Optimal. Leaf size=295 \[ \frac{2 a \left (5 a^2 A+21 a b B+18 A b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left (9 a^2 A b+3 a^3 B+15 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left (5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 \left (9 a^2 A b+3 a^3 B+15 a b^2 B+5 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d} \]
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Rubi [A] time = 0.597226, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2960, 4026, 4076, 4047, 3768, 3771, 2639, 4046, 2641} \[ \frac{2 a \left (5 a^2 A+21 a b B+18 A b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left (9 a^2 A b+3 a^3 B+15 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left (5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 \left (9 a^2 A b+3 a^3 B+15 a b^2 B+5 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 2960
Rule 4026
Rule 4076
Rule 4047
Rule 3768
Rule 3771
Rule 2639
Rule 4046
Rule 2641
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx &=\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)} F\left (\left .\frac{1}{2} (c+d x)\right |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*}
Mathematica [A] time = 3.50839, size = 225, normalized size = 0.76 \[ \frac{2 \sqrt{\sec (c+d x)} \left (21 \left (9 a^2 A b+3 a^3 B+15 a b^2 B+5 A b^3\right ) \sin (c+d x)+5 a \left (5 a^2 A+21 a b B+21 A b^2\right ) \tan (c+d x)+5 \left (5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-21 \left (9 a^2 A b+3 a^3 B+15 a b^2 B+5 A b^3\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+21 a^2 (a B+3 A b) \tan (c+d x) \sec (c+d x)+15 a^3 A \tan (c+d x) \sec ^2(c+d x)\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 13.595, size = 944, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{3} \cos \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sec \left (d x + c\right )^{\frac{9}{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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