Optimal. Leaf size=236 \[ \frac{2 \left (3 a^2 A b+a^3 B+9 a b^2 B+3 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}+\frac{2 \left (3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^2 (5 a B+9 A b) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 (a A-5 b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.461241, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4025, 4074, 4047, 3771, 2641, 4046, 2639} \[ \frac{2 \left (3 a^2 A b+a^3 B+9 a b^2 B+3 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 \left (3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^2 (5 a B+9 A b) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 (a A-5 b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4025
Rule 4074
Rule 4047
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx &=\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2}{5} \int \frac{(a+b \sec (c+d x)) \left (-\frac{1}{2} a (9 A b+5 a B)-\frac{1}{2} \left (3 a^2 A+5 A b^2+10 a b B\right ) \sec (c+d x)+\frac{1}{2} b (a A-5 b B) \sec ^2(c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (9 A b+5 a B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4}{15} \int \frac{\frac{3}{4} a \left (3 a^2 A+14 A b^2+15 a b B\right )+\frac{5}{4} \left (3 a^2 A b+3 A b^3+a^3 B+9 a b^2 B\right ) \sec (c+d x)-\frac{3}{4} b^2 (a A-5 b B) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a^2 (9 A b+5 a B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4}{15} \int \frac{\frac{3}{4} a \left (3 a^2 A+14 A b^2+15 a b B\right )-\frac{3}{4} b^2 (a A-5 b B) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (3 a^2 A b+3 A b^3+a^3 B+9 a b^2 B\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 a^2 (9 A b+5 a B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 (a A-5 b B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{5} \left (3 a^3 A+15 a A b^2+15 a^2 b B-5 b^3 B\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (\left (3 a^2 A b+3 A b^3+a^3 B+9 a 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{2 \left (3 a^2 A b+3 A b^3+a^3 B+9 a b^2 B\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 a^2 (9 A b+5 a B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 (a A-5 b B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{5} \left (\left (3 a^3 A+15 a A b^2+15 a^2 b B-5 b^3 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^3 A+15 a A b^2+15 a^2 b B-5 b^3 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 (3 a^2 A b+3 A b^3+a^3 B+9 a b^2 B\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 a^2 (9 A b+5 a B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 (a A-5 b B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 1.58586, size = 172, normalized size = 0.73 \[ \frac{\sqrt{\sec (c+d x)} \left (20 \left (3 a^2 A b+a^3 B+9 a b^2 B+3 A b^3\right ) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+2 \sin (c+d x) \left (3 \left (a^3 A \cos (2 (c+d x))+a^3 A+10 b^3 B\right )+10 a^2 (a B+3 A b) \cos (c+d x)\right )+12 \left (3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{30 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.298, size = 867, normalized size = 3.7 \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(-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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B b^{3} \sec \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} \sec \left (d x + c\right )^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \sec \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac{5}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \sec{\left (c + d x \right )}\right ) \left (a + b \sec{\left (c + d x \right )}\right )^{3}}{\sec ^{\frac{5}{2}}{\left (c + d x \right )}}\, dx \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 \frac{{\left (B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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