Optimal. Leaf size=271 \[ \frac{4 a^3 (13 A+21 B+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}-\frac{4 a^3 (41 A+42 B-35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (7 A+9 B+5 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (7 A+9 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 (6 A+7 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{7 d \sec ^{\frac{5}{2}}(c+d x)} \]
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Rubi [A] time = 0.632989, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4086, 4017, 3997, 3787, 3771, 2639, 2641} \[ -\frac{4 a^3 (41 A+42 B-35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (7 A+9 B+5 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (13 A+21 B+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^3 (7 A+9 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 (6 A+7 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{7 d \sec ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4086
Rule 4017
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+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx &=\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \int \frac{(a+a \sec (c+d x))^3 \left (\frac{1}{2} a (6 A+7 B)-\frac{1}{2} a (A-7 C) \sec (c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{7 a}\\ &=\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 \int \frac{(a+a \sec (c+d x))^2 \left (\frac{7}{4} a^2 (7 A+9 B+5 C)-\frac{1}{4} a^2 (11 A+7 B-35 C) \sec (c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{35 a}\\ &=\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 A+9 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{8 \int \frac{(a+a \sec (c+d x)) \left (\frac{1}{4} a^3 (106 A+147 B+140 C)-\frac{1}{4} a^3 (41 A+42 B-35 C) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{105 a}\\ &=-\frac{4 a^3 (41 A+42 B-35 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 A+9 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{16 \int \frac{\frac{21}{8} a^4 (7 A+9 B+5 C)+\frac{5}{8} a^4 (13 A+21 B+35 C) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{105 a}\\ &=-\frac{4 a^3 (41 A+42 B-35 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 A+9 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{1}{5} \left (2 a^3 (7 A+9 B+5 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (2 a^3 (13 A+21 B+35 C)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=-\frac{4 a^3 (41 A+42 B-35 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 A+9 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{1}{5} \left (2 a^3 (7 A+9 B+5 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 (13 A+21 B+35 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 (7 A+9 B+5 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 (13 A+21 B+35 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{4 a^3 (41 A+42 B-35 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 A+9 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 3.08927, size = 266, normalized size = 0.98 \[ \frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (-112 i (7 A+9 B+5 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+80 (13 A+21 B+35 C) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+126 A \sin (c+d x)+550 A \sin (2 (c+d x))+126 A \sin (3 (c+d x))+15 A \sin (4 (c+d x))+2352 i A \cos (c+d x)+42 B \sin (c+d x)+420 B \sin (2 (c+d x))+42 B \sin (3 (c+d x))+3024 i B \cos (c+d x)+840 C \sin (c+d x)+140 C \sin (2 (c+d x))+1680 i C \cos (c+d x)\right )}{420 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.711, size = 727, normalized size = 2.7 \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{7}{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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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