Optimal. Leaf size=135 \[ \frac{2 a (3 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d} \]
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Rubi [A] time = 0.14369, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3997, 3787, 3771, 2641, 3768, 2639} \[ \frac{2 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a (3 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3997
Rule 3787
Rule 3771
Rule 2641
Rule 3768
Rule 2639
Rubi steps
\begin{align*} \int \sqrt{\sec (c+d x)} (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=\frac{2 a B \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac{2}{3} \int \sqrt{\sec (c+d x)} \left (\frac{1}{2} a (3 A+B)+\frac{3}{2} a (A+B) \sec (c+d x)\right ) \, dx\\ &=\frac{2 a B \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+(a (A+B)) \int \sec ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{3} (a (3 A+B)) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 a (A+B) \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a B \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}-(a (A+B)) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (a (3 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 a (3 A+B) \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 (A+B) \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a B \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}-\left (a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 a (A+B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{d}+\frac{2 a (3 A+B) \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 (A+B) \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a B \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.515784, size = 94, normalized size = 0.7 \[ \frac{a \sec ^{\frac{3}{2}}(c+d x) \left (2 (3 A+B) \cos ^{\frac{3}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-6 (A+B) \cos ^{\frac{3}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) (3 (A+B) \cos (c+d x)+B)\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.205, size = 427, 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 \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )} \sqrt{\sec \left (d x + c\right )}\,{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 a \sec \left (d x + c\right )^{2} +{\left (A + B\right )} a \sec \left (d x + c\right ) + A a\right )} \sqrt{\sec \left (d x + c\right )}, 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 \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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