Optimal. Leaf size=237 \[ \frac{8 a^2 (7 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{2 a^2 (35 A+33 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{4 a^2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{8 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{35 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d} \]
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Rubi [A] time = 0.413934, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4089, 4018, 3997, 3787, 3771, 2641, 3768, 2639} \[ \frac{2 a^2 (35 A+33 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a^2 (7 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{4 a^2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{8 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{35 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 4089
Rule 4018
Rule 3997
Rule 3787
Rule 3771
Rule 2641
Rule 3768
Rule 2639
Rubi steps
\begin{align*} \int \sqrt{\sec (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{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2 \int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2 \left (\frac{1}{2} a (7 A+C)+2 a C \sec (c+d x)\right ) \, dx}{7 a}\\ &=\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{8 C \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac{4 \int \sqrt{\sec (c+d x)} (a+a \sec (c+d x)) \left (\frac{1}{4} a^2 (35 A+9 C)+\frac{1}{4} a^2 (35 A+33 C) \sec (c+d x)\right ) \, dx}{35 a}\\ &=\frac{2 a^2 (35 A+33 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{8 C \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac{8 \int \sqrt{\sec (c+d x)} \left (\frac{5}{2} a^3 (7 A+3 C)+\frac{21}{4} a^3 (5 A+3 C) \sec (c+d x)\right ) \, dx}{105 a}\\ &=\frac{2 a^2 (35 A+33 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{8 C \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac{1}{5} \left (2 a^2 (5 A+3 C)\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{21} \left (4 a^2 (7 A+3 C)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{4 a^2 (5 A+3 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a^2 (35 A+33 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{8 C \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac{1}{5} \left (2 a^2 (5 A+3 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (4 a^2 (7 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{8 a^2 (7 A+3 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^2 (5 A+3 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a^2 (35 A+33 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{8 C \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac{1}{5} \left (2 a^2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{4 a^2 (5 A+3 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{8 a^2 (7 A+3 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^2 (5 A+3 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a^2 (35 A+33 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{8 C \sec ^{\frac{3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}\\ \end{align*}
Mathematica [C] time = 5.88483, size = 436, normalized size = 1.84 \[ \frac{a^2 \csc (c) e^{-i d x} \cos ^4(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (\sec (c+d x)+1)^2 \left (A+C \sec ^2(c+d x)\right ) \left (7 \sqrt{2} \left (-1+e^{2 i c}\right ) (5 A+3 C) e^{2 i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+20 (7 A+3 C) \sin (c) e^{i d x} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-\frac{\left (-1+e^{2 i c}\right ) e^{-i (c-d x)} \sqrt{\sec (c+d x)} \left (35 A \left (6 e^{i (c+d x)}+e^{2 i (c+d x)}+6 e^{3 i (c+d x)}-1\right ) \left (1+e^{2 i (c+d x)}\right )^2+6 C \left (7 e^{i (c+d x)}-20 e^{2 i (c+d x)}+63 e^{3 i (c+d x)}+20 e^{4 i (c+d x)}+77 e^{5 i (c+d x)}+10 e^{6 i (c+d x)}+21 e^{7 i (c+d x)}-10\right )\right )}{2 \left (1+e^{2 i (c+d x)}\right )^3}\right )}{105 d (A \cos (2 (c+d x))+A+2 C)} \]
Antiderivative was successfully verified.
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Maple [B] time = 7.912, size = 919, normalized size = 3.9 \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 )^{4} + 2 \, C a^{2} \sec \left (d x + c\right )^{3} +{\left (A + C\right )} a^{2} \sec \left (d x + c\right )^{2} + 2 \, A a^{2} \sec \left (d x + c\right ) + A a^{2}\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} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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