Integrand size = 43, antiderivative size = 307 \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 a^3 (27 A+21 B+17 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (21 A+13 B+11 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (27 A+21 B+17 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (42 A+41 B+32 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 (3 B+2 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac {2 (63 A+99 B+73 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d} \] Output:
-4/15*a^3*(27*A+21*B+17*C)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2 ^(1/2))*sec(d*x+c)^(1/2)/d+4/21*a^3*(21*A+13*B+11*C)*cos(d*x+c)^(1/2)*Inve rseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+4/15*a^3*(27*A+21*B+ 17*C)*sec(d*x+c)^(1/2)*sin(d*x+c)/d+4/105*a^3*(42*A+41*B+32*C)*sec(d*x+c)^ (3/2)*sin(d*x+c)/d+2/9*C*sec(d*x+c)^(3/2)*(a+a*sec(d*x+c))^3*sin(d*x+c)/d+ 2/21*(3*B+2*C)*sec(d*x+c)^(3/2)*(a^2+a^2*sec(d*x+c))^2*sin(d*x+c)/a/d+2/31 5*(63*A+99*B+73*C)*sec(d*x+c)^(3/2)*(a^3+a^3*sec(d*x+c))*sin(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.16 (sec) , antiderivative size = 1267, normalized size of antiderivative = 4.13 \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(3*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I )*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A* Cos[2*c + 2*d*x])) + (7*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]* Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)* (c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*( A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (17*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c] *(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hype rgeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*( a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(45*Sqrt[2] *d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (A*Sqrt[ Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (13*B*Sqrt[Cos[c + d*x ]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^...
Time = 1.91 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.99, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 4576, 27, 3042, 4506, 27, 3042, 4506, 27, 3042, 4485, 27, 3042, 4274, 3042, 4255, 3042, 4258, 3042, 3119, 3120}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4576 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^3 (a (9 A+C)+3 a (3 B+2 C) \sec (c+d x))dx}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^3 (a (9 A+C)+3 a (3 B+2 C) \sec (c+d x))dx}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (9 A+C)+3 a (3 B+2 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {1}{2} \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^2 \left ((63 A+9 B+13 C) a^2+(63 A+99 B+73 C) \sec (c+d x) a^2\right )dx+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \int \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^2 \left ((63 A+9 B+13 C) a^2+(63 A+99 B+73 C) \sec (c+d x) a^2\right )dx+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((63 A+9 B+13 C) a^2+(63 A+99 B+73 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int 3 \sqrt {\sec (c+d x)} (\sec (c+d x) a+a) \left ((63 A+24 B+23 C) a^3+3 (42 A+41 B+32 C) \sec (c+d x) a^3\right )dx+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \sqrt {\sec (c+d x)} (\sec (c+d x) a+a) \left ((63 A+24 B+23 C) a^3+3 (42 A+41 B+32 C) \sec (c+d x) a^3\right )dx+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((63 A+24 B+23 C) a^3+3 (42 A+41 B+32 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4485 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\frac {2}{3} \int \frac {3}{2} \sqrt {\sec (c+d x)} \left (5 (21 A+13 B+11 C) a^4+7 (27 A+21 B+17 C) \sec (c+d x) a^4\right )dx+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \sqrt {\sec (c+d x)} \left (5 (21 A+13 B+11 C) a^4+7 (27 A+21 B+17 C) \sec (c+d x) a^4\right )dx+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (21 A+13 B+11 C) a^4+7 (27 A+21 B+17 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (7 a^4 (27 A+21 B+17 C) \int \sec ^{\frac {3}{2}}(c+d x)dx+5 a^4 (21 A+13 B+11 C) \int \sqrt {\sec (c+d x)}dx+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+13 B+11 C) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+7 a^4 (27 A+21 B+17 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+13 B+11 C) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+7 a^4 (27 A+21 B+17 C) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+13 B+11 C) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+7 a^4 (27 A+21 B+17 C) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+13 B+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+7 a^4 (27 A+21 B+17 C) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+13 B+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a^4 (27 A+21 B+17 C) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+13 B+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}+7 a^4 (27 A+21 B+17 C) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}+\frac {6}{5} \left (\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}+\frac {10 a^4 (21 A+13 B+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+7 a^4 (27 A+21 B+17 C) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
Input:
Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[ c + d*x]^2),x]
Output:
(2*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d) + ((6*( 3*B + 2*C)*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(7* d) + ((2*(63*A + 99*B + 73*C)*Sec[c + d*x]^(3/2)*(a^4 + a^4*Sec[c + d*x])* Sin[c + d*x])/(5*d) + (6*((10*a^4*(21*A + 13*B + 11*C)*Sqrt[Cos[c + d*x]]* EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a^4*(42*A + 41*B + 32 *C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/d + 7*a^4*(27*A + 21*B + 17*C)*((-2*S qrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*Sqr t[Sec[c + d*x]]*Sin[c + d*x])/d)))/5)/7)/(9*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[ e + f*x]*((d*Csc[e + f*x])^n/(f*(n + 1))), x] + Simp[1/(n + 1) Int[(d*Csc [e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x ], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && !LeQ[ n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x] )^n*Simp[a*A*d*(m + n) + B*(b*d*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))* Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Cs c[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(b*(m + n + 1)) Int[(a + b*Cs c[e + f*x])^m*(d*Csc[e + f*x])^n*Simp[A*b*(m + n + 1) + b*C*n + (a*C*m + b* B*(m + n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m , n}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1237\) vs. \(2(278)=556\).
Time = 16.12 (sec) , antiderivative size = 1238, normalized size of antiderivative = 4.03
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1238\) |
| parts | \(\text {Expression too large to display}\) | \(1576\) |
Input:
int(sec(d*x+c)^(1/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
Output:
-a^3*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A*(sin(1 /2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1 /2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+ 2*C*(-1/144*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c) ^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^5-7/180*cos(1/2*d*x+1/2*c)*(-2*sin(1/ 2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^3-14 /15*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)* sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2* d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2) *EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*( -2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2* c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1 /2*c),2^(1/2))))+16*(3/8*A+1/8*B)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2* c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d* x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(c os(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2))+16*(1/8*B+3/8*C)* (-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^( 1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+ 1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+5/21*(si n(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2...
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.99 \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (21 \, A + 13 \, B + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (21 \, A + 13 \, B + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (27 \, A + 21 \, B + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (27 \, A + 21 \, B + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (42 \, {\left (27 \, A + 21 \, B + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 15 \, {\left (21 \, A + 26 \, B + 22 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 7 \, {\left (9 \, A + 27 \, B + 34 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 45 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 35 \, C a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{315 \, d \cos \left (d x + c\right )^{4}} \] Input:
integrate(sec(d*x+c)^(1/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
Output:
-2/315*(15*I*sqrt(2)*(21*A + 13*B + 11*C)*a^3*cos(d*x + c)^4*weierstrassPI nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(21*A + 13*B + 11*C)*a^3*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin( d*x + c)) + 21*I*sqrt(2)*(27*A + 21*B + 17*C)*a^3*cos(d*x + c)^4*weierstra ssZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*I*sqrt(2)*(27*A + 21*B + 17*C)*a^3*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (42*(27*A + 21*B + 17*C)*a^3*cos(d*x + c)^4 + 15*(21*A + 26*B + 22*C)*a^3*cos(d*x + c)^3 + 7*(9*A + 27*B + 34*C)*a^3*cos(d*x + c)^2 + 45*(B + 3*C)*a^3*cos(d*x + c) + 35*C*a^3)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^4)
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**(1/2)*(a+a*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+ c)**2),x)
Output:
Timed out
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)^(1/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
Output:
Timed out
\[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )} \,d x } \] Input:
integrate(sec(d*x+c)^(1/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^3*s qrt(sec(d*x + c)), x)
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \] Input:
int((a + a/cos(c + d*x))^3*(1/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/ cos(c + d*x)^2),x)
Output:
int((a + a/cos(c + d*x))^3*(1/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/ cos(c + d*x)^2), x)
\[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^{3} \left (\left (\int \sqrt {\sec \left (d x +c \right )}d x \right ) a +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{4}d x \right ) b +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{4}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}d x \right ) a +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}d x \right ) b +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}d x \right ) c +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}d x \right ) a +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}d x \right ) b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}d x \right ) c +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) a +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) b \right ) \] Input:
int(sec(d*x+c)^(1/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
a**3*(int(sqrt(sec(c + d*x)),x)*a + int(sqrt(sec(c + d*x))*sec(c + d*x)**5 ,x)*c + int(sqrt(sec(c + d*x))*sec(c + d*x)**4,x)*b + 3*int(sqrt(sec(c + d *x))*sec(c + d*x)**4,x)*c + int(sqrt(sec(c + d*x))*sec(c + d*x)**3,x)*a + 3*int(sqrt(sec(c + d*x))*sec(c + d*x)**3,x)*b + 3*int(sqrt(sec(c + d*x))*s ec(c + d*x)**3,x)*c + 3*int(sqrt(sec(c + d*x))*sec(c + d*x)**2,x)*a + 3*in t(sqrt(sec(c + d*x))*sec(c + d*x)**2,x)*b + int(sqrt(sec(c + d*x))*sec(c + d*x)**2,x)*c + 3*int(sqrt(sec(c + d*x))*sec(c + d*x),x)*a + int(sqrt(sec( c + d*x))*sec(c + d*x),x)*b)