Optimal. Leaf size=253 \[ \frac{4 a^3 (35 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{8 a^3 (70 A+53 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (5 A+7 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}{15 d}-\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{12 C \sin (c+d x) \sqrt{\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{35 a d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d} \]
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Rubi [A] time = 0.562782, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4089, 4018, 3997, 3787, 3771, 2639, 2641} \[ \frac{8 a^3 (70 A+53 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (5 A+7 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac{4 a^3 (35 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{12 C \sin (c+d x) \sqrt{\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{35 a d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d} \]
Antiderivative was successfully verified.
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Rule 4089
Rule 4018
Rule 3997
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx &=\frac{2 C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{2 \int \frac{(a+a \sec (c+d x))^3 \left (\frac{1}{2} a (7 A-C)+3 a C \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{7 a}\\ &=\frac{2 C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{12 C \sqrt{\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac{4 \int \frac{(a+a \sec (c+d x))^2 \left (\frac{1}{4} a^2 (35 A-11 C)+\frac{7}{4} a^2 (5 A+7 C) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{35 a}\\ &=\frac{2 C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{12 C \sqrt{\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac{2 (5 A+7 C) \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{8 \int \frac{(a+a \sec (c+d x)) \left (\frac{1}{4} a^3 (35 A-41 C)+\frac{1}{2} a^3 (70 A+53 C) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{105 a}\\ &=\frac{8 a^3 (70 A+53 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{12 C \sqrt{\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac{2 (5 A+7 C) \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{16 \int \frac{-\frac{21}{8} a^4 (5 A+7 C)+\frac{5}{8} a^4 (35 A+13 C) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{105 a}\\ &=\frac{8 a^3 (70 A+53 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{12 C \sqrt{\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac{2 (5 A+7 C) \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac{1}{5} \left (2 a^3 (5 A+7 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (2 a^3 (35 A+13 C)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{8 a^3 (70 A+53 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{12 C \sqrt{\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac{2 (5 A+7 C) \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac{1}{5} \left (2 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (2 a^3 (35 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{4 a^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 d}+\frac{4 a^3 (35 A+13 C) \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{8 a^3 (70 A+53 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{12 C \sqrt{\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac{2 (5 A+7 C) \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}\\ \end{align*}
Mathematica [C] time = 3.49733, size = 280, normalized size = 1.11 \[ \frac{a^3 e^{-i d x} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left (14 i (5 A+7 C) e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{7/2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+80 (35 A+13 C) \cos ^{\frac{7}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+70 A \sin (c+d x)+630 A \sin (2 (c+d x))+70 A \sin (3 (c+d x))+315 A \sin (4 (c+d x))-840 i A \cos (2 (c+d x))-210 i A \cos (4 (c+d x))-630 i A+380 C \sin (c+d x)+840 C \sin (2 (c+d x))+260 C \sin (3 (c+d x))+294 C \sin (4 (c+d x))-1176 i C \cos (2 (c+d x))-294 i C \cos (4 (c+d x))-882 i C\right )}{420 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 7.518, size = 1014, normalized size = 4. \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{C a^{3} \sec \left (d x + c\right )^{5} + 3 \, C a^{3} \sec \left (d x + c\right )^{4} +{\left (A + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{3} +{\left (3 \, A + C\right )} a^{3} \sec \left (d x + c\right )^{2} + 3 \, A a^{3} \sec \left (d x + c\right ) + A a^{3}}{\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 \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\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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