Integrand size = 45, antiderivative size = 226 \[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {16 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {8 a (16 A+18 B+21 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (16 A+18 B+21 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (A+9 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{9 d} \] Output:
16/315*a*(16*A+18*B+21*C)*sin(d*x+c)/d/cos(d*x+c)^(1/2)/(a+a*sec(d*x+c))^( 1/2)+8/315*a*(16*A+18*B+21*C)*cos(d*x+c)^(1/2)*sin(d*x+c)/d/(a+a*sec(d*x+c ))^(1/2)+2/105*a*(16*A+18*B+21*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d/(a+a*sec(d *x+c))^(1/2)+2/63*a*(A+9*B)*cos(d*x+c)^(5/2)*sin(d*x+c)/d/(a+a*sec(d*x+c)) ^(1/2)+2/9*A*cos(d*x+c)^(7/2)*(a+a*sec(d*x+c))^(1/2)*sin(d*x+c)/d
Time = 0.40 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.56 \[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\sqrt {\cos (c+d x)} (1321 A+1368 B+1596 C+(752 A+846 B+672 C) \cos (c+d x)+4 (83 A+54 B+63 C) \cos (2 (c+d x))+80 A \cos (3 (c+d x))+90 B \cos (3 (c+d x))+35 A \cos (4 (c+d x))) \sqrt {a (1+\sec (c+d x))} \sin (c+d x)}{1260 d (1+\cos (c+d x))} \] Input:
Integrate[Cos[c + d*x]^(9/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(Sqrt[Cos[c + d*x]]*(1321*A + 1368*B + 1596*C + (752*A + 846*B + 672*C)*Co s[c + d*x] + 4*(83*A + 54*B + 63*C)*Cos[2*(c + d*x)] + 80*A*Cos[3*(c + d*x )] + 90*B*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d* x])]*Sin[c + d*x])/(1260*d*(1 + Cos[c + d*x]))
Time = 1.49 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {3042, 4753, 3042, 4574, 27, 3042, 4503, 3042, 4292, 3042, 4292, 3042, 4291}
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.
| \(\displaystyle \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \sec (c+d x)+a} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{9/2} \sqrt {a \sec (c+d x)+a} \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4753 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sqrt {\sec (c+d x) a+a} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right )}{\sec ^{\frac {9}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a} \left (C \csc \left (c+d x+\frac {\pi }{2}\right )^2+B \csc \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \int \frac {\sqrt {\sec (c+d x) a+a} (a (A+9 B)+3 a (2 A+3 C) \sec (c+d x))}{2 \sec ^{\frac {7}{2}}(c+d x)}dx}{9 a}+\frac {2 A \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\sqrt {\sec (c+d x) a+a} (a (A+9 B)+3 a (2 A+3 C) \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)}dx}{9 a}+\frac {2 A \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a} \left (a (A+9 B)+3 a (2 A+3 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx}{9 a}+\frac {2 A \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4503 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3}{7} a (16 A+18 B+21 C) \int \frac {\sqrt {\sec (c+d x) a+a}}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^2 (A+9 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3}{7} a (16 A+18 B+21 C) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a^2 (A+9 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4292 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3}{7} a (16 A+18 B+21 C) \left (\frac {4}{5} \int \frac {\sqrt {\sec (c+d x) a+a}}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}\right )+\frac {2 a^2 (A+9 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3}{7} a (16 A+18 B+21 C) \left (\frac {4}{5} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}\right )+\frac {2 a^2 (A+9 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4292 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3}{7} a (16 A+18 B+21 C) \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\sqrt {\sec (c+d x) a+a}}{\sqrt {\sec (c+d x)}}dx+\frac {2 a \sin (c+d x)}{3 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}\right )+\frac {2 a^2 (A+9 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3}{7} a (16 A+18 B+21 C) \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sin (c+d x)}{3 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}\right )+\frac {2 a^2 (A+9 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4291 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {2 a^2 (A+9 B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {3}{7} a (16 A+18 B+21 C) \left (\frac {2 a \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {4}{5} \left (\frac {4 a \sin (c+d x) \sqrt {\sec (c+d x)}}{3 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a \sin (c+d x)}{3 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}\right )\right )}{9 a}+\frac {2 A \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
Input:
Int[Cos[c + d*x]^(9/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Se c[c + d*x]^2),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((2*a^2*(A + 9*B)*Sin[c + d*x])/(7*d*S ec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (3*a*(16*A + 18*B + 21*C)*(( 2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (4*( (2*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4* a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])))/5))/7) /(9*a))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[-2*a*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*S qrt[d*Csc[e + f*x]])), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[a*((2*n + 1)/(2*b*d*n)) Int[Sqrt[a + b*Csc [e + f*x]]*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)] && IntegerQ[2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*b^2*Co t[e + f*x]*((d*Csc[e + f*x])^n/(a*f*n*Sqrt[a + b*Csc[e + f*x]])), x] + Simp [(A*b*(2*n + 1) + 2*a*B*n)/(2*a*d*n) Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[ e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a *B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] && LtQ[n, 0]
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])
Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m Int[ActivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[u, x ]
Time = 1.64 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.62
| method | result | size |
| default | \(\frac {2 \sin \left (d x +c \right ) \left (\left (35 \cos \left (d x +c \right )^{4}+40 \cos \left (d x +c \right )^{3}+48 \cos \left (d x +c \right )^{2}+64 \cos \left (d x +c \right )+128\right ) A +\left (45 \cos \left (d x +c \right )^{3}+54 \cos \left (d x +c \right )^{2}+72 \cos \left (d x +c \right )+144\right ) B +\left (63 \cos \left (d x +c \right )^{2}+84 \cos \left (d x +c \right )+168\right ) C \right ) \sqrt {\cos \left (d x +c \right )}\, \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{315 d \left (\cos \left (d x +c \right )+1\right )}\) | \(139\) |
Input:
int(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x,method=_RETURNVERBOSE)
Output:
2/315/d*sin(d*x+c)*((35*cos(d*x+c)^4+40*cos(d*x+c)^3+48*cos(d*x+c)^2+64*co s(d*x+c)+128)*A+(45*cos(d*x+c)^3+54*cos(d*x+c)^2+72*cos(d*x+c)+144)*B+(63* cos(d*x+c)^2+84*cos(d*x+c)+168)*C)*cos(d*x+c)^(1/2)*(a*(1+sec(d*x+c)))^(1/ 2)/(cos(d*x+c)+1)
Time = 0.09 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.55 \[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (35 \, A \cos \left (d x + c\right )^{4} + 5 \, {\left (8 \, A + 9 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (16 \, A + 18 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (16 \, A + 18 \, B + 21 \, C\right )} \cos \left (d x + c\right ) + 128 \, A + 144 \, B + 168 \, C\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \] Input:
integrate(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d* x+c)^2),x, algorithm="fricas")
Output:
2/315*(35*A*cos(d*x + c)^4 + 5*(8*A + 9*B)*cos(d*x + c)^3 + 3*(16*A + 18*B + 21*C)*cos(d*x + c)^2 + 4*(16*A + 18*B + 21*C)*cos(d*x + c) + 128*A + 14 4*B + 168*C)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*si n(d*x + c)/(d*cos(d*x + c) + d)
Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(9/2)*(a+a*sec(d*x+c))**(1/2)*(A+B*sec(d*x+c)+C*sec( d*x+c)**2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (196) = 392\).
Time = 0.31 (sec) , antiderivative size = 668, normalized size of antiderivative = 2.96 \[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d* x+c)^2),x, algorithm="maxima")
Output:
1/5040*(sqrt(2)*(1890*cos(8/9*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c)))*sin(9/2*d*x + 9/2*c) + 420*cos(2/3*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c)))*sin(9/2*d*x + 9/2*c) + 252*cos(4/9*arctan2(sin(9/2* d*x + 9/2*c), cos(9/2*d*x + 9/2*c)))*sin(9/2*d*x + 9/2*c) + 45*cos(2/9*arc tan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c)))*sin(9/2*d*x + 9/2*c) - 1 890*cos(9/2*d*x + 9/2*c)*sin(8/9*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))) - 420*cos(9/2*d*x + 9/2*c)*sin(2/3*arctan2(sin(9/2*d*x + 9/2*c ), cos(9/2*d*x + 9/2*c))) - 252*cos(9/2*d*x + 9/2*c)*sin(4/9*arctan2(sin(9 /2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))) - 45*cos(9/2*d*x + 9/2*c)*sin(2/9* arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))) + 70*sin(9/2*d*x + 9/ 2*c) + 45*sin(7/9*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))) + 2 52*sin(5/9*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))) + 420*sin( 1/3*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))) + 1890*sin(1/9*ar ctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))))*A*sqrt(a) - 18*sqrt(2) *(7*(15*sin(3*d*x + 3*c) + 5*sin(2*d*x + 2*c) + sin(d*x + c))*cos(7/2*arct an2(sin(d*x + c), cos(d*x + c))) - (105*cos(3*d*x + 3*c) + 35*cos(2*d*x + 2*c) + 7*cos(d*x + c) + 10)*sin(7/2*arctan2(sin(d*x + c), cos(d*x + c))) - 7*sin(5/2*arctan2(sin(d*x + c), cos(d*x + c))) - 35*sin(3/2*arctan2(sin(d *x + c), cos(d*x + c))) - 105*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) )*B*sqrt(a) - 84*sqrt(2)*(5*(6*sin(2*d*x + 2*c) + sin(d*x + c))*cos(5/2...
Time = 0.30 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.34 \[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left ({\left ({\left ({\left (\sqrt {2} {\left (107 \, A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 81 \, B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 147 \, C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 18 \, \sqrt {2} {\left (18 \, A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 29 \, B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 28 \, C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 126 \, \sqrt {2} {\left (7 \, A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 6 \, B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 7 \, C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 210 \, \sqrt {2} {\left (2 \, A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 3 \, B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 4 \, C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 315 \, \sqrt {2} {\left (A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{315 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {9}{2}} d} \] Input:
integrate(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d* x+c)^2),x, algorithm="giac")
Output:
2/315*((((sqrt(2)*(107*A*a^5*sgn(cos(d*x + c)) + 81*B*a^5*sgn(cos(d*x + c) ) + 147*C*a^5*sgn(cos(d*x + c)))*tan(1/2*d*x + 1/2*c)^2 + 18*sqrt(2)*(18*A *a^5*sgn(cos(d*x + c)) + 29*B*a^5*sgn(cos(d*x + c)) + 28*C*a^5*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 + 126*sqrt(2)*(7*A*a^5*sgn(cos(d*x + c)) + 6*B*a^5*sgn(cos(d*x + c)) + 7*C*a^5*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2 *c)^2 + 210*sqrt(2)*(2*A*a^5*sgn(cos(d*x + c)) + 3*B*a^5*sgn(cos(d*x + c)) + 4*C*a^5*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 + 315*sqrt(2)*(A*a^5 *sgn(cos(d*x + c)) + B*a^5*sgn(cos(d*x + c)) + C*a^5*sgn(cos(d*x + c))))*t an(1/2*d*x + 1/2*c)/((a*tan(1/2*d*x + 1/2*c)^2 + a)^(9/2)*d)
Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \] Input:
int(cos(c + d*x)^(9/2)*(a + a/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/ cos(c + d*x)^2),x)
Output:
int(cos(c + d*x)^(9/2)*(a + a/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/ cos(c + d*x)^2), x)
\[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\sqrt {a}\, \left (\left (\int \sqrt {\sec \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\sec \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a \right ) \] Input:
int(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x)
Output:
sqrt(a)*(int(sqrt(sec(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x)**4*sec (c + d*x)**2,x)*c + int(sqrt(sec(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d*x),x)*b + int(sqrt(sec(c + d*x) + 1)*sqrt(cos(c + d*x))* cos(c + d*x)**4,x)*a)