Optimal. Leaf size=663 \[ \frac{2 \sqrt{\sec (c+d x)} \left (-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)+80 a^3 b^2 B+5 a^5 B-80 a b^4 B+128 A b^5\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{15 a^5 d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \left (-a^2 b^2 (71 A-15 C)+a^4 (3 A-35 C)+50 a^3 b B-30 a b^3 B+48 A b^4\right ) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left (a^2-b^2\right )^2 \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left (-2 a^2 b^2 (6 A-C)+9 a^3 b B-6 a^4 C-5 a b^3 B+8 A b^4\right )}{3 a^2 d \left (a^2-b^2\right )^2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \sin (c+d x) \left (-2 a^2 b^3 (49 A-10 C)+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-5 a^5 B-40 a b^4 B+64 A b^5\right ) \sqrt{a+b \sec (c+d x)}}{15 a^4 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)}}+\frac{2 \left (5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)+140 a^3 b^3 B-40 a^5 b B-80 a b^5 B+128 A b^6\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{15 a^5 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.22685, antiderivative size = 663, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac{2 \sin (c+d x) \left (-a^2 b^2 (71 A-15 C)+a^4 (3 A-35 C)+50 a^3 b B-30 a b^3 B+48 A b^4\right ) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left (a^2-b^2\right )^2 \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left (-2 a^2 b^2 (6 A-C)+9 a^3 b B-6 a^4 C-5 a b^3 B+8 A b^4\right )}{3 a^2 d \left (a^2-b^2\right )^2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \sin (c+d x) \left (-2 a^2 b^3 (49 A-10 C)+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-5 a^5 B-40 a b^4 B+64 A b^5\right ) \sqrt{a+b \sec (c+d x)}}{15 a^4 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\sec (c+d x)} \left (-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)+80 a^3 b^2 B+5 a^5 B-80 a b^4 B+128 A b^5\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{15 a^5 d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)+140 a^3 b^3 B-40 a^5 b B-80 a b^5 B+128 A b^6\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{15 a^5 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4100
Rule 4104
Rule 4035
Rule 3856
Rule 2655
Rule 2653
Rule 3858
Rule 2663
Rule 2661
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx &=\frac{2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (8 A b^2-5 a b B-a^2 (3 A-5 C)\right )+\frac{3}{2} a (A b-a B+b C) \sec (c+d x)-3 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac{2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{4 \int \frac{\frac{1}{4} \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right )+\frac{1}{4} a \left (2 A b^3+3 a^3 B+a b^2 B-2 a^2 b (3 A+2 C)\right ) \sec (c+d x)-\left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{8 \int \frac{\frac{3}{8} \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right )+\frac{1}{8} a \left (16 A b^4+30 a^3 b B-10 a b^3 B-3 a^4 (3 A+5 C)-a^2 b^2 (27 A+5 C)\right ) \sec (c+d x)-\frac{1}{4} b \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{15 a^3 \left (a^2-b^2\right )^2}\\ &=\frac{2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{16 \int \frac{\frac{3}{16} \left (128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B+5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)\right )+\frac{3}{16} a \left (32 A b^5+5 a^5 B+35 a^3 b^2 B-20 a b^4 B-2 a^2 b^3 (22 A-5 C)-2 a^4 b (4 A+15 C)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{45 a^4 \left (a^2-b^2\right )^2}\\ &=\frac{2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{\left (128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B+5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)\right ) \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right )^2}+\frac{\left (128 A b^5+5 a^5 B+80 a^3 b^2 B-80 a b^4 B-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right )}\\ &=\frac{2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{\left (\left (128 A b^5+5 a^5 B+80 a^3 b^2 B-80 a b^4 B-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)\right ) \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{b+a \cos (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B+5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{15 a^5 \left (a^2-b^2\right )^2 \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ &=\frac{2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{\left (\left (128 A b^5+5 a^5 B+80 a^3 b^2 B-80 a b^4 B-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{15 a^5 \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B+5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}} \, dx}{15 a^5 \left (a^2-b^2\right )^2 \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}\\ &=\frac{2 \left (128 A b^5+5 a^5 B+80 a^3 b^2 B-80 a b^4 B-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{\sec (c+d x)}}{15 a^5 \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B+5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{a+b \sec (c+d x)}}{15 a^5 \left (a^2-b^2\right )^2 d \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}+\frac{2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 10.3919, size = 9192, normalized size = 13.86 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.879, size = 11337, normalized size = 17.1 \begin{align*} \text{output 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{{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt{b \sec \left (d x + c\right ) + a} \sqrt{\sec \left (d x + c\right )}}{b^{3} \sec \left (d x + c\right )^{6} + 3 \, a b^{2} \sec \left (d x + c\right )^{5} + 3 \, a^{2} b \sec \left (d x + c\right )^{4} + a^{3} \sec \left (d x + c\right )^{3}}, 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{C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]