Integrand size = 37, antiderivative size = 219 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {4 a^2 (136 A+189 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (136 A+189 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (52 A+63 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \]
2/9*A*(a+a*cos(d*x+c))^(3/2)*sec(d*x+c)^(9/2)*sin(d*x+c)/d+2/315*a^2*(136* A+189*C)*sec(d*x+c)^(3/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/315*a^2*(5 2*A+63*C)*sec(d*x+c)^(5/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/21*a*A*se c(d*x+c)^(7/2)*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d+4/315*a^2*(136*A+189*C) *sin(d*x+c)*sec(d*x+c)^(1/2)/d/(a+a*cos(d*x+c))^(1/2)
Time = 0.83 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.56 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} (752 A+693 C+(748 A+567 C) \cos (c+d x)+(748 A+882 C) \cos (2 (c+d x))+136 A \cos (3 (c+d x))+189 C \cos (3 (c+d x))+136 A \cos (4 (c+d x))+189 C \cos (4 (c+d x))) \sec ^{\frac {9}{2}}(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{630 d} \]
(a*Sqrt[a*(1 + Cos[c + d*x])]*(752*A + 693*C + (748*A + 567*C)*Cos[c + d*x ] + (748*A + 882*C)*Cos[2*(c + d*x)] + 136*A*Cos[3*(c + d*x)] + 189*C*Cos[ 3*(c + d*x)] + 136*A*Cos[4*(c + d*x)] + 189*C*Cos[4*(c + d*x)])*Sec[c + d* x]^(9/2)*Tan[(c + d*x)/2])/(630*d)
Time = 1.41 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {3042, 4709, 3042, 3523, 27, 3042, 3454, 27, 3042, 3459, 3042, 3251, 3042, 3250}
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.
| \(\displaystyle \int \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^{11/2} (a \cos (c+d x)+a)^{3/2} \left (A+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(\cos (c+d x) a+a)^{3/2} \left (C \cos ^2(c+d x)+A\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \int \frac {(\cos (c+d x) a+a)^{3/2} (3 a A+a (4 A+9 C) \cos (c+d x))}{2 \cos ^{\frac {9}{2}}(c+d x)}dx}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {(\cos (c+d x) a+a)^{3/2} (3 a A+a (4 A+9 C) \cos (c+d x))}{\cos ^{\frac {9}{2}}(c+d x)}dx}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (3 a A+a (4 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {2}{7} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((52 A+63 C) a^2+(40 A+63 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((52 A+63 C) a^2+(40 A+63 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left ((52 A+63 C) a^2+(40 A+63 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {6 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {3}{5} a^2 (136 A+189 C) \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^3 (52 A+63 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {6 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {3}{5} a^2 (136 A+189 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a^3 (52 A+63 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {6 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {3}{5} a^2 (136 A+189 C) \left (\frac {2}{3} \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (52 A+63 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {6 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {3}{5} a^2 (136 A+189 C) \left (\frac {2}{3} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (52 A+63 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {6 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3250 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {6 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (\frac {2 a^3 (52 A+63 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {3}{5} a^2 (136 A+189 C) \left (\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4 a \sin (c+d x)}{3 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )\right )}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*(a + a*Cos[c + d*x])^(3/2)*Sin [c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((6*a^2*A*Sqrt[a + a*Cos[c + d*x]]*S in[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*a^3*(52*A + 63*C)*Sin[c + d*x] )/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (3*a^2*(136*A + 189* C)*((2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])))/5) /7)/(9*a))
3.13.13.3.1 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[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(3/2), x_Symbol] :> Simp[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sq rt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0 ])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1) *(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b* c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d)) Int[Sqrt[a + b*Sin[e + f*x] ]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a *d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* (c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Time = 1.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.52
| method | result | size |
| default | \(-\frac {2 a \left (\cos \left (d x +c \right )-1\right ) \left (\left (272 \left (\cos ^{4}\left (d x +c \right )\right )+136 \left (\cos ^{3}\left (d x +c \right )\right )+102 \left (\cos ^{2}\left (d x +c \right )\right )+85 \cos \left (d x +c \right )+35\right ) A +\left (\cos ^{2}\left (d x +c \right )\right ) \left (378 \left (\cos ^{2}\left (d x +c \right )\right )+189 \cos \left (d x +c \right )+63\right ) C \right ) \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\sec ^{\frac {11}{2}}\left (d x +c \right )\right ) \cot \left (d x +c \right )}{315 d}\) | \(114\) |
| parts | \(-\frac {2 A a \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\sec ^{\frac {11}{2}}\left (d x +c \right )\right ) \left (272 \left (\cos ^{5}\left (d x +c \right )\right )-136 \left (\cos ^{4}\left (d x +c \right )\right )-34 \left (\cos ^{3}\left (d x +c \right )\right )-17 \left (\cos ^{2}\left (d x +c \right )\right )-50 \cos \left (d x +c \right )-35\right ) \cot \left (d x +c \right )}{315 d}-\frac {2 C a \left (\sec ^{\frac {11}{2}}\left (d x +c \right )\right ) \left (6 \left (\cos ^{3}\left (d x +c \right )\right )-3 \left (\cos ^{2}\left (d x +c \right )\right )-2 \cos \left (d x +c \right )-1\right ) \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\cos ^{2}\left (d x +c \right )\right ) \cot \left (d x +c \right )}{5 d}\) | \(156\) |
-2/315*a/d*(cos(d*x+c)-1)*((272*cos(d*x+c)^4+136*cos(d*x+c)^3+102*cos(d*x+ c)^2+85*cos(d*x+c)+35)*A+cos(d*x+c)^2*(378*cos(d*x+c)^2+189*cos(d*x+c)+63) *C)*((1+cos(d*x+c))*a)^(1/2)*sec(d*x+c)^(11/2)*cot(d*x+c)
Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.54 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {2 \, {\left (2 \, {\left (136 \, A + 189 \, C\right )} a \cos \left (d x + c\right )^{4} + {\left (136 \, A + 189 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (34 \, A + 21 \, C\right )} a \cos \left (d x + c\right )^{2} + 85 \, A a \cos \left (d x + c\right ) + 35 \, A a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )} \sqrt {\cos \left (d x + c\right )}} \]
2/315*(2*(136*A + 189*C)*a*cos(d*x + c)^4 + (136*A + 189*C)*a*cos(d*x + c) ^3 + 3*(34*A + 21*C)*a*cos(d*x + c)^2 + 85*A*a*cos(d*x + c) + 35*A*a)*sqrt (a*cos(d*x + c) + a)*sin(d*x + c)/((d*cos(d*x + c)^5 + d*cos(d*x + c)^4)*s qrt(cos(d*x + c)))
Timed out. \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (189) = 378\).
Time = 0.50 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.83 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {4 \, {\left (\frac {{\left (\frac {315 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {840 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1344 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1242 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {517 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {94 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}\right )} A {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{{\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}} {\left (\frac {4 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {\sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 1\right )}} + \frac {63 \, {\left (\frac {5 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {20 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {32 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {26 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {11 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {2 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}\right )} C {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{{\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}} {\left (\frac {4 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {\sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 1\right )}}\right )}}{315 \, d} \]
4/315*((315*sqrt(2)*a^(3/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 840*sqrt(2)* a^(3/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1344*sqrt(2)*a^(3/2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1242*sqrt(2)*a^(3/2)*sin(d*x + c)^7/(cos(d* x + c) + 1)^7 + 517*sqrt(2)*a^(3/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 94*sqrt(2)*a^(3/2)*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)*A*(sin(d*x + c)^ 2/(cos(d*x + c) + 1)^2 + 1)^4/((sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2 )*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2)*(4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*sin(d*x + c)^6/(c os(d*x + c) + 1)^6 + sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 1)) + 63*(5*sqr t(2)*a^(3/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 20*sqrt(2)*a^(3/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 32*sqrt(2)*a^(3/2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 26*sqrt(2)*a^(3/2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 11*s qrt(2)*a^(3/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 2*sqrt(2)*a^(3/2)*sin (d*x + c)^11/(cos(d*x + c) + 1)^11)*C*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^4/((sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2)*(-sin(d*x + c)/(cos( d*x + c) + 1) + 1)^(11/2)*(4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*sin(d *x + c)^4/(cos(d*x + c) + 1)^4 + 4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + s in(d*x + c)^8/(cos(d*x + c) + 1)^8 + 1)))/d
Timed out. \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out} \]
Time = 7.03 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.46 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {\sqrt {\frac {1}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (-\frac {8\,C\,a\,{\mathrm {e}}^{\frac {c\,9{}\mathrm {i}}{2}+\frac {d\,x\,9{}\mathrm {i}}{2}}\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{d}+\frac {8\,a\,{\mathrm {e}}^{\frac {c\,9{}\mathrm {i}}{2}+\frac {d\,x\,9{}\mathrm {i}}{2}}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,A+13\,C\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{5\,d}+\frac {8\,a\,{\mathrm {e}}^{\frac {c\,9{}\mathrm {i}}{2}+\frac {d\,x\,9{}\mathrm {i}}{2}}\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,\left (68\,A+77\,C\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{35\,d}+\frac {8\,a\,{\mathrm {e}}^{\frac {c\,9{}\mathrm {i}}{2}+\frac {d\,x\,9{}\mathrm {i}}{2}}\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,\left (136\,A+189\,C\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{315\,d}\right )}{12\,{\mathrm {e}}^{\frac {c\,9{}\mathrm {i}}{2}+\frac {d\,x\,9{}\mathrm {i}}{2}}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+8\,{\mathrm {e}}^{\frac {c\,9{}\mathrm {i}}{2}+\frac {d\,x\,9{}\mathrm {i}}{2}}\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )+8\,{\mathrm {e}}^{\frac {c\,9{}\mathrm {i}}{2}+\frac {d\,x\,9{}\mathrm {i}}{2}}\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )+2\,{\mathrm {e}}^{\frac {c\,9{}\mathrm {i}}{2}+\frac {d\,x\,9{}\mathrm {i}}{2}}\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )+2\,{\mathrm {e}}^{\frac {c\,9{}\mathrm {i}}{2}+\frac {d\,x\,9{}\mathrm {i}}{2}}\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )} \]
((1/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((8*a*exp((c*9i )/2 + (d*x*9i)/2)*sin(c/2 + (d*x)/2)*(12*A + 13*C)*(a + a*cos(c + d*x))^(1 /2))/(5*d) - (8*C*a*exp((c*9i)/2 + (d*x*9i)/2)*sin((3*c)/2 + (3*d*x)/2)*(a + a*cos(c + d*x))^(1/2))/d + (8*a*exp((c*9i)/2 + (d*x*9i)/2)*sin((5*c)/2 + (5*d*x)/2)*(68*A + 77*C)*(a + a*cos(c + d*x))^(1/2))/(35*d) + (8*a*exp(( c*9i)/2 + (d*x*9i)/2)*sin((9*c)/2 + (9*d*x)/2)*(136*A + 189*C)*(a + a*cos( c + d*x))^(1/2))/(315*d)))/(12*exp((c*9i)/2 + (d*x*9i)/2)*cos(c/2 + (d*x)/ 2) + 8*exp((c*9i)/2 + (d*x*9i)/2)*cos((3*c)/2 + (3*d*x)/2) + 8*exp((c*9i)/ 2 + (d*x*9i)/2)*cos((5*c)/2 + (5*d*x)/2) + 2*exp((c*9i)/2 + (d*x*9i)/2)*co s((7*c)/2 + (7*d*x)/2) + 2*exp((c*9i)/2 + (d*x*9i)/2)*cos((9*c)/2 + (9*d*x )/2))