Optimal. Leaf size=192 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (a (3 A+C)+b B)}{3 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (5 a B+5 A b+3 b C)}{5 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (5 a B+5 A b+3 b C)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 b C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d} \]
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Rubi [A] time = 0.234659, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4076, 4047, 3768, 3771, 2639, 4046, 2641} \[ \frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (5 a B+5 A b+3 b C)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (a (3 A+C)+b B)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (5 a B+5 A b+3 b C)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 b C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 4076
Rule 4047
Rule 3768
Rule 3771
Rule 2639
Rule 4046
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 b C \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2}{5} \int \sqrt{\sec (c+d x)} \left (\frac{5 a A}{2}+\frac{1}{2} (5 A b+5 a B+3 b C) \sec (c+d x)+\frac{5}{2} (b B+a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 b C \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2}{5} \int \sqrt{\sec (c+d x)} \left (\frac{5 a A}{2}+\frac{5}{2} (b B+a C) \sec ^2(c+d x)\right ) \, dx+\frac{1}{5} (5 A b+5 a B+3 b C) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 (5 A b+5 a B+3 b C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (b B+a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac{2 b C \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{5} (-5 A b-5 a B-3 b C) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} (b B+a (3 A+C)) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 (5 A b+5 a B+3 b C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (b B+a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac{2 b C \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{5} \left ((-5 A b-5 a B-3 b C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{3} \left ((b B+a (3 A+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 (5 A b+5 a B+3 b 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{2 (b B+a (3 A+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 (5 A b+5 a B+3 b C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (b B+a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac{2 b C \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [C] time = 7.0227, size = 1140, normalized size = 5.94 \[ \frac{4 a A \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^{\frac{7}{2}}(c+d x)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{4 b B \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^{\frac{7}{2}}(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{4 a C \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^{\frac{7}{2}}(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} A b 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)}} \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 ) (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^3(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} a B 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)}} \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 ) (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^3(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} b 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)}} \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 ) (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^3(c+d x)}{5 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{4 b C \sec (c) \sin (d x) \sec ^2(c+d x)}{5 d}+\frac{4 \sec (c) (3 b C \sin (c)+5 b B \sin (d x)+5 a C \sin (d x)) \sec (c+d x)}{15 d}+\frac{4 (5 A b+3 C b+5 a B) \cos (d x) \csc (c)}{5 d}+\frac{4 (b B+a C) \tan (c)}{3 d}\right )}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 6.992, size = 742, 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 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 )\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} + 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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