Optimal. Leaf size=143 \[ \frac{2 a (3 A+3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}+\frac{2 a (A-B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a (B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d} \]
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Rubi [A] time = 0.219038, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {4076, 4047, 3771, 2641, 4046, 2639} \[ \frac{2 a (3 A+3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 a (A-B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a (B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4076
Rule 4047
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx &=\frac{2 a C \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac{2}{3} \int \frac{\frac{3 a A}{2}+\frac{1}{2} a (3 A+3 B+C) \sec (c+d x)+\frac{3}{2} a (B+C) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a C \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac{2}{3} \int \frac{\frac{3 a A}{2}+\frac{3}{2} a (B+C) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} (a (3 A+3 B+C)) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 a (B+C) \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a C \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+(a (A-B-C)) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (a (3 A+3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 a (3 A+3 B+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 (B+C) \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a C \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\left (a (A-B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 a (A-B-C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{d}+\frac{2 a (3 A+3 B+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 (B+C) \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a C \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [C] time = 1.84693, size = 208, normalized size = 1.45 \[ \frac{a e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left (-i (A-B-C) \left (1+e^{2 i (c+d x)}\right )^{3/2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+2 (3 A+3 B+C) \cos ^{\frac{3}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+3 i A \cos (2 (c+d x))+3 i A+3 B \sin (2 (c+d x))-3 i B \cos (2 (c+d x))-3 i B+2 C \sin (c+d x)+3 C \sin (2 (c+d x))-3 i C \cos (2 (c+d x))-3 i C\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.522, size = 516, normalized size = 3.6 \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] 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 )}}{\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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C a \sec \left (d x + c\right )^{3} +{\left (B + C\right )} a \sec \left (d x + c\right )^{2} +{\left (A + B\right )} a \sec \left (d x + c\right ) + A a}{\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 \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 )}}{\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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