Optimal. Leaf size=343 \[ \frac{4 a^3 (95 A+105 B+121 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{231 d}+\frac{4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (145 A+195 B+143 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (175 A+195 B+221 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{195 d}+\frac{2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{143 a d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac{11}{2}}(c+d x)} \]
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Rubi [A] time = 0.717735, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4086, 4017, 3996, 3787, 3769, 3771, 2639, 2641} \[ \frac{4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (145 A+195 B+143 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (95 A+105 B+121 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a^3 (175 A+195 B+221 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{195 d}+\frac{2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{143 a d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac{11}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4086
Rule 4017
Rule 3996
Rule 3787
Rule 3769
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{13}{2}}(c+d x)} \, dx &=\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}+\frac{2 \int \frac{(a+a \sec (c+d x))^3 \left (\frac{1}{2} a (6 A+13 B)+\frac{1}{2} a (5 A+13 C) \sec (c+d x)\right )}{\sec ^{\frac{11}{2}}(c+d x)} \, dx}{13 a}\\ &=\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac{9}{2}}(c+d x)}+\frac{4 \int \frac{(a+a \sec (c+d x))^2 \left (\frac{1}{4} a^2 (145 A+195 B+143 C)+\frac{1}{4} a^2 (85 A+65 B+143 C) \sec (c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx}{143 a}\\ &=\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{8 \int \frac{(a+a \sec (c+d x)) \left (\frac{5}{4} a^3 (236 A+273 B+286 C)+\frac{1}{4} a^3 (745 A+780 B+1001 C) \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx}{1287 a}\\ &=\frac{20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{16 \int \frac{-\frac{77}{8} a^4 (175 A+195 B+221 C)-\frac{117}{8} a^4 (95 A+105 B+121 C) \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{9009 a}\\ &=\frac{20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{77} \left (2 a^3 (95 A+105 B+121 C)\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{117} \left (2 a^3 (175 A+195 B+221 C)\right ) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{231} \left (2 a^3 (95 A+105 B+121 C)\right ) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{195} \left (2 a^3 (175 A+195 B+221 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{231} \left (2 a^3 (95 A+105 B+121 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{195} \left (2 a^3 (175 A+195 B+221 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{4 a^3 (175 A+195 B+221 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{195 d}+\frac{4 a^3 (95 A+105 B+121 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{231 d}+\frac{20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [C] time = 6.50075, size = 300, normalized size = 0.87 \[ \frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (-4928 i (175 A+195 B+221 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+12480 (95 A+105 B+121 C) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\cos (c+d x) (780 (1811 A+1953 B+2134 C) \sin (c+d x)+77 (7825 A+7800 B+7592 C) \sin (2 (c+d x))+251550 A \sin (3 (c+d x))+90860 A \sin (4 (c+d x))+24570 A \sin (5 (c+d x))+3465 A \sin (6 (c+d x))+2587200 i A+221130 B \sin (3 (c+d x))+60060 B \sin (4 (c+d x))+8190 B \sin (5 (c+d x))+2882880 i B+154440 C \sin (3 (c+d x))+20020 C \sin (4 (c+d x))+3267264 i C)\right )}{720720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.474, size = 576, normalized size = 1.7 \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{C a^{3} \sec \left (d x + c\right )^{5} +{\left (B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{4} +{\left (A + 3 \, B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{3} +{\left (3 \, A + 3 \, B + C\right )} a^{3} \sec \left (d x + c\right )^{2} +{\left (3 \, A + B\right )} a^{3} \sec \left (d x + c\right ) + A a^{3}}{\sec \left (d x + c\right )^{\frac{13}{2}}}, 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} + B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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