Optimal. Leaf size=171 \[ \frac{2 \left (a^2 B+2 a A b+3 b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}+\frac{2 \left (3 a^2 A+10 a b B+5 A b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^2 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (a B+2 A b) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.260082, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4024, 4047, 3771, 2639, 4045, 2641} \[ \frac{2 \left (a^2 B+2 a A b+3 b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 \left (3 a^2 A+10 a b B+5 A b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^2 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (a B+2 A b) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4024
Rule 4047
Rule 3771
Rule 2639
Rule 4045
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx &=\frac{2 a^2 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2}{5} \int \frac{-\frac{5}{2} a (2 A b+a B)+\left (A \left (-\frac{3 a^2}{2}-\frac{5 b^2}{2}\right )-5 a b B\right ) \sec (c+d x)-\frac{5}{2} b^2 B \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2}{5} \int \frac{-\frac{5}{2} a (2 A b+a B)-\frac{5}{2} b^2 B \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx-\frac{1}{5} \left (-3 a^2 A-5 A b^2-10 a b B\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a^2 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (2 A b+a B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}-\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 (\left (-3 a^2 A-5 A b^2-10 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 (3 a^2 A+5 A b^2+10 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 a^2 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (2 A b+a B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}-\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{2 \left (3 a^2 A+5 A b^2+10 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 (2 a A b+a^2 B+3 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 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (2 A b+a B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.971852, size = 128, normalized size = 0.75 \[ \frac{\sqrt{\sec (c+d x)} \left (10 \left (a^2 B+2 a A b+3 b^2 B\right ) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+6 \left (3 a^2 A+10 a b B+5 A b^2\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+a \sin (2 (c+d x)) (3 a A \cos (c+d x)+5 a B+10 A b)\right )}{15 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.87, size = 487, normalized size = 2.9 \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^{2} \sec \left (d x + c\right )^{3} + A a^{2} +{\left (2 \, B a b + A b^{2}\right )} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a 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 )^{2}}{\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 )}^{2}}{\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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