Optimal. Leaf size=319 \[ \frac{4 a^3 (143 A+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{231 d}+\frac{8 a^3 (44 A+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{385 d}+\frac{4 a^3 (143 A+105 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{2 (33 A+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{231 d}+\frac{4 a^3 (7 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{4 a^3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{33 a d}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^3}{11 d} \]
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Rubi [A] time = 0.618146, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{8 a^3 (44 A+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{385 d}+\frac{4 a^3 (143 A+105 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{2 (33 A+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{231 d}+\frac{4 a^3 (7 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^3 (143 A+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}-\frac{4 a^3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{33 a d}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^3}{11 d} \]
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))^3 \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))^3 \sin (c+d x)}{11 d}+\frac{2 \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (\frac{1}{2} a (11 A+3 C)+3 a C \sec (c+d x)\right ) \, dx}{11 a}\\ &=\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac{4 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac{4 \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (\frac{9}{4} a^2 (11 A+5 C)+\frac{3}{4} a^2 (33 A+35 C) \sec (c+d x)\right ) \, dx}{99 a}\\ &=\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac{4 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac{2 (33 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}+\frac{8 \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \left (\frac{45}{4} a^3 (11 A+7 C)+\frac{9}{2} a^3 (44 A+35 C) \sec (c+d x)\right ) \, dx}{693 a}\\ &=\frac{8 a^3 (44 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac{4 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac{2 (33 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}+\frac{16 \int \sec ^{\frac{3}{2}}(c+d x) \left (\frac{693}{8} a^4 (7 A+5 C)+\frac{45}{8} a^4 (143 A+105 C) \sec (c+d x)\right ) \, dx}{3465 a}\\ &=\frac{8 a^3 (44 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac{4 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac{2 (33 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}+\frac{1}{5} \left (2 a^3 (7 A+5 C)\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{77} \left (2 a^3 (143 A+105 C)\right ) \int \sec ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{4 a^3 (7 A+5 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{4 a^3 (143 A+105 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{8 a^3 (44 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac{4 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac{2 (33 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}-\frac{1}{5} \left (2 a^3 (7 A+5 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{231} \left (2 a^3 (143 A+105 C)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{4 a^3 (7 A+5 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{4 a^3 (143 A+105 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{8 a^3 (44 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac{4 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac{2 (33 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}-\frac{1}{5} \left (2 a^3 (7 A+5 C) \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 (143 A+105 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+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 (143 A+105 C) \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{4 a^3 (7 A+5 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{4 a^3 (143 A+105 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{8 a^3 (44 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac{4 C \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac{2 (33 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}\\ \end{align*}
Mathematica [C] time = 7.00122, size = 863, normalized size = 2.71 \[ \frac{7 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 ^5(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)^3 \left (C \sec ^2(c+d x)+A\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C)}+\frac{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 ^5(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)^3 \left (C \sec ^2(c+d x)+A\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C)}+\frac{13 A \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (\sec (c+d x) a+a)^3 \left (C \sec ^2(c+d x)+A\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{21 d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{9}{2}}(c+d x)}+\frac{5 C \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (\sec (c+d x) a+a)^3 \left (C \sec ^2(c+d x)+A\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{11 d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{9}{2}}(c+d x)}+\frac{(\sec (c+d x) a+a)^3 \left (C \sec ^2(c+d x)+A\right ) \left (\frac{C \sec (c) \sin (d x) \sec ^5(c+d x)}{22 d}+\frac{\sec (c) (3 C \sin (c)+11 C \sin (d x)) \sec ^4(c+d x)}{66 d}+\frac{\sec (c) (77 C \sin (c)+33 A \sin (d x)+126 C \sin (d x)) \sec ^3(c+d x)}{462 d}+\frac{\sec (c) (165 A \sin (c)+630 C \sin (c)+693 A \sin (d x)+770 C \sin (d x)) \sec ^2(c+d x)}{2310 d}+\frac{\sec (c) (693 A \sin (c)+770 C \sin (c)+1430 A \sin (d x)+1050 C \sin (d x)) \sec (c+d x)}{2310 d}+\frac{(7 A+5 C) \cos (d x) \csc (c)}{5 d}+\frac{(143 A+105 C) \tan (c)}{231 d}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{(\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{9}{2}}(c+d x)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 10.123, size = 1409, normalized size = 4.4 \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^{3} \sec \left (d x + c\right )^{6} + 3 \, C a^{3} \sec \left (d x + c\right )^{5} +{\left (A + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{4} +{\left (3 \, A + C\right )} a^{3} \sec \left (d x + c\right )^{3} + 3 \, A a^{3} \sec \left (d x + c\right )^{2} + A a^{3} \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 )}^{3} \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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