Optimal. Leaf size=232 \[ -\frac{(3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 a d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(5 A+7 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}-\frac{(3 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{3 (5 A+7 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d}-\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d} \]
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Rubi [A] time = 0.236459, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4085, 3787, 3768, 3771, 2641, 2639} \[ -\frac{(A+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(5 A+7 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}-\frac{(3 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{3 (5 A+7 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}-\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d} \]
Antiderivative was successfully verified.
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Rule 4085
Rule 3787
Rule 3768
Rule 3771
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{5}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx &=-\frac{(A+C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{\int \sec ^{\frac{5}{2}}(c+d x) \left (\frac{1}{2} a (3 A+5 C)-\frac{1}{2} a (5 A+7 C) \sec (c+d x)\right ) \, dx}{a^2}\\ &=-\frac{(A+C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{(3 A+5 C) \int \sec ^{\frac{5}{2}}(c+d x) \, dx}{2 a}+\frac{(5 A+7 C) \int \sec ^{\frac{7}{2}}(c+d x) \, dx}{2 a}\\ &=-\frac{(3 A+5 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d}+\frac{(5 A+7 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 a d}-\frac{(A+C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{(3 A+5 C) \int \sqrt{\sec (c+d x)} \, dx}{6 a}+\frac{(3 (5 A+7 C)) \int \sec ^{\frac{3}{2}}(c+d x) \, dx}{10 a}\\ &=\frac{3 (5 A+7 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 a d}-\frac{(3 A+5 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d}+\frac{(5 A+7 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 a d}-\frac{(A+C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{(3 (5 A+7 C)) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{10 a}-\frac{\left ((3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a}\\ &=-\frac{(3 A+5 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 a d}+\frac{3 (5 A+7 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 a d}-\frac{(3 A+5 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d}+\frac{(5 A+7 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 a d}-\frac{(A+C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{\left (3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{10 a}\\ &=-\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(3 A+5 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 a d}+\frac{3 (5 A+7 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 a d}-\frac{(3 A+5 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d}+\frac{(5 A+7 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 a d}-\frac{(A+C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}\\ \end{align*}
Mathematica [C] time = 6.24486, size = 342, normalized size = 1.47 \[ -\frac{e^{-i d x} \cos \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \left (\cos \left (\frac{1}{2} (c+3 d x)\right )+i \sin \left (\frac{1}{2} (c+3 d x)\right )\right ) \left (-3 i (5 A+7 C) e^{-2 i (c+d x)} \left (1+e^{i (c+d x)}\right ) \left (1+e^{2 i (c+d x)}\right )^{5/2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+40 (3 A+5 C) \cos \left (\frac{1}{2} (c+d x)\right ) \cos ^{\frac{5}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-i \sin \left (\frac{1}{2} (c+d x)\right )\right )+2 i (2 (45 A+56 C) \cos (c+d x)+6 (5 A+7 C) \cos (2 (c+d x))+15 i A \sin (c+d x)+15 i A \sin (3 (c+d x))+30 A \cos (3 (c+d x))+30 A+31 i C \sin (c+d x)-4 i C \sin (2 (c+d x))+19 i C \sin (3 (c+d x))+44 C \cos (3 (c+d x))+54 C)\right )}{60 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 7.168, size = 803, normalized size = 3.5 \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{{\left (C \sec \left (d x + c\right )^{4} + A \sec \left (d x + c\right )^{2}\right )} \sqrt{\sec \left (d x + c\right )}}{a \sec \left (d x + c\right ) + a}, 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 \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}{a \sec \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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