Integrand size = 43, antiderivative size = 343 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\frac {4 a^3 (175 A+195 B+221 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {4 a^3 (95 A+105 B+121 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)} \] Output:
4/195*a^3*(175*A+195*B+221*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+4/231*a^3*(95*A+105*B+121*C)*cos(d*x+c)^(1/2 )*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+20/9009*a^3*(2 36*A+273*B+286*C)*sin(d*x+c)/d/sec(d*x+c)^(5/2)+4/585*a^3*(175*A+195*B+221 *C)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+4/231*a^3*(95*A+105*B+121*C)*sin(d*x+c)/ d/sec(d*x+c)^(1/2)+2/13*A*(a+a*sec(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(11/2 )+2/143*(6*A+13*B)*(a^2+a^2*sec(d*x+c))^2*sin(d*x+c)/a/d/sec(d*x+c)^(9/2)+ 2/1287*(145*A+195*B+143*C)*(a^3+a^3*sec(d*x+c))*sin(d*x+c)/d/sec(d*x+c)^(7 /2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.51 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.87 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (12480 (95 A+105 B+121 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-4928 i (175 A+195 B+221 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (2587200 i A+2882880 i B+3267264 i C+780 (1811 A+1953 B+2134 C) \sin (c+d x)+77 (7825 A+7800 B+7592 C) \sin (2 (c+d x))+251550 A \sin (3 (c+d x))+221130 B \sin (3 (c+d x))+154440 C \sin (3 (c+d x))+90860 A \sin (4 (c+d x))+60060 B \sin (4 (c+d x))+20020 C \sin (4 (c+d x))+24570 A \sin (5 (c+d x))+8190 B \sin (5 (c+d x))+3465 A \sin (6 (c+d x)))\right )}{720720 d} \] Input:
Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)) /Sec[c + d*x]^(13/2),x]
Output:
(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(12480*(95*A + 105*B + 121 *C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (4928*I)*(175*A + 195*B + 221*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[ 1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((2587200*I)*A + (2882 880*I)*B + (3267264*I)*C + 780*(1811*A + 1953*B + 2134*C)*Sin[c + d*x] + 7 7*(7825*A + 7800*B + 7592*C)*Sin[2*(c + d*x)] + 251550*A*Sin[3*(c + d*x)] + 221130*B*Sin[3*(c + d*x)] + 154440*C*Sin[3*(c + d*x)] + 90860*A*Sin[4*(c + d*x)] + 60060*B*Sin[4*(c + d*x)] + 20020*C*Sin[4*(c + d*x)] + 24570*A*S in[5*(c + d*x)] + 8190*B*Sin[5*(c + d*x)] + 3465*A*Sin[6*(c + d*x)])))/(72 0720*d*E^(I*d*x))
Time = 2.06 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 4574, 27, 3042, 4505, 27, 3042, 4505, 3042, 4484, 27, 3042, 4274, 3042, 4256, 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.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \frac {2 \int \frac {(\sec (c+d x) a+a)^3 (a (6 A+13 B)+a (5 A+13 C) \sec (c+d x))}{2 \sec ^{\frac {11}{2}}(c+d x)}dx}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\sec (c+d x) a+a)^3 (a (6 A+13 B)+a (5 A+13 C) \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)}dx}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (6 A+13 B)+a (5 A+13 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {2}{11} \int \frac {(\sec (c+d x) a+a)^2 \left ((145 A+195 B+143 C) a^2+(85 A+65 B+143 C) \sec (c+d x) a^2\right )}{2 \sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \int \frac {(\sec (c+d x) a+a)^2 \left ((145 A+195 B+143 C) a^2+(85 A+65 B+143 C) \sec (c+d x) a^2\right )}{\sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((145 A+195 B+143 C) a^2+(85 A+65 B+143 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \frac {(\sec (c+d x) a+a) \left (5 (236 A+273 B+286 C) a^3+(745 A+780 B+1001 C) \sec (c+d x) a^3\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (5 (236 A+273 B+286 C) a^3+(745 A+780 B+1001 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {10 a^4 (236 A+273 B+286 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2}{7} \int -\frac {77 (175 A+195 B+221 C) a^4+117 (95 A+105 B+121 C) \sec (c+d x) a^4}{2 \sec ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \int \frac {77 (175 A+195 B+221 C) a^4+117 (95 A+105 B+121 C) \sec (c+d x) a^4}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {10 a^4 (236 A+273 B+286 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \int \frac {77 (175 A+195 B+221 C) a^4+117 (95 A+105 B+121 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {10 a^4 (236 A+273 B+286 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^4 (175 A+195 B+221 C) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)}dx+117 a^4 (95 A+105 B+121 C) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {10 a^4 (236 A+273 B+286 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^4 (175 A+195 B+221 C) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+117 a^4 (95 A+105 B+121 C) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {10 a^4 (236 A+273 B+286 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^4 (175 A+195 B+221 C) \left (\frac {3}{5} \int \frac {1}{\sqrt {\sec (c+d x)}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+117 a^4 (95 A+105 B+121 C) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {10 a^4 (236 A+273 B+286 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^4 (175 A+195 B+221 C) \left (\frac {3}{5} \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+117 a^4 (95 A+105 B+121 C) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {10 a^4 (236 A+273 B+286 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^4 (175 A+195 B+221 C) \left (\frac {3}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+117 a^4 (95 A+105 B+121 C) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {10 a^4 (236 A+273 B+286 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^4 (175 A+195 B+221 C) \left (\frac {3}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+117 a^4 (95 A+105 B+121 C) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {10 a^4 (236 A+273 B+286 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (117 a^4 (95 A+105 B+121 C) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+77 a^4 (175 A+195 B+221 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )+\frac {10 a^4 (236 A+273 B+286 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2}{9} \left (\frac {10 a^4 (236 A+273 B+286 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{7} \left (117 a^4 (95 A+105 B+121 C) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+77 a^4 (175 A+195 B+221 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )\right )\right )+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)}\) |
Input:
Int[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2),x]
Output:
(2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2)) + ((2 *(6*A + 13*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^ (9/2)) + ((2*(145*A + 195*B + 143*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x] )/(9*d*Sec[c + d*x]^(7/2)) + (2*((10*a^4*(236*A + 273*B + 286*C)*Sin[c + d *x])/(7*d*Sec[c + d*x]^(5/2)) + (77*a^4*(175*A + 195*B + 221*C)*((6*Sqrt[C os[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*Sin[ c + d*x])/(5*d*Sec[c + d*x]^(3/2))) + 117*a^4*(95*A + 105*B + 121*C)*((2*S qrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2 *Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])))/7))/9)/11)/(13*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[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Simp[1/(d*n) Int[(d*Csc[e + f*x])^( n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*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[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] - Sim p[b/(a*d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Sim p[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, 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[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[ e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] , x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 57.83 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.68
| method | result | size |
| default | \(-\frac {4 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{3} \left (-221760 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (1058400 A +131040 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2122400 A -567840 B -80080 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (2331040 A +1004640 B +314600 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1535860 A -939120 B -487916 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (633710 A +510510 B +386386 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-121230 A -114660 B -105534 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+18525 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-40425 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+20475 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-45045 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+23595 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-51051 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{45045 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(576\) |
| parts | \(\text {Expression too large to display}\) | \(1319\) |
Input:
int((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(13/2),x ,method=_RETURNVERBOSE)
Output:
-4/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(-221 760*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^14+(1058400*A+131040*B)*sin(1/ 2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-2122400*A-567840*B-80080*C)*sin(1/2*d *x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(2331040*A+1004640*B+314600*C)*sin(1/2*d*x +1/2*c)^8*cos(1/2*d*x+1/2*c)+(-1535860*A-939120*B-487916*C)*sin(1/2*d*x+1/ 2*c)^6*cos(1/2*d*x+1/2*c)+(633710*A+510510*B+386386*C)*sin(1/2*d*x+1/2*c)^ 4*cos(1/2*d*x+1/2*c)+(-121230*A-114660*B-105534*C)*sin(1/2*d*x+1/2*c)^2*co s(1/2*d*x+1/2*c)+18525*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c )^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-40425*A*(sin(1/2*d*x+1/ 2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c ),2^(1/2))+20475*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1) ^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-45045*B*(sin(1/2*d*x+1/2*c)^2 )^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1 /2))+23595*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2) *EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-51051*C*(sin(1/2*d*x+1/2*c)^2)^(1/2 )*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))/ (-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2 *cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.86 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (195 i \, \sqrt {2} {\left (95 \, A + 105 \, B + 121 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 195 i \, \sqrt {2} {\left (95 \, A + 105 \, B + 121 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (175 \, A + 195 \, B + 221 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (175 \, A + 195 \, B + 221 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (3465 \, A a^{3} \cos \left (d x + c\right )^{6} + 4095 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (50 \, A + 39 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 585 \, {\left (38 \, A + 42 \, B + 33 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 154 \, {\left (175 \, A + 195 \, B + 221 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 390 \, {\left (95 \, A + 105 \, B + 121 \, C\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{45045 \, d} \] Input:
integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 3/2),x, algorithm="fricas")
Output:
-2/45045*(195*I*sqrt(2)*(95*A + 105*B + 121*C)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 195*I*sqrt(2)*(95*A + 105*B + 121*C)* a^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*I*sqrt (2)*(175*A + 195*B + 221*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse (-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 231*I*sqrt(2)*(175*A + 195*B + 2 21*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (3465*A*a^3*cos(d*x + c)^6 + 4095*(3*A + B)*a^3*cos(d* x + c)^5 + 385*(50*A + 39*B + 13*C)*a^3*cos(d*x + c)^4 + 585*(38*A + 42*B + 33*C)*a^3*cos(d*x + c)^3 + 154*(175*A + 195*B + 221*C)*a^3*cos(d*x + c)^ 2 + 390*(95*A + 105*B + 121*C)*a^3*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)* *(13/2),x)
Output:
Timed out
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 3/2),x, algorithm="maxima")
Output:
Timed out
\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 3/2),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 ec(d*x + c)^(13/2), x)
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}} \,d x \] Input:
int(((a + a/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co s(c + d*x))^(13/2),x)
Output:
int(((a + a/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co s(c + d*x))^(13/2), x)
\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=a^{3} \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{7}}d x \right ) a +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{6}}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{6}}d x \right ) b +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x \right ) a +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) a +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) b +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) b +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) c \right ) \] Input:
int((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(13/2),x )
Output:
a**3*(int(sqrt(sec(c + d*x))/sec(c + d*x)**7,x)*a + 3*int(sqrt(sec(c + d*x ))/sec(c + d*x)**6,x)*a + int(sqrt(sec(c + d*x))/sec(c + d*x)**6,x)*b + 3* int(sqrt(sec(c + d*x))/sec(c + d*x)**5,x)*a + 3*int(sqrt(sec(c + d*x))/sec (c + d*x)**5,x)*b + 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)*a + 3*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)*b + 3*int(sqrt(sec(c + d*x))/sec(c + d*x)**3,x )*c + int(sqrt(sec(c + d*x))/sec(c + d*x)**2,x)*c)