Optimal. Leaf size=244 \[ \frac{4 a^3 (21 A+13 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{4 a^3 (42 A+41 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (7 A+11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{35 d}+\frac{4 a^3 (9 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{4 a^3 (9 A+7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d} \]
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Rubi [A] time = 0.438735, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4018, 3997, 3787, 3771, 2641, 3768, 2639} \[ \frac{4 a^3 (42 A+41 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (7 A+11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{35 d}+\frac{4 a^3 (9 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^3 (21 A+13 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{4 a^3 (9 A+7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 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))^3 (A+B \sec (c+d x)) \, dx &=\frac{2 a B \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2}{7} \int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2 \left (\frac{1}{2} a (7 A+B)+\frac{1}{2} a (7 A+11 B) \sec (c+d x)\right ) \, dx\\ &=\frac{2 a B \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2 (7 A+11 B) \sec ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac{4}{35} \int \sqrt{\sec (c+d x)} (a+a \sec (c+d x)) \left (\frac{1}{2} a^2 (21 A+8 B)+\frac{1}{2} a^2 (42 A+41 B) \sec (c+d x)\right ) \, dx\\ &=\frac{4 a^3 (42 A+41 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2 (7 A+11 B) \sec ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac{8}{105} \int \sqrt{\sec (c+d x)} \left (\frac{5}{4} a^3 (21 A+13 B)+\frac{21}{4} a^3 (9 A+7 B) \sec (c+d x)\right ) \, dx\\ &=\frac{4 a^3 (42 A+41 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2 (7 A+11 B) \sec ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac{1}{5} \left (2 a^3 (9 A+7 B)\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{21} \left (2 a^3 (21 A+13 B)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{4 a^3 (9 A+7 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{4 a^3 (42 A+41 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2 (7 A+11 B) \sec ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac{1}{5} \left (2 a^3 (9 A+7 B)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (2 a^3 (21 A+13 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a^3 (21 A+13 B) \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 (9 A+7 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{4 a^3 (42 A+41 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2 (7 A+11 B) \sec ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac{1}{5} \left (2 a^3 (9 A+7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{4 a^3 (9 A+7 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^3 (21 A+13 B) \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 (9 A+7 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{4 a^3 (42 A+41 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2 (7 A+11 B) \sec ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}\\ \end{align*}
Mathematica [C] time = 5.13715, size = 465, normalized size = 1.91 \[ \frac{a^3 \csc (c) e^{-i d x} \cos ^4(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (\sec (c+d x)+1)^3 (A+B \sec (c+d x)) \left (7 \sqrt{2} \left (-1+e^{2 i c}\right ) (9 A+7 B) e^{2 i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-\frac{\left (-1+e^{2 i c}\right ) e^{-i (c-d x)} \sqrt{\sec (c+d x)} \left (10 i (21 A+13 B) \left (1+e^{2 i (c+d x)}\right )^3 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+21 A \left (16 e^{i (c+d x)}-5 e^{2 i (c+d x)}+54 e^{3 i (c+d x)}+5 e^{4 i (c+d x)}+56 e^{5 i (c+d x)}+5 e^{6 i (c+d x)}+18 e^{7 i (c+d x)}-5\right )+2 B \left (84 e^{i (c+d x)}-95 e^{2 i (c+d x)}+441 e^{3 i (c+d x)}+95 e^{4 i (c+d x)}+504 e^{5 i (c+d x)}+65 e^{6 i (c+d x)}+147 e^{7 i (c+d x)}-65\right )\right )}{2 \left (1+e^{2 i (c+d x)}\right )^3}\right )}{420 d (A \cos (c+d x)+B)} \]
Antiderivative was successfully verified.
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Maple [B] time = 6.857, size = 931, normalized size = 3.8 \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 ({\left (B a^{3} \sec \left (d x + c\right )^{4} +{\left (A + 3 \, B\right )} a^{3} \sec \left (d x + c\right )^{3} + 3 \,{\left (A + B\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}\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 (B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{3} \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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