Optimal. Leaf size=214 \[ \frac{4 a^2 (A+2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}-\frac{2 a^2 (7 A+5 B-15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (4 A+5 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (4 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.493026, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{2 a^2 (7 A+5 B-15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (A+2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 (4 A+5 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (4 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)} \]
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))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx &=\frac{2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \int \frac{(a+a \sec (c+d x))^2 \left (\frac{1}{2} a (4 A+5 B)-\frac{1}{2} a (A-5 C) \sec (c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{5 a}\\ &=\frac{2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (4 A+5 B) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 \int \frac{(a+a \sec (c+d x)) \left (\frac{1}{4} a^2 (17 A+25 B+15 C)-\frac{1}{4} a^2 (7 A+5 B-15 C) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{15 a}\\ &=-\frac{2 a^2 (7 A+5 B-15 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (4 A+5 B) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{8 \int \frac{\frac{3}{4} a^3 (4 A+5 B)+\frac{5}{4} a^3 (A+2 B+3 C) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{15 a}\\ &=-\frac{2 a^2 (7 A+5 B-15 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (4 A+5 B) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{1}{5} \left (2 a^2 (4 A+5 B)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (2 a^2 (A+2 B+3 C)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=-\frac{2 a^2 (7 A+5 B-15 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (4 A+5 B) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{1}{5} \left (2 a^2 (4 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{3} \left (2 a^2 (A+2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a^2 (4 A+5 B) \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^2 (A+2 B+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 d}-\frac{2 a^2 (7 A+5 B-15 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (4 A+5 B) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 2.06259, size = 187, normalized size = 0.87 \[ \frac{a^2 \sqrt{\sec (c+d x)} \left (40 (A+2 B+3 C) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-8 i (4 A+5 B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+3 A \sin (c+d x)+20 A \sin (2 (c+d x))+3 A \sin (3 (c+d x))+96 i A \cos (c+d x)+10 B \sin (2 (c+d x))+120 i B \cos (c+d x)+60 C \sin (c+d x)\right )}{30 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.448, size = 595, normalized size = 2.8 \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{C a^{2} \sec \left (d x + c\right )^{4} +{\left (B + 2 \, C\right )} a^{2} \sec \left (d x + c\right )^{3} +{\left (A + 2 \, B + C\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{5}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int \frac{A}{\sec ^{\frac{5}{2}}{\left (c + d x \right )}}\, dx + \int \frac{2 A}{\sec ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx + \int \frac{A}{\sqrt{\sec{\left (c + d x \right )}}}\, dx + \int \frac{B}{\sec ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx + \int \frac{2 B}{\sqrt{\sec{\left (c + d x \right )}}}\, dx + \int B \sqrt{\sec{\left (c + d x \right )}}\, dx + \int \frac{C}{\sqrt{\sec{\left (c + d x \right )}}}\, dx + \int 2 C \sqrt{\sec{\left (c + d x \right )}}\, dx + \int C \sec ^{\frac{3}{2}}{\left (c + d x \right )}\, dx\right ) \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 )}^{2}}{\sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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