Optimal. Leaf size=234 \[ \frac{4 a^2 (5 A+6 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{4 a^2 (8 A+9 B) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (11 A+9 B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (5 A+6 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (8 A+9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.320409, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{4 a^2 (8 A+9 B) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (11 A+9 B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (5 A+6 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (5 A+6 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^2 (8 A+9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
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))^2 (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{2 A \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2}{9} \int \frac{(a+a \sec (c+d x)) \left (\frac{1}{2} a (11 A+9 B)+\frac{1}{2} a (5 A+9 B) \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (11 A+9 B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{4}{63} \int \frac{-\frac{7}{2} a^2 (8 A+9 B)-\frac{9}{2} a^2 (5 A+6 B) \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (11 A+9 B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{7} \left (2 a^2 (5 A+6 B)\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{9} \left (2 a^2 (8 A+9 B)\right ) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (11 A+9 B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (8 A+9 B) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (5 A+6 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 A \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{21} \left (2 a^2 (5 A+6 B)\right ) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{15} \left (2 a^2 (8 A+9 B)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a^2 (11 A+9 B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (8 A+9 B) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (5 A+6 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 A \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{21} \left (2 a^2 (5 A+6 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{15} \left (2 a^2 (8 A+9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{4 a^2 (8 A+9 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^2 (5 A+6 B) \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{2 a^2 (11 A+9 B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (8 A+9 B) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (5 A+6 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 A \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [C] time = 2.9442, size = 217, normalized size = 0.93 \[ \frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (-112 i (8 A+9 B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+240 (5 A+6 B) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\cos (c+d x) (30 (46 A+51 B) \sin (c+d x)+14 (37 A+36 B) \sin (2 (c+d x))+180 A \sin (3 (c+d x))+35 A \sin (4 (c+d x))+2688 i A+90 B \sin (3 (c+d x))+3024 i B)\right )}{1260 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.755, size = 413, normalized size = 1.8 \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^{2} \sec \left (d x + c\right )^{3} +{\left (A + 2 \, B\right )} a^{2} \sec \left (d x + c\right )^{2} +{\left (2 \, A + B\right )} a^{2} \sec \left (d x + c\right ) + A a^{2}}{\sec \left (d x + c\right )^{\frac{9}{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 )}^{2}}{\sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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