Optimal. Leaf size=241 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (3 a^2 B+2 a b (3 A+C)+b^2 B\right )}{3 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (4 a^2 C+10 a b B+5 A b^2+3 b^2 C\right )}{5 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-5 a^2 (A-C)+10 a b B+b^2 (5 A+3 C)\right )}{5 d}+\frac{2 b (4 a C+5 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d} \]
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Rubi [A] time = 0.51274, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4096, 4076, 4047, 3771, 2641, 4046, 2639} \[ \frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (4 a^2 C+10 a b B+5 A b^2+3 b^2 C\right )}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 B+2 a b (3 A+C)+b^2 B\right )}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-5 a^2 (A-C)+10 a b B+b^2 (5 A+3 C)\right )}{5 d}+\frac{2 b (4 a C+5 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d} \]
Antiderivative was successfully verified.
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Rule 4096
Rule 4076
Rule 4047
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx &=\frac{2 C \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{2}{5} \int \frac{(a+b \sec (c+d x)) \left (\frac{1}{2} a (5 A-C)+\frac{1}{2} (5 A b+5 a B+3 b C) \sec (c+d x)+\frac{1}{2} (5 b B+4 a C) \sec ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 b (5 b B+4 a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 C \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{4}{15} \int \frac{\frac{3}{4} a^2 (5 A-C)+\frac{5}{4} \left (3 a^2 B+b^2 B+2 a b (3 A+C)\right ) \sec (c+d x)+\frac{3}{4} \left (5 A b^2+10 a b B+4 a^2 C+3 b^2 C\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 b (5 b B+4 a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 C \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{4}{15} \int \frac{\frac{3}{4} a^2 (5 A-C)+\frac{3}{4} \left (5 A b^2+10 a b B+4 a^2 C+3 b^2 C\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (3 a^2 B+b^2 B+2 a b (3 A+C)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 \left (5 A b^2+10 a b B+4 a^2 C+3 b^2 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 b (5 b B+4 a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 C \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{1}{5} \left (-10 a b B+5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (\left (3 a^2 B+b^2 B+2 a b (3 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (3 a^2 B+b^2 B+2 a b (3 A+C)\right ) \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 \left (5 A b^2+10 a b B+4 a^2 C+3 b^2 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 b (5 b B+4 a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 C \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{1}{5} \left (\left (-10 a b B+5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 \left (10 a b B-5 a^2 (A-C)+b^2 (5 A+3 C)\right ) \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 \left (3 a^2 B+b^2 B+2 a b (3 A+C)\right ) \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 \left (5 A b^2+10 a b B+4 a^2 C+3 b^2 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 b (5 b B+4 a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 C \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 2.08799, size = 271, normalized size = 1.12 \[ \frac{4 (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (5 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (3 a^2 B+2 a b (3 A+C)+b^2 B\right )+3 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (5 a^2 (A-C)-10 a b B-b^2 (5 A+3 C)\right )+15 a^2 C \sin (c+d x)+30 a b B \sin (c+d x)+10 a b C \tan (c+d x)+15 A b^2 \sin (c+d x)+5 b^2 B \tan (c+d x)+9 b^2 C \sin (c+d x)+3 b^2 C \tan (c+d x) \sec (c+d x)\right )}{15 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [B] time = 7.509, size = 1000, normalized size = 4.2 \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 (\frac{C b^{2} \sec \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \sec \left (d x + c\right )^{3} + A a^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, 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 )}^{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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