Optimal. Leaf size=152 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (3 a B+3 A b+b C)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (b B-a (A-C))}{d}+\frac{2 (a C+b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d} \]
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Rubi [A] time = 0.223363, antiderivative size = 152, 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} \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (3 a B+3 A b+b C)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (b B-a (A-C))}{d}+\frac{2 (a C+b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d} \]
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+b \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 b 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} (3 A b+3 a B+b C) \sec (c+d x)+\frac{3}{2} (b B+a C) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 b 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} (b B+a C) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} (3 A b+3 a B+b C) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 (b B+a C) \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 b C \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+(-b B+a (A-C)) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left ((3 A b+3 a B+b C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (3 A b+3 a B+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 (b B+a C) \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 b C \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\left ((-b B+a (A-C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 (b B-a (A-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 (3 A b+3 a B+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 (b B+a C) \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 b C \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [C] time = 2.51004, size = 223, normalized size = 1.47 \[ \frac{e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left (i \left (1+e^{2 i (c+d x)}\right )^{3/2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right ) (a (C-A)+b B)+2 \cos ^{\frac{3}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (3 a B+3 A b+b C)+3 i a A \cos (2 (c+d x))+3 i a A+3 a C \sin (2 (c+d x))-3 i a C \cos (2 (c+d x))-3 i a C+3 b B \sin (2 (c+d x))-3 i b B \cos (2 (c+d x))-3 i b B+2 b C \sin (c+d x)\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.243, size = 666, 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] 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 (b \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 b \sec \left (d x + c\right )^{3} +{\left (C a + B b\right )} \sec \left (d x + c\right )^{2} + A a +{\left (B a + A b\right )} \sec \left (d x + c\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 \frac{{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (b \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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