Optimal. Leaf size=426 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (4 a^3 b (5 A+7 C)+42 a^2 b^2 B+5 a^4 B+28 a b^3 (A+3 C)+21 b^4 B\right )}{21 d}+\frac{2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{315 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (7 a^2 (7 A+9 C)+162 a b B+3 b^2 (41 A-105 C)\right )}{315 d}+\frac{2 a \sin (c+d x) \left (a^2 (202 A b+294 b C)+75 a^3 B+261 a b^2 B+64 A b^3\right )}{315 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+36 a^3 b B+60 a b^3 B+15 b^4 (A-C)\right )}{15 d}+\frac{2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.31303, antiderivative size = 426, 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, 2641, 4046, 2639} \[ \frac{2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{315 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (7 a^2 (7 A+9 C)+162 a b B+3 b^2 (41 A-105 C)\right )}{315 d}+\frac{2 a \sin (c+d x) \left (a^2 (202 A b+294 b C)+75 a^3 B+261 a b^2 B+64 A b^3\right )}{315 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (5 A+7 C)+42 a^2 b^2 B+5 a^4 B+28 a b^3 (A+3 C)+21 b^4 B\right )}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+36 a^3 b B+60 a b^3 B+15 b^4 (A-C)\right )}{15 d}+\frac{2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4094
Rule 4074
Rule 4047
Rule 3771
Rule 2641
Rule 4046
Rule 2639
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{9}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2}{9} \int \frac{(a+b \sec (c+d x))^3 \left (\frac{1}{2} (8 A b+9 a B)+\frac{1}{2} (7 a A+9 b B+9 a C) \sec (c+d x)-\frac{1}{2} b (A-9 C) \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{4}{63} \int \frac{(a+b \sec (c+d x))^2 \left (\frac{1}{4} \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right )+\frac{1}{4} \left (82 a A b+45 a^2 B+63 b^2 B+126 a b C\right ) \sec (c+d x)-\frac{3}{4} b (5 A b+3 a B-21 b C) \sec ^2(c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{8}{315} \int \frac{(a+b \sec (c+d x)) \left (\frac{3}{8} \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right )+\frac{1}{8} \left (531 a^2 b B+315 b^3 B+21 a^3 (7 A+9 C)+a b^2 (479 A+945 C)\right ) \sec (c+d x)-\frac{1}{8} b \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)}}+\frac{2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{16}{945} \int \frac{-\frac{3}{16} \left (192 A b^4+756 a^3 b B+1098 a b^3 B+21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)\right )-\frac{45}{16} \left (5 a^4 B+42 a^2 b^2 B+21 b^4 B+28 a b^3 (A+3 C)+4 a^3 b (5 A+7 C)\right ) \sec (c+d x)+\frac{3}{16} b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)}}+\frac{2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{16}{945} \int \frac{-\frac{3}{16} \left (192 A b^4+756 a^3 b B+1098 a b^3 B+21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)\right )+\frac{3}{16} b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx-\frac{1}{21} \left (-5 a^4 B-42 a^2 b^2 B-21 b^4 B-28 a b^3 (A+3 C)-4 a^3 b (5 A+7 C)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d}+\frac{2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{1}{15} \left (-36 a^3 b B-60 a b^3 B-15 b^4 (A-C)-18 a^2 b^2 (3 A+5 C)-a^4 (7 A+9 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx-\frac{1}{21} \left (\left (-5 a^4 B-42 a^2 b^2 B-21 b^4 B-28 a b^3 (A+3 C)-4 a^3 b (5 A+7 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 (5 a^4 B+42 a^2 b^2 B+21 b^4 B+28 a b^3 (A+3 C)+4 a^3 b (5 A+7 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 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d}+\frac{2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{1}{15} \left (\left (-36 a^3 b B-60 a b^3 B-15 b^4 (A-C)-18 a^2 b^2 (3 A+5 C)-a^4 (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 (36 a^3 b B+60 a b^3 B+15 b^4 (A-C)+18 a^2 b^2 (3 A+5 C)+a^4 (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 (5 a^4 B+42 a^2 b^2 B+21 b^4 B+28 a b^3 (A+3 C)+4 a^3 b (5 A+7 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 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d}+\frac{2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 7.33785, size = 517, normalized size = 1.21 \[ \frac{2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (100 a^3 A b+210 a^2 b^2 B+140 a^3 b C+25 a^4 B+140 a A b^3+420 a b^3 C+105 b^4 B\right )+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (378 a^2 A b^2+49 a^4 A+630 a^2 b^2 C+252 a^3 b B+63 a^4 C+420 a b^3 B+105 A b^4-105 b^4 C\right )}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right )}{105 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac{1}{180} a^2 \sin (3 (c+d x)) \left (43 a^2 A+36 a^2 C+144 a b B+216 A b^2\right )+\frac{1}{21} a \sin (2 (c+d x)) \left (52 a^2 A b+56 a^2 b C+13 a^3 B+84 a b^2 B+56 A b^3\right )+\frac{1}{90} \sin (c+d x) \left (108 a^2 A b^2+19 a^4 A+72 a^3 b B+18 a^4 C+360 b^4 C\right )+\frac{1}{14} a^3 (a B+4 A b) \sin (4 (c+d x))+\frac{1}{36} a^4 A \sin (5 (c+d x))\right )}{d \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 3.615, size = 1652, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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{9}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]