Integrand size = 33, antiderivative size = 295 \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {2 \left (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (5 a^3 A+21 a A b^2+21 a^2 b B+21 b^3 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a \left (5 a^2 A+18 A b^2+21 a b B\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (11 A b+7 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^2 \sin (c+d x)}{7 d} \]
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Time = 0.71 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3039, 4111, 4161, 4132, 3853, 3856, 2719, 4131, 2720} \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {2 a \left (5 a^2 A+21 a b B+18 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {2 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d} \]
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Rule 2719
Rule 2720
Rule 3039
Rule 3853
Rule 3856
Rule 4111
Rule 4131
Rule 4132
Rule 4161
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\sec (c+d x)} (b+a \sec (c+d x))^3 (B+A \sec (c+d x)) \, dx \\ & = \frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2}{7} \int \sqrt {\sec (c+d x)} (b+a \sec (c+d x)) \left (\frac {1}{2} b (a A+7 b B)+\frac {1}{2} \left (5 a^2 A+7 b (A b+2 a B)\right ) \sec (c+d x)+\frac {1}{2} a (11 A b+7 a B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 a^2 (11 A b+7 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {4}{35} \int \sqrt {\sec (c+d x)} \left (\frac {5}{4} b^2 (a A+7 b B)+\frac {7}{4} \left (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \sec (c+d x)+\frac {5}{4} a \left (5 a^2 A+18 A b^2+21 a b B\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 a^2 (11 A b+7 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {4}{35} \int \sqrt {\sec (c+d x)} \left (\frac {5}{4} b^2 (a A+7 b B)+\frac {5}{4} a \left (5 a^2 A+18 A b^2+21 a b B\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{5} \left (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 \left (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a \left (5 a^2 A+18 A b^2+21 a b B\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (11 A b+7 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{5} \left (-9 a^2 A b-5 A b^3-3 a^3 B-15 a b^2 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (5 a^3 A+21 a A b^2+21 a^2 b B+21 b^3 B\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {2 \left (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a \left (5 a^2 A+18 A b^2+21 a b B\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (11 A b+7 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{5} \left (\left (-9 a^2 A b-5 A b^3-3 a^3 B-15 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (\left (5 a^3 A+21 a A b^2+21 a^2 b B+21 b^3 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 (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 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 a^3 A+21 a A b^2+21 a^2 b B+21 b^3 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 (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a \left (5 a^2 A+18 A b^2+21 a b B\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (11 A b+7 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^2 \sin (c+d x)}{7 d} \\ \end{align*}
Time = 3.82 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.76 \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {2 \sqrt {\sec (c+d x)} \left (-21 \left (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 \left (5 a^3 A+21 a A b^2+21 a^2 b B+21 b^3 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+21 \left (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \sin (c+d x)+5 a \left (5 a^2 A+21 A b^2+21 a b B\right ) \tan (c+d x)+21 a^2 (3 A b+a B) \sec (c+d x) \tan (c+d x)+15 a^3 A \sec ^2(c+d x) \tan (c+d x)\right )}{105 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(916\) vs. \(2(319)=638\).
Time = 1876.70 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.11
method | result | size |
default | \(\text {Expression too large to display}\) | \(917\) |
parts | \(\text {Expression too large to display}\) | \(1175\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.23 \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {5 \, \sqrt {2} {\left (5 i \, A a^{3} + 21 i \, B a^{2} b + 21 i \, A a b^{2} + 21 i \, B b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-5 i \, A a^{3} - 21 i \, B a^{2} b - 21 i \, A a b^{2} - 21 i \, B b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (3 i \, B a^{3} + 9 i \, A a^{2} b + 15 i \, B a b^{2} + 5 i \, A b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-3 i \, B a^{3} - 9 i \, A a^{2} b - 15 i \, B a b^{2} - 5 i \, A b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (15 \, A a^{3} + 21 \, {\left (3 \, B a^{3} + 9 \, A a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (5 \, A a^{3} + 21 \, B a^{2} b + 21 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 21 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \]
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Timed out. \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
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\[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int \left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
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