Integrand size = 27, antiderivative size = 170 \[ \int (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a^{5/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {2 a^3 (49 A+32 C) \tan (c+d x)}{21 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (7 A+8 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{21 d}+\frac {2 a C (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d} \]
2*a^(5/2)*A*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d+2/7*a*C*(a +a*sec(d*x+c))^(3/2)*tan(d*x+c)/d+2/7*C*(a+a*sec(d*x+c))^(5/2)*tan(d*x+c)/ d+2/21*a^3*(49*A+32*C)*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+2/21*a^2*(7*A+8 *C)*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d
Time = 4.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.89 \[ \int (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 \left (42 A \arctan \left (\sqrt {-1+\sec (c+d x)}\right ) \cos ^3(c+d x)+(7 A+29 C+(84 A+93 C) \cos (c+d x)+(7 A+23 C) \cos (2 (c+d x))+28 A \cos (3 (c+d x))+23 C \cos (3 (c+d x))) \sqrt {-1+\sec (c+d x)}\right ) \sec ^3(c+d x) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{21 d \sqrt {-1+\sec (c+d x)}} \]
(a^2*(42*A*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[c + d*x]^3 + (7*A + 29*C + (84*A + 93*C)*Cos[c + d*x] + (7*A + 23*C)*Cos[2*(c + d*x)] + 28*A*Cos[3*(c + d*x)] + 23*C*Cos[3*(c + d*x)])*Sqrt[-1 + Sec[c + d*x]])*Sec[c + d*x]^3* Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(21*d*Sqrt[-1 + Sec[c + d*x]] )
Time = 1.02 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 4543, 27, 3042, 4405, 27, 3042, 4405, 27, 3042, 4403, 3042, 4261, 216, 4279}
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.
| \(\displaystyle \int (a \sec (c+d x)+a)^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2} \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4543 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (\sec (c+d x) a+a)^{5/2} (7 a A+5 a C \sec (c+d x))dx}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (\sec (c+d x) a+a)^{5/2} (7 a A+5 a C \sec (c+d x))dx}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (7 a A+5 a C \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 4405 |
| \(\displaystyle \frac {\frac {2}{5} \int \frac {5}{2} (\sec (c+d x) a+a)^{3/2} \left (7 A a^2+(7 A+8 C) \sec (c+d x) a^2\right )dx+\frac {2 a^2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (\sec (c+d x) a+a)^{3/2} \left (7 A a^2+(7 A+8 C) \sec (c+d x) a^2\right )dx+\frac {2 a^2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (7 A a^2+(7 A+8 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {2 a^2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 4405 |
| \(\displaystyle \frac {\frac {2}{3} \int \frac {1}{2} \sqrt {\sec (c+d x) a+a} \left (21 A a^3+(49 A+32 C) \sec (c+d x) a^3\right )dx+\frac {2 a^3 (7 A+8 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}+\frac {2 a^2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \int \sqrt {\sec (c+d x) a+a} \left (21 A a^3+(49 A+32 C) \sec (c+d x) a^3\right )dx+\frac {2 a^3 (7 A+8 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}+\frac {2 a^2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a} \left (21 A a^3+(49 A+32 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {2 a^3 (7 A+8 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}+\frac {2 a^2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 4403 |
| \(\displaystyle \frac {\frac {1}{3} \left (a^3 (49 A+32 C) \int \sec (c+d x) \sqrt {\sec (c+d x) a+a}dx+21 a^3 A \int \sqrt {\sec (c+d x) a+a}dx\right )+\frac {2 a^3 (7 A+8 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}+\frac {2 a^2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (a^3 (49 A+32 C) \int \csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+21 a^3 A \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx\right )+\frac {2 a^3 (7 A+8 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}+\frac {2 a^2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {\frac {1}{3} \left (a^3 (49 A+32 C) \int \csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-\frac {42 a^4 A \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{\sec (c+d x) a+a}+a}d\left (-\frac {a \tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}\right )+\frac {2 a^3 (7 A+8 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}+\frac {2 a^2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {1}{3} \left (a^3 (49 A+32 C) \int \csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {42 a^{7/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}\right )+\frac {2 a^3 (7 A+8 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}+\frac {2 a^2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 4279 |
| \(\displaystyle \frac {\frac {2 a^3 (7 A+8 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}+\frac {2 a^2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}+\frac {1}{3} \left (\frac {42 a^{7/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {2 a^4 (49 A+32 C) \tan (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )}{7 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}\) |
(2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d) + ((2*a^3*(7*A + 8*C)* Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d) + (2*a^2*C*(a + a*Sec[c + d*x ])^(3/2)*Tan[c + d*x])/d + ((42*a^(7/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sq rt[a + a*Sec[c + d*x]]])/d + (2*a^4*(49*A + 32*C)*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]))/3)/(7*a)
3.2.76.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*b*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]])), x] /; Free Q[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_ .) + (c_)), x_Symbol] :> Simp[c Int[Sqrt[a + b*Csc[e + f*x]], x], x] + Si mp[d Int[Sqrt[a + b*Csc[e + f*x]]*Csc[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d _.) + (c_)), x_Symbol] :> Simp[(-b)*d*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Simp[1/m Int[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*c*m + (b*c*m + a*d*(2*m - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && EqQ[a^2 - b^2, 0] && IntegerQ[2 *m]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_. ) + (a_))^(m_.), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^ m/(f*(m + 1))), x] + Simp[1/(b*(m + 1)) Int[(a + b*Csc[e + f*x])^m*Simp[A *b*(m + 1) + a*C*m*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Time = 9.74 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.34
| method | result | size |
| default | \(\frac {2 a^{2} \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (21 A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+21 A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+56 A \sin \left (d x +c \right )+46 C \sin \left (d x +c \right )+7 A \tan \left (d x +c \right )+23 C \tan \left (d x +c \right )+12 C \sec \left (d x +c \right ) \tan \left (d x +c \right )+3 C \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}\right )}{21 d \left (\cos \left (d x +c \right )+1\right )}\) | \(228\) |
| parts | \(\frac {2 A \,a^{2} \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (3 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+3 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+8 \sin \left (d x +c \right )+\tan \left (d x +c \right )\right )}{3 d \left (\cos \left (d x +c \right )+1\right )}+\frac {2 C \,a^{2} \left (46 \cos \left (d x +c \right )^{3}+23 \cos \left (d x +c \right )^{2}+12 \cos \left (d x +c \right )+3\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}}{21 d \left (\cos \left (d x +c \right )+1\right )}\) | \(249\) |
2/21*a^2/d*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)*(21*A*(-cos(d*x+c)/(cos (d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+ c)+1))^(1/2))*cos(d*x+c)+21*A*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(s in(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+56*A*sin(d*x+ c)+46*C*sin(d*x+c)+7*A*tan(d*x+c)+23*C*tan(d*x+c)+12*C*sec(d*x+c)*tan(d*x+ c)+3*C*tan(d*x+c)*sec(d*x+c)^2)
Time = 0.29 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.40 \[ \int (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\left [\frac {21 \, {\left (A a^{2} \cos \left (d x + c\right )^{4} + A a^{2} \cos \left (d x + c\right )^{3}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (2 \, {\left (28 \, A + 23 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (7 \, A + 23 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 12 \, C a^{2} \cos \left (d x + c\right ) + 3 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{21 \, {\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}}, -\frac {2 \, {\left (21 \, {\left (A a^{2} \cos \left (d x + c\right )^{4} + A a^{2} \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (2 \, {\left (28 \, A + 23 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (7 \, A + 23 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 12 \, C a^{2} \cos \left (d x + c\right ) + 3 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{21 \, {\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}}\right ] \]
[1/21*(21*(A*a^2*cos(d*x + c)^4 + A*a^2*cos(d*x + c)^3)*sqrt(-a)*log((2*a* cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d* x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*(2*(28*A + 23*C)*a^2*cos(d*x + c)^3 + (7*A + 23*C)*a^2*cos(d*x + c)^2 + 12*C*a^2*c os(d*x + c) + 3*C*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c ))/(d*cos(d*x + c)^4 + d*cos(d*x + c)^3), -2/21*(21*(A*a^2*cos(d*x + c)^4 + A*a^2*cos(d*x + c)^3)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) - (2*(28*A + 23*C)*a^2*cos(d*x + c)^3 + (7*A + 23*C)*a^2*cos(d*x + c)^2 + 12*C*a^2*cos(d*x + c) + 3*C*a^2) *sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 + d*cos(d*x + c)^3)]
\[ \int (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}} \left (A + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \]
\[ \int (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
-1/42*(8*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1 )^(1/4)*(7*(3*A*a^2*sin(6*d*x + 6*c) + 5*(2*A + C)*a^2*sin(4*d*x + 4*c) + (11*A + 10*C)*a^2*sin(2*d*x + 2*c))*cos(7/2*arctan2(sin(2*d*x + 2*c), cos( 2*d*x + 2*c) + 1)) - (21*A*a^2*cos(6*d*x + 6*c) + 35*(2*A + C)*a^2*cos(4*d *x + 4*c) + 7*(11*A + 10*C)*a^2*cos(2*d*x + 2*c) + (28*A + 23*C)*a^2)*sin( 7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))*sqrt(a) + 21*((A*a^2 *cos(2*d*x + 2*c)^4 + A*a^2*sin(2*d*x + 2*c)^4 + 4*A*a^2*cos(2*d*x + 2*c)^ 3 + 6*A*a^2*cos(2*d*x + 2*c)^2 + 4*A*a^2*cos(2*d*x + 2*c) + A*a^2 + 2*(A*a ^2*cos(2*d*x + 2*c)^2 + 2*A*a^2*cos(2*d*x + 2*c) + A*a^2)*sin(2*d*x + 2*c) ^2)*arctan2((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)), (cos( 2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/ 2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + 1) - (A*a^2*cos(2*d*x + 2*c)^4 + A*a^2*sin(2*d*x + 2*c)^4 + 4*A*a^2*cos(2*d*x + 2*c)^3 + 6*A*a^ 2*cos(2*d*x + 2*c)^2 + 4*A*a^2*cos(2*d*x + 2*c) + A*a^2 + 2*(A*a^2*cos(2*d *x + 2*c)^2 + 2*A*a^2*cos(2*d*x + 2*c) + A*a^2)*sin(2*d*x + 2*c)^2)*arctan 2((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4) *sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)), (cos(2*d*x + 2* c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2( sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - 1) - 2*(A*a^2*d*cos(2*d*x + ...
\[ \int (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]