5 i \, B b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (5 i \, A a^{2} + 6 i \, B a b + 3 i \, A b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-5 i \, A a^{2} - 6 i \, B a b - 3 i \, A b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (21 \, {\left (5 \, A a^{2} + 6 \, B a b + 3 \, A b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, B b^{2} + 5 \, {\left (7 \, B a^{2} + 14 \, A a b + 5 \, B b^{2}\right )} \cos \left (d x + c\right )^{2} + 21 \, {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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\[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^2\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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