Optimal. Leaf size=397 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right )}{21 d}+\frac{2 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{315 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (54 a^2 b B+8 a^3 C+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{63 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right )}{15 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right )}{15 d}+\frac{2 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}{21 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d} \]
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Rubi [A] time = 0.866576, antiderivative size = 397, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{2 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{315 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (54 a^2 b B+8 a^3 C+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{63 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right )}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right )}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right )}{15 d}+\frac{2 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}{21 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d} \]
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))^3 \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))^3 \sin (c+d x)}{9 d}+\frac{2}{9} \int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 \left (\frac{1}{2} a (9 A+C)+\frac{1}{2} (9 A b+9 a B+7 b C) \sec (c+d x)+\frac{3}{2} (3 b B+2 a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 (3 b B+2 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{4}{63} \int \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (\frac{1}{4} a (63 a A+9 b B+13 a C)+\frac{1}{4} \left (126 a A b+63 a^2 B+45 b^2 B+86 a b C\right ) \sec (c+d x)+\frac{1}{4} \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (3 b B+2 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{8}{315} \int \sqrt{\sec (c+d x)} \left (\frac{5}{8} a^2 (63 a A+9 b B+13 a C)+\frac{21}{8} \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \sec (c+d x)+\frac{15}{8} \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (3 b B+2 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{8}{315} \int \sqrt{\sec (c+d x)} \left (\frac{5}{8} a^2 (63 a A+9 b B+13 a C)+\frac{15}{8} \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sec ^2(c+d x)\right ) \, dx+\frac{1}{15} \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (3 b B+2 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{1}{21} \left (21 a^2 b B+5 b^3 B+7 a^3 (3 A+C)+3 a b^2 (7 A+5 C)\right ) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{15} \left (-15 a^3 B-27 a b^2 B-9 a^2 b (5 A+3 C)-b^3 (9 A+7 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (3 b B+2 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac{1}{21} \left (\left (21 a^2 b B+5 b^3 B+7 a^3 (3 A+C)+3 a 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{1}{15} \left (\left (-15 a^3 B-27 a b^2 B-9 a^2 b (5 A+3 C)-b^3 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 \left (21 a^2 b B+5 b^3 B+7 a^3 (3 A+C)+3 a 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 (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (3 b B+2 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 7.07986, size = 566, normalized size = 1.43 \[ \frac{2 \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (105 a^3 A+105 a^2 b B+35 a^3 C+105 a A b^2+75 a b^2 C+25 b^3 B\right )+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-315 a^2 A b-189 a^2 b C-105 a^3 B-189 a b^2 B-63 A b^3-49 b^3 C\right )}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right )}{105 d (a \cos (c+d x)+b)^3 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac{4}{15} \sin (c+d x) \left (45 a^2 A b+27 a^2 b C+15 a^3 B+27 a b^2 B+9 A b^3+7 b^3 C\right )+\frac{4}{45} \sec ^2(c+d x) \left (27 a^2 b C \sin (c+d x)+27 a b^2 B \sin (c+d x)+9 A b^3 \sin (c+d x)+7 b^3 C \sin (c+d x)\right )+\frac{4}{21} \sec (c+d x) \left (21 a^2 b B \sin (c+d x)+7 a^3 C \sin (c+d x)+21 a A b^2 \sin (c+d x)+15 a b^2 C \sin (c+d x)+5 b^3 B \sin (c+d x)\right )+\frac{4}{7} \sec ^3(c+d x) \left (3 a b^2 C \sin (c+d x)+b^3 B \sin (c+d x)\right )+\frac{4}{9} b^3 C \tan (c+d x) \sec ^3(c+d x)\right )}{d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^3 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [B] time = 12.11, size = 1292, 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^{3} \sec \left (d x + c\right )^{5} +{\left (3 \, C a b^{2} + B b^{3}\right )} \sec \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \sec \left (d x + c\right )^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} \sec \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} 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 )}^{3} \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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