Integrand size = 33, antiderivative size = 277 \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {4 a^3 (15 A+17 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (105 A+121 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (15 A+17 B) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (105 A+121 B) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)} \] Output:
4/15*a^3*(15*A+17*B)*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*(105*A+121*B)*cos(d*x+c)^(1/2)*InverseJacob iAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+20/693*a^3*(21*A+22*B)*sin(d *x+c)/d/sec(d*x+c)^(5/2)+4/45*a^3*(15*A+17*B)*sin(d*x+c)/d/sec(d*x+c)^(3/2 )+4/231*a^3*(105*A+121*B)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/11*a*A*(a+a*sec( d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(9/2)+2/99*(15*A+11*B)*(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 = 3.90 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.86 \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (480 (105 A+121 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-2464 i (15 A+17 B) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (110880 i A+125664 i B+30 (1953 A+2134 B) \sin (c+d x)+308 (75 A+73 B) \sin (2 (c+d x))+8505 A \sin (3 (c+d x))+5940 B \sin (3 (c+d x))+2310 A \sin (4 (c+d x))+770 B \sin (4 (c+d x))+315 A \sin (5 (c+d x)))\right )}{27720 d} \] Input:
Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2 ),x]
Output:
(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(480*(105*A + 121*B)*Sqrt[ Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2464*I)*(15*A + 17*B)*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]*((110880*I)*A + (125664*I)*B + 30*(1953*A + 2134*B)*Sin[c + d*x] + 308*(75*A + 73*B)*Sin[2*(c + d*x)] + 8505*A*Sin[ 3*(c + d*x)] + 5940*B*Sin[3*(c + d*x)] + 2310*A*Sin[4*(c + d*x)] + 770*B*S in[4*(c + d*x)] + 315*A*Sin[5*(c + d*x)])))/(27720*d*E^(I*d*x))
Time = 1.50 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {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.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {2}{11} \int \frac {(\sec (c+d x) a+a)^2 (a (15 A+11 B)+a (5 A+11 B) \sec (c+d x))}{2 \sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \frac {(\sec (c+d x) a+a)^2 (a (15 A+11 B)+a (5 A+11 B) \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a (15 A+11 B)+a (5 A+11 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {(\sec (c+d x) a+a) \left (5 (21 A+22 B) a^2+(60 A+77 B) \sec (c+d x) a^2\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (5 (21 A+22 B) a^2+(60 A+77 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \left (\frac {10 a^3 (21 A+22 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2}{7} \int -\frac {77 (15 A+17 B) a^3+9 (105 A+121 B) \sec (c+d x) a^3}{2 \sec ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \int \frac {77 (15 A+17 B) a^3+9 (105 A+121 B) \sec (c+d x) a^3}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {10 a^3 (21 A+22 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \int \frac {77 (15 A+17 B) a^3+9 (105 A+121 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {10 a^3 (21 A+22 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^3 (15 A+17 B) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)}dx+9 a^3 (105 A+121 B) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {10 a^3 (21 A+22 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^3 (15 A+17 B) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 a^3 (105 A+121 B) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {10 a^3 (21 A+22 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^3 (15 A+17 B) \left (\frac {3}{5} \int \frac {1}{\sqrt {\sec (c+d x)}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+9 a^3 (105 A+121 B) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {10 a^3 (21 A+22 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^3 (15 A+17 B) \left (\frac {3}{5} \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+9 a^3 (105 A+121 B) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {10 a^3 (21 A+22 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^3 (15 A+17 B) \left (\frac {3}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+9 a^3 (105 A+121 B) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {10 a^3 (21 A+22 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^3 (15 A+17 B) \left (\frac {3}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+9 a^3 (105 A+121 B) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {10 a^3 (21 A+22 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (9 a^3 (105 A+121 B) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+77 a^3 (15 A+17 B) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )+\frac {10 a^3 (21 A+22 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2}{9} \left (\frac {10 a^3 (21 A+22 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{7} \left (9 a^3 (105 A+121 B) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+77 a^3 (15 A+17 B) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )\right )\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
Input:
Int[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2),x]
Output:
(2*a*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (( 2*(15*A + 11*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(9*d*Sec[c + d*x]^( 7/2)) + (2*((10*a^3*(21*A + 22*B)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (77*a^3*(15*A + 17*B)*((6*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sq rt[Sec[c + d*x]])/(5*d) + (2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))) + 9*a ^3*(105*A + 121*B)*((2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[S ec[c + d*x]])/(3*d) + (2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])))/7))/9)/1 1
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]
Time = 30.22 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.59
| 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 (10080 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-43680 A -6160 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (77280 A +24200 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-72240 A -37532 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (39270 A +29722 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-8820 A -8118 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1575 A \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \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}-3465 A \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \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}+1815 B \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \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}-3927 B \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \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}\right )}{3465 \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}\) | \(441\) |
| parts | \(\text {Expression too large to display}\) | \(1059\) |
Input:
int((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x,method=_RETURN VERBOSE)
Output:
-4/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(10080 *A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-43680*A-6160*B)*sin(1/2*d*x+ 1/2*c)^10*cos(1/2*d*x+1/2*c)+(77280*A+24200*B)*sin(1/2*d*x+1/2*c)^8*cos(1/ 2*d*x+1/2*c)+(-72240*A-37532*B)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(3 9270*A+29722*B)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-8820*A-8118*B)*s in(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+1575*A*EllipticF(cos(1/2*d*x+1/2*c) ,2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)-34 65*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2 *sin(1/2*d*x+1/2*c)^2-1)^(1/2)+1815*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2) )*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)-3927*B*Ell ipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(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)/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 = 251, normalized size of antiderivative = 0.91 \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (105 \, A + 121 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (105 \, A + 121 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (15 \, A + 17 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (15 \, A + 17 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (315 \, A a^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{4} + 135 \, {\left (14 \, A + 11 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 154 \, {\left (15 \, A + 17 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 30 \, {\left (105 \, A + 121 \, B\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{3465 \, d} \] Input:
integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x, algorit hm="fricas")
Output:
-2/3465*(15*I*sqrt(2)*(105*A + 121*B)*a^3*weierstrassPInverse(-4, 0, cos(d *x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(105*A + 121*B)*a^3*weierstrassPI nverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*I*sqrt(2)*(15*A + 17*B) *a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*si n(d*x + c))) + 231*I*sqrt(2)*(15*A + 17*B)*a^3*weierstrassZeta(-4, 0, weie rstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (315*A*a^3*cos(d* x + c)^5 + 385*(3*A + B)*a^3*cos(d*x + c)^4 + 135*(14*A + 11*B)*a^3*cos(d* x + c)^3 + 154*(15*A + 17*B)*a^3*cos(d*x + c)^2 + 30*(105*A + 121*B)*a^3*c os(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))**3*(A+B*sec(d*x+c))/sec(d*x+c)**(11/2),x)
Output:
Timed out
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x, algorit hm="maxima")
Output:
Timed out
\[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x, algorit hm="giac")
Output:
integrate((B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^3/sec(d*x + c)^(11/2), x)
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \] Input:
int(((A + B/cos(c + d*x))*(a + a/cos(c + d*x))^3)/(1/cos(c + d*x))^(11/2), x)
Output:
int(((A + B/cos(c + d*x))*(a + a/cos(c + d*x))^3)/(1/cos(c + d*x))^(11/2), x)
\[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=a^{3} \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{6}}d x \right ) a +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x \right ) b +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) a +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) a +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) b \right ) \] Input:
int((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x)
Output:
a**3*(int(sqrt(sec(c + d*x))/sec(c + d*x)**6,x)*a + 3*int(sqrt(sec(c + d*x ))/sec(c + d*x)**5,x)*a + int(sqrt(sec(c + d*x))/sec(c + d*x)**5,x)*b + 3* 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 + int(sqrt(sec(c + d*x))/sec(c + d*x)**3,x)*a + 3*int(sq rt(sec(c + d*x))/sec(c + d*x)**3,x)*b + int(sqrt(sec(c + d*x))/sec(c + d*x )**2,x)*b)