Integrand size = 33, antiderivative size = 221 \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x) \, dx=-\frac {2 \left (3 a^2 A+5 b (A b+2 a B)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (2 a A b+a^2 B+3 b^2 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 d}+\frac {2 \left (3 a^2 A+5 b (A b+2 a B)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (7 A b+5 a B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{5 d} \]
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Time = 0.42 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3039, 4111, 4132, 3853, 3856, 2719, 4131, 2720} \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x) \, dx=\frac {2 \left (a^2 B+2 a A b+3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \left (3 a^2 A+5 b (2 a B+A b)\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}-\frac {2 \left (3 a^2 A+5 b (2 a B+A b)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a (5 a B+7 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)}{5 d} \]
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Rule 2719
Rule 2720
Rule 3039
Rule 3853
Rule 3856
Rule 4111
Rule 4131
Rule 4132
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\sec (c+d x)} (b+a \sec (c+d x))^2 (B+A \sec (c+d x)) \, dx \\ & = \frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{5 d}+\frac {2}{5} \int \sqrt {\sec (c+d x)} \left (\frac {1}{2} b (a A+5 b B)+\frac {1}{2} \left (3 a^2 A+5 b (A b+2 a B)\right ) \sec (c+d x)+\frac {1}{2} a (7 A b+5 a B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{5 d}+\frac {2}{5} \int \sqrt {\sec (c+d x)} \left (\frac {1}{2} b (a A+5 b B)+\frac {1}{2} a (7 A b+5 a B) \sec ^2(c+d x)\right ) \, dx+\frac {1}{5} \left (3 a^2 A+5 b (A b+2 a B)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 \left (3 a^2 A+5 b (A b+2 a B)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (7 A b+5 a B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{5 d}+\frac {1}{3} \left (2 a A b+a^2 B+3 b^2 B\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{5} \left (-3 a^2 A-5 b (A b+2 a B)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {2 \left (3 a^2 A+5 b (A b+2 a B)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (7 A b+5 a B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{5 d}+\frac {1}{3} \left (\left (2 a A b+a^2 B+3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{5} \left (\left (-3 a^2 A-5 b (A b+2 a B)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {2 \left (3 a^2 A+5 b (A b+2 a 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 (2 a A b+a^2 B+3 b^2 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 d}+\frac {2 \left (3 a^2 A+5 b (A b+2 a B)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (7 A b+5 a B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{5 d} \\ \end{align*}
Time = 2.94 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.77 \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x) \, dx=\frac {\sec ^{\frac {5}{2}}(c+d x) \left (-12 \left (3 a^2 A+5 A b^2+10 a b B\right ) \cos ^{\frac {5}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+20 \left (2 a A b+a^2 B+3 b^2 B\right ) \cos ^{\frac {5}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (15 \left (a^2 A+A b^2+2 a b B\right )+10 a (2 A b+a B) \cos (c+d x)+3 \left (3 a^2 A+5 A b^2+10 a b B\right ) \cos (2 (c+d x))\right ) \sin (c+d x)\right )}{30 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(722\) vs. \(2(249)=498\).
Time = 490.58 (sec) , antiderivative size = 723, normalized size of antiderivative = 3.27
method | result | size |
default | \(\text {Expression too large to display}\) | \(723\) |
parts | \(\text {Expression too large to display}\) | \(914\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.29 \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x) \, dx=-\frac {5 \, \sqrt {2} {\left (i \, B a^{2} + 2 i \, A a b + 3 i \, B b^{2}\right )} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, B a^{2} - 2 i \, A a b - 3 i \, B b^{2}\right )} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 3 \, \sqrt {2} {\left (3 i \, A a^{2} + 10 i \, B a b + 5 i \, A b^{2}\right )} \cos \left (d x + c\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (-3 i \, A a^{2} - 10 i \, B a b - 5 i \, A b^{2}\right )} \cos \left (d x + c\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (3 \, A a^{2} + 3 \, {\left (3 \, A a^{2} + 10 \, B a b + 5 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{15 \, d \cos \left (d x + c\right )^{2}} \]
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Timed out. \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
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\[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x) \, dx=\int \left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2 \,d x \]
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