Optimal. Leaf size=270 \[ \frac{4 a^2 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{2 a^2 (21 A+19 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^2 (7 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{16 a^2 (3 A+2 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{16 a^2 (3 A+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{8 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{63 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2}{9 d} \]
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Rubi [A] time = 0.438974, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4089, 4018, 3997, 3787, 3768, 3771, 2639, 2641} \[ \frac{2 a^2 (21 A+19 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^2 (7 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{16 a^2 (3 A+2 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{16 a^2 (3 A+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{8 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{63 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 4089
Rule 4018
Rule 3997
Rule 3787
Rule 3768
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (\frac{3}{2} a (3 A+C)+2 a C \sec (c+d x)\right ) \, dx}{9 a}\\ &=\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{8 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{4 \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \left (\frac{3}{4} a^2 (21 A+11 C)+\frac{3}{4} a^2 (21 A+19 C) \sec (c+d x)\right ) \, dx}{63 a}\\ &=\frac{2 a^2 (21 A+19 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{8 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{8 \int \sec ^{\frac{3}{2}}(c+d x) \left (21 a^3 (3 A+2 C)+\frac{45}{4} a^3 (7 A+5 C) \sec (c+d x)\right ) \, dx}{315 a}\\ &=\frac{2 a^2 (21 A+19 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{8 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{1}{15} \left (8 a^2 (3 A+2 C)\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{7} \left (2 a^2 (7 A+5 C)\right ) \int \sec ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{16 a^2 (3 A+2 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^2 (7 A+5 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a^2 (21 A+19 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{8 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{63 d}-\frac{1}{15} \left (8 a^2 (3 A+2 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (2 a^2 (7 A+5 C)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{16 a^2 (3 A+2 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^2 (7 A+5 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a^2 (21 A+19 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{8 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{63 d}-\frac{1}{15} \left (8 a^2 (3 A+2 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^2 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{16 a^2 (3 A+2 C) \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 (7 A+5 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{16 a^2 (3 A+2 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^2 (7 A+5 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a^2 (21 A+19 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{8 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{63 d}\\ \end{align*}
Mathematica [C] time = 6.80225, size = 821, normalized size = 3.04 \[ \frac{4 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{8 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{45 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{2 A \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{7}{2}}(c+d x)}+\frac{10 C \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{21 d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{7}{2}}(c+d x)}+\frac{(\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \left (\frac{C \sec (c) \sin (d x) \sec ^4(c+d x)}{9 d}+\frac{\sec (c) (7 C \sin (c)+18 C \sin (d x)) \sec ^3(c+d x)}{63 d}+\frac{\sec (c) (90 C \sin (c)+63 A \sin (d x)+112 C \sin (d x)) \sec ^2(c+d x)}{315 d}+\frac{\sec (c) (63 A \sin (c)+112 C \sin (c)+210 A \sin (d x)+150 C \sin (d x)) \sec (c+d x)}{315 d}+\frac{8 (3 A+2 C) \cos (d x) \csc (c)}{15 d}+\frac{2 (7 A+5 C) \tan (c)}{21 d}\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{(\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 9.129, size = 1168, normalized size = 4.3 \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 ({\left (C a^{2} \sec \left (d x + c\right )^{5} + 2 \, C a^{2} \sec \left (d x + c\right )^{4} +{\left (A + C\right )} a^{2} \sec \left (d x + c\right )^{3} + 2 \, A a^{2} \sec \left (d x + c\right )^{2} + A a^{2} \sec \left (d x + c\right )\right )} \sqrt{\sec \left (d x + c\right )}, 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{\left (C \sec \left (d x + c\right )^{2} + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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