Optimal. Leaf size=277 \[ \frac{4 a^3 (105 A+121 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{231 d}+\frac{4 a^3 (15 A+17 B) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (105 A+121 B) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (15 A+17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]
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Rubi [A] time = 0.510461, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4017, 3996, 3787, 3769, 3771, 2639, 2641} \[ \frac{4 a^3 (15 A+17 B) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (105 A+121 B) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (105 A+121 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a^3 (15 A+17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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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 (A+B \sec (c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2}{11} \int \frac{(a+a \sec (c+d x))^2 \left (\frac{1}{2} a (15 A+11 B)+\frac{1}{2} a (5 A+11 B) \sec (c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{4}{99} \int \frac{(a+a \sec (c+d x)) \left (\frac{5}{2} a^2 (21 A+22 B)+\frac{1}{2} a^2 (60 A+77 B) \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{8}{693} \int \frac{-\frac{77}{4} a^3 (15 A+17 B)-\frac{9}{4} a^3 (105 A+121 B) \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{9} \left (2 a^3 (15 A+17 B)\right ) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx+\frac{1}{77} \left (2 a^3 (105 A+121 B)\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (15 A+17 B) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (105 A+121 B) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{15} \left (2 a^3 (15 A+17 B)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{231} \left (2 a^3 (105 A+121 B)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (15 A+17 B) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (105 A+121 B) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{15} \left (2 a^3 (15 A+17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{231} \left (2 a^3 (105 A+121 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a^3 (15 A+17 B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^3 (105 A+121 B) \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 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (15 A+17 B) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (105 A+121 B) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [C] time = 3.49177, size = 239, normalized size = 0.86 \[ \frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (-2464 i (15 A+17 B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+480 (105 A+121 B) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\cos (c+d x) (30 (1953 A+2134 B) \sin (c+d x)+308 (75 A+73 B) \sin (2 (c+d x))+8505 A \sin (3 (c+d x))+2310 A \sin (4 (c+d x))+315 A \sin (5 (c+d x))+110880 i A+5940 B \sin (3 (c+d x))+770 B \sin (4 (c+d x))+125664 i B)\right )}{27720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.659, size = 441, normalized size = 1.6 \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 a^{3} \sec \left (d x + c\right )^{4} +{\left (A + 3 \, B\right )} a^{3} \sec \left (d x + c\right )^{3} + 3 \,{\left (A + B\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{11}{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 (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{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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