Optimal. Leaf size=289 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right )}{21 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right )}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right )}{5 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right )}{5 d}+\frac{2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d} \]
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Rubi [A] time = 0.529411, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4096, 4076, 4047, 3768, 3771, 2639, 4046, 2641} \[ \frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right )}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right )}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right )}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right )}{5 d}+\frac{2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 4096
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))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2}{7} \int \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (\frac{1}{2} a (7 A+C)+\frac{1}{2} (7 A b+7 a B+5 b C) \sec (c+d x)+\frac{1}{2} (7 b B+4 a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 b (7 b B+4 a C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{4}{35} \int \sqrt{\sec (c+d x)} \left (\frac{5}{4} a^2 (7 A+C)+\frac{7}{4} \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sec (c+d x)+\frac{5}{4} \left (7 A b^2+14 a b B+4 a^2 C+5 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 b (7 b B+4 a C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{4}{35} \int \sqrt{\sec (c+d x)} \left (\frac{5}{4} a^2 (7 A+C)+\frac{5}{4} \left (7 A b^2+14 a b B+4 a^2 C+5 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx+\frac{1}{5} \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 \left (7 A b^2+14 a b B+4 a^2 C+5 b^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 b (7 b B+4 a C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{1}{5} \left (-10 a A b-5 a^2 B-3 b^2 B-6 a b C\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (14 a b B+7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 \left (7 A b^2+14 a b B+4 a^2 C+5 b^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 b (7 b B+4 a C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{1}{5} \left (\left (-10 a A b-5 a^2 B-3 b^2 B-6 a b C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (\left (14 a b B+7 a^2 (3 A+C)+b^2 (7 A+5 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 (10 a A b+5 a^2 B+3 b^2 B+6 a b 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 (14 a b B+7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \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{2 \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 \left (7 A b^2+14 a b B+4 a^2 C+5 b^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 b (7 b B+4 a C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 2.41429, size = 333, normalized size = 1.15 \[ \frac{4 (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (5 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right )-21 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (5 a^2 B+2 a b (5 A+3 C)+3 b^2 B\right )+105 a^2 B \sin (c+d x)+35 a^2 C \tan (c+d x)+210 a A b \sin (c+d x)+70 a b B \tan (c+d x)+126 a b C \sin (c+d x)+42 a b C \tan (c+d x) \sec (c+d x)+35 A b^2 \tan (c+d x)+63 b^2 B \sin (c+d x)+21 b^2 B \tan (c+d x) \sec (c+d x)+25 b^2 C \tan (c+d x)+15 b^2 C \tan (c+d x) \sec ^2(c+d x)\right )}{105 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [B] time = 9.167, size = 947, normalized size = 3.3 \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^{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 )\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 )}^{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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