Integrand size = 35, antiderivative size = 253 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {4 a^3 (17 A+27 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (11 A+21 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {8 a^3 (16 A+21 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (73 A+63 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)} \]
2/9*A*(a+a*sec(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(7/2)+4/21*A*(a^2+a^2*sec (d*x+c))^2*sin(d*x+c)/a/d/sec(d*x+c)^(5/2)+2/315*(73*A+63*C)*(a^3+a^3*sec( d*x+c))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+8/105*a^3*(16*A+21*C)*sin(d*x+c)/d/s ec(d*x+c)^(1/2)+4/15*a^3*(17*A+27*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2* d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+ c)^(1/2)/d+4/21*a^3*(11*A+21*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1 /2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1 /2)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 4.57 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.81 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (240 (11 A+21 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-112 i (17 A+27 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (5712 i A+9072 i C+30 (97 A+84 C) \sin (c+d x)+14 (73 A+18 C) \sin (2 (c+d x))+270 A \sin (3 (c+d x))+35 A \sin (4 (c+d x)))\right )}{1260 d} \]
(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(240*(11*A + 21*C)*Sqrt[Co s[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(17*A + 27*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]*((5712*I)*A + (9072*I)*C + 30*(97*A + 84*C)* Sin[c + d*x] + 14*(73*A + 18*C)*Sin[2*(c + d*x)] + 270*A*Sin[3*(c + d*x)] + 35*A*Sin[4*(c + d*x)])))/(1260*d*E^(I*d*x))
Time = 1.62 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.04, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.543, Rules used = {3042, 4575, 27, 3042, 4505, 27, 3042, 4505, 27, 3042, 4484, 27, 3042, 4274, 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.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^3 \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4575 |
| \(\displaystyle \frac {2 \int \frac {(\sec (c+d x) a+a)^3 (6 a A+a (A+9 C) \sec (c+d x))}{2 \sec ^{\frac {7}{2}}(c+d x)}dx}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\sec (c+d x) a+a)^3 (6 a A+a (A+9 C) \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)}dx}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (6 a A+a (A+9 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {(\sec (c+d x) a+a)^2 \left ((73 A+63 C) a^2+(13 A+63 C) \sec (c+d x) a^2\right )}{2 \sec ^{\frac {5}{2}}(c+d x)}dx+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {(\sec (c+d x) a+a)^2 \left ((73 A+63 C) a^2+(13 A+63 C) \sec (c+d x) a^2\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((73 A+63 C) a^2+(13 A+63 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {3 (\sec (c+d x) a+a) \left (6 (16 A+21 C) a^3+(23 A+63 C) \sec (c+d x) a^3\right )}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {(\sec (c+d x) a+a) \left (6 (16 A+21 C) a^3+(23 A+63 C) \sec (c+d x) a^3\right )}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (6 (16 A+21 C) a^3+(23 A+63 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \sqrt {\sec (c+d x)}}-\frac {2}{3} \int -\frac {3 \left (7 (17 A+27 C) a^4+5 (11 A+21 C) \sec (c+d x) a^4\right )}{2 \sqrt {\sec (c+d x)}}dx\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \frac {7 (17 A+27 C) a^4+5 (11 A+21 C) \sec (c+d x) a^4}{\sqrt {\sec (c+d x)}}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \sqrt {\sec (c+d x)}}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \frac {7 (17 A+27 C) a^4+5 (11 A+21 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \sqrt {\sec (c+d x)}}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (7 a^4 (17 A+27 C) \int \frac {1}{\sqrt {\sec (c+d x)}}dx+5 a^4 (11 A+21 C) \int \sqrt {\sec (c+d x)}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \sqrt {\sec (c+d x)}}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (7 a^4 (17 A+27 C) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+5 a^4 (11 A+21 C) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \sqrt {\sec (c+d x)}}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (11 A+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+7 a^4 (17 A+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \sqrt {\sec (c+d x)}}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (11 A+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a^4 (17 A+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \sqrt {\sec (c+d x)}}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (11 A+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \sqrt {\sec (c+d x)}}+\frac {14 a^4 (17 A+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6}{5} \left (\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \sqrt {\sec (c+d x)}}+\frac {10 a^4 (11 A+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {14 a^4 (17 A+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {12 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
(2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((12* A*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((2* (73*A + 63*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/ 2)) + (6*((14*a^4*(17*A + 27*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (10*a^4*(11*A + 21*C)*Sqrt[Cos[c + d*x]]*Ellipt icF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^4*(16*A + 21*C)*Sin[c + d *x])/(d*Sqrt[Sec[c + d*x]])))/5)/7)/(9*a)
3.3.28.3.1 Defintions of rubi rules used
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*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_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Co t[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 *(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 7.82 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.61
| method | result | size |
| default | \(-\frac {4 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{3} \left (-560 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+2200 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-3412 A -252 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (2702 A +882 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-738 A -378 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+165 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-357 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+315 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-567 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{315 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(408\) |
| parts | \(\text {Expression too large to display}\) | \(1094\) |
-4/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(-560*A *cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+2200*A*cos(1/2*d*x+1/2*c)*sin(1/ 2*d*x+1/2*c)^8+(-3412*A-252*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(27 02*A+882*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-738*A-378*C)*sin(1/2 *d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+165*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))-357*A*(si n(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))+315*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))-567*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 higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.90 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (11 \, A + 21 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (11 \, A + 21 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 i \, \sqrt {2} {\left (17 \, A + 27 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} {\left (17 \, A + 27 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (35 \, A a^{3} \cos \left (d x + c\right )^{4} + 135 \, A a^{3} \cos \left (d x + c\right )^{3} + 7 \, {\left (34 \, A + 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (22 \, A + 21 \, C\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{315 \, d} \]
-2/315*(15*I*sqrt(2)*(11*A + 21*C)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(11*A + 21*C)*a^3*weierstrassPInvers e(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*I*sqrt(2)*(17*A + 27*C)*a^3*w eierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*I*sqrt(2)*(17*A + 27*C)*a^3*weierstrassZeta(-4, 0, weierstrass PInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (35*A*a^3*cos(d*x + c)^4 + 135*A*a^3*cos(d*x + c)^3 + 7*(34*A + 9*C)*a^3*cos(d*x + c)^2 + 15*(22*A + 21*C)*a^3*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]