Optimal. Leaf size=444 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)\right )}{231 d}+\frac{2 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left (2 a^2 b (673 A+891 C)+539 a^3 B+1353 a b^2 B+192 A b^3\right )}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+682 a b^3 B+64 A b^4\right )}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac{2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]
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Rubi [A] time = 1.31581, antiderivative size = 444, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4094, 4074, 4047, 3771, 2639, 4045, 2641} \[ \frac{2 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left (2 a^2 b (673 A+891 C)+539 a^3 B+1353 a b^2 B+192 A b^3\right )}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+682 a b^3 B+64 A b^4\right )}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)\right )}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac{2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4094
Rule 4074
Rule 4047
Rule 3771
Rule 2639
Rule 4045
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2}{11} \int \frac{(a+b \sec (c+d x))^3 \left (\frac{1}{2} (8 A b+11 a B)+\frac{1}{2} (9 a A+11 b B+11 a C) \sec (c+d x)+\frac{1}{2} b (A+11 C) \sec ^2(c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{4}{99} \int \frac{(a+b \sec (c+d x))^2 \left (\frac{3}{4} \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right )+\frac{1}{4} \left (146 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \sec (c+d x)+\frac{1}{4} b (17 A b+11 a B+99 b C) \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{8}{693} \int \frac{(a+b \sec (c+d x)) \left (\frac{1}{8} \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right )+\frac{1}{8} \left (1441 a^2 b B+693 b^3 B+45 a^3 (9 A+11 C)+a b^2 (1381 A+2079 C)\right ) \sec (c+d x)+\frac{1}{8} b \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \sec ^2(c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}-\frac{16 \int \frac{-\frac{15}{16} \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right )-\frac{231}{16} \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sec (c+d x)-\frac{5}{16} b^2 \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{3465}\\ &=\frac{2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}-\frac{16 \int \frac{-\frac{15}{16} \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right )-\frac{5}{16} b^2 \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{3465}-\frac{1}{15} \left (-7 a^4 B-54 a^2 b^2 B-15 b^4 B-12 a b^3 (3 A+5 C)-4 a^3 b (7 A+9 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}-\frac{1}{231} \left (-220 a^3 b B-308 a b^3 B-77 b^4 (A+3 C)-66 a^2 b^2 (5 A+7 C)-5 a^4 (9 A+11 C)\right ) \int \sqrt{\sec (c+d x)} \, dx-\frac{1}{15} \left (\left (-7 a^4 B-54 a^2 b^2 B-15 b^4 B-12 a b^3 (3 A+5 C)-4 a^3 b (7 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 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 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}-\frac{1}{231} \left (\left (-220 a^3 b B-308 a b^3 B-77 b^4 (A+3 C)-66 a^2 b^2 (5 A+7 C)-5 a^4 (9 A+11 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 (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 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 (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{231 d}+\frac{2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 7.06534, size = 580, normalized size = 1.31 \[ \frac{2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (1650 a^2 A b^2+225 a^4 A+2310 a^2 b^2 C+1100 a^3 b B+275 a^4 C+1540 a b^3 B+385 A b^4+1155 b^4 C\right )+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (2156 a^3 A b+4158 a^2 b^2 B+2772 a^3 b C+539 a^4 B+2772 a A b^3+4620 a b^3 C+1155 b^4 B\right )}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right )}{1155 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac{1}{154} a^2 \sin (4 (c+d x)) \left (16 a^2 A+11 a^2 C+44 a b B+66 A b^2\right )+\frac{1}{90} a \sin (c+d x) \left (76 a^2 A b+72 a^2 b C+19 a^3 B+108 a b^2 B+72 A b^3\right )+\frac{1}{180} a \sin (3 (c+d x)) \left (172 a^2 A b+144 a^2 b C+43 a^3 B+216 a b^2 B+144 A b^3\right )+\frac{\sin (2 (c+d x)) \left (6864 a^2 A b^2+1041 a^4 A+7392 a^2 b^2 C+4576 a^3 b B+1144 a^4 C+4928 a b^3 B+1232 A b^4\right )}{1848}+\frac{1}{36} a^3 (a B+4 A b) \sin (5 (c+d x))+\frac{1}{88} a^4 A \sin (6 (c+d x))\right )}{d \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.626, size = 1273, normalized size = 2.9 \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 (\frac{C b^{4} \sec \left (d x + c\right )^{6} +{\left (4 \, C a b^{3} + B b^{4}\right )} \sec \left (d x + c\right )^{5} + A a^{4} +{\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \sec \left (d x + c\right )^{4} + 2 \,{\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \sec \left (d x + c\right )^{3} +{\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \sec \left (d x + c\right )^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac{11}{2}}}, 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 )}^{4}}{\sec \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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