Integrand size = 33, antiderivative size = 205 \[ \int \frac {(a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 a (7 A+9 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 a (5 A+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a A \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a (7 A+9 C) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a (5 A+7 C) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}} \] Output:
2/15*a*(7*A+9*C)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*se c(d*x+c)^(1/2)/d+2/21*a*(5*A+7*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+2/9*a*A*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2 /7*a*A*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/45*a*(7*A+9*C)*sin(d*x+c)/d/sec(d*x +c)^(3/2)+2/21*a*(5*A+7*C)*sin(d*x+c)/d/sec(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.20 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00 \[ \int \frac {(a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {a e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (120 (5 A+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-56 i (7 A+9 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (1176 i A+1512 i C+30 (23 A+28 C) \sin (c+d x)+14 (19 A+18 C) \sin (2 (c+d x))+90 A \sin (3 (c+d x))+35 A \sin (4 (c+d x)))\right )}{1260 d} \] Input:
Integrate[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2) ,x]
Output:
(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(120*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(7*A + 9*C)*E^(I*(c + d*x))*Sqr t[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((1176*I)*A + (1512*I)*C + 30*(23*A + 28*C)*Sin[c + d*x] + 14*(19*A + 18*C)*Sin[2*(c + d*x)] + 90*A*Sin[3*(c + d*x)] + 35*A*S in[4*(c + d*x)])))/(1260*d*E^(I*d*x))
Time = 1.24 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.98, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 4563, 27, 3042, 4535, 3042, 4256, 3042, 4258, 3042, 3119, 4533, 3042, 4256, 3042, 4258, 3042, 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.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a) \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right ) \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4563 |
| \(\displaystyle \frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {2}{9} \int -\frac {9 a C \sec ^2(c+d x)+a (7 A+9 C) \sec (c+d x)+9 a A}{2 \sec ^{\frac {7}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {9 a C \sec ^2(c+d x)+a (7 A+9 C) \sec (c+d x)+9 a A}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {9 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+a (7 A+9 C) \csc \left (c+d x+\frac {\pi }{2}\right )+9 a A}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{9} \left (a (7 A+9 C) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)}dx+\int \frac {9 a C \sec ^2(c+d x)+9 a A}{\sec ^{\frac {7}{2}}(c+d x)}dx\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (a (7 A+9 C) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\int \frac {9 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {1}{9} \left (\int \frac {9 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+a (7 A+9 C) \left (\frac {3}{5} \int \frac {1}{\sqrt {\sec (c+d x)}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \frac {9 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+a (7 A+9 C) \left (\frac {3}{5} \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\int \frac {9 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+a (7 A+9 C) \left (\frac {3}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \frac {9 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+a (7 A+9 C) \left (\frac {3}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\int \frac {9 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+a (7 A+9 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} a (5 A+7 C) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)}dx+a (7 A+9 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a A \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} a (5 A+7 C) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+a (7 A+9 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a A \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} a (5 A+7 C) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+a (7 A+9 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a A \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} a (5 A+7 C) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+a (7 A+9 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a A \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} a (5 A+7 C) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+a (7 A+9 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a A \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} a (5 A+7 C) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+a (7 A+9 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a A \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} a (5 A+7 C) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+a (7 A+9 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a A \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
Input:
Int[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]
Output:
(2*a*A*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((18*a*A*Sin[c + d*x])/(7* d*Sec[c + d*x]^(5/2)) + a*(7*A + 9*C)*((6*Sqrt[Cos[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))) + (9*a*(5*A + 7*C)*((2*Sqrt[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
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_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Simp[(C*m + A*(m + 1))/(b^2*m) Int[(b*Csc[e + f*x])^(m + 2), x], x] /; Fr eeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(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[A*b*n + a*(C*n + A*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x ]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && LtQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(405\) vs. \(2(180)=360\).
Time = 11.79 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.98
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a \left (-1120 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+2960 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-3152 A -504 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (1792 A +924 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-408 A -336 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-147 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+75 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-189 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+105 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{315 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(406\) |
| parts | \(\text {Expression too large to display}\) | \(798\) |
Input:
int((a+a*sec(d*x+c))*(A+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x,method=_RETURNV ERBOSE)
Output:
-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a*(-1120*A* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+2960*A*cos(1/2*d*x+1/2*c)*sin(1/2 *d*x+1/2*c)^8+(-3152*A-504*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(179 2*A+924*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-408*A-336*C)*sin(1/2* d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-147*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))+75*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))-189*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))+105*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)))/(-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 = 0.10 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.03 \[ \int \frac {(a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {-15 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (7 \, A + 9 \, C\right )} a {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (7 \, A + 9 \, C\right )} a {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + \frac {2 \, {\left (35 \, A a \cos \left (d x + c\right )^{4} + 45 \, A a \cos \left (d x + c\right )^{3} + 7 \, {\left (7 \, A + 9 \, C\right )} a \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, A + 7 \, C\right )} a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \] Input:
integrate((a+a*sec(d*x+c))*(A+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorith m="fricas")
Output:
1/315*(-15*I*sqrt(2)*(5*A + 7*C)*a*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*I*sqrt(2)*(5*A + 7*C)*a*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*I*sqrt(2)*(7*A + 9*C)*a*weierstrassZe ta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21* I*sqrt(2)*(7*A + 9*C)*a*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(35*A*a*cos(d*x + c)^4 + 45*A*a*cos(d* x + c)^3 + 7*(7*A + 9*C)*a*cos(d*x + c)^2 + 15*(5*A + 7*C)*a*cos(d*x + c)) *sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int \frac {(a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))*(A+C*sec(d*x+c)**2)/sec(d*x+c)**(9/2),x)
Output:
Timed out
\[ \int \frac {(a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))*(A+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorith m="maxima")
Output:
integrate((C*sec(d*x + c)^2 + A)*(a*sec(d*x + c) + a)/sec(d*x + c)^(9/2), x)
\[ \int \frac {(a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))*(A+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorith m="giac")
Output:
integrate((C*sec(d*x + c)^2 + A)*(a*sec(d*x + c) + a)/sec(d*x + c)^(9/2), x)
Timed out. \[ \int \frac {(a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \] Input:
int(((A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x)))/(1/cos(c + d*x))^(9/2),x )
Output:
int(((A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x)))/(1/cos(c + d*x))^(9/2), x)
\[ \int \frac {(a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=a \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) c \right ) \] Input:
int((a+a*sec(d*x+c))*(A+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x)
Output:
a*(int(sqrt(sec(c + d*x))/sec(c + d*x)**5,x)*a + int(sqrt(sec(c + d*x))/se c(c + d*x)**4,x)*a + int(sqrt(sec(c + d*x))/sec(c + d*x)**3,x)*c + int(sqr t(sec(c + d*x))/sec(c + d*x)**2,x)*c)