Integrand size = 37, antiderivative size = 230 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {2 a^{5/2} C \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{d}+\frac {2 a^3 (32 A+49 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{21 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (8 A+7 C) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d} \]
2/7*a*A*(a+a*cos(d*x+c))^(3/2)*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/7*A*(a+a*co s(d*x+c))^(5/2)*sec(d*x+c)^(7/2)*sin(d*x+c)/d+2/21*a^2*(8*A+7*C)*sec(d*x+c )^(3/2)*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d+2*a^(5/2)*C*arcsin(sin(d*x+c)* a^(1/2)/(a+a*cos(d*x+c))^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+2/21*a ^3*(32*A+49*C)*sin(d*x+c)*sec(d*x+c)^(1/2)/d/(a+a*cos(d*x+c))^(1/2)
Time = 1.50 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.66 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \left (84 \sqrt {2} C \arcsin \left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^{\frac {7}{2}}(c+d x)+4 (29 A+7 C+(93 A+84 C) \cos (c+d x)+(23 A+7 C) \cos (2 (c+d x))+23 A \cos (3 (c+d x))+28 C \cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{84 d} \]
(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(7/2)*(84*Sq rt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(7/2) + 4*(29*A + 7* C + (93*A + 84*C)*Cos[c + d*x] + (23*A + 7*C)*Cos[2*(c + d*x)] + 23*A*Cos[ 3*(c + d*x)] + 28*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(84*d)
Time = 1.48 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.05, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {3042, 4709, 3042, 3523, 27, 3042, 3454, 27, 3042, 3454, 27, 3042, 3459, 3042, 3253, 223}
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.
| \(\displaystyle \int \sec ^{\frac {9}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^{9/2} (a \cos (c+d x)+a)^{5/2} \left (A+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(\cos (c+d x) a+a)^{5/2} \left (C \cos ^2(c+d x)+A\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \int \frac {(\cos (c+d x) a+a)^{5/2} (5 a A+7 a C \cos (c+d x))}{2 \cos ^{\frac {7}{2}}(c+d x)}dx}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {(\cos (c+d x) a+a)^{5/2} (5 a A+7 a C \cos (c+d x))}{\cos ^{\frac {7}{2}}(c+d x)}dx}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (5 a A+7 a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {2}{5} \int \frac {5 (\cos (c+d x) a+a)^{3/2} \left ((8 A+7 C) a^2+7 C \cos (c+d x) a^2\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {(\cos (c+d x) a+a)^{3/2} \left ((8 A+7 C) a^2+7 C \cos (c+d x) a^2\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left ((8 A+7 C) a^2+7 C \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {2}{3} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((32 A+49 C) a^3+21 C \cos (c+d x) a^3\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a^3 (8 A+7 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((32 A+49 C) a^3+21 C \cos (c+d x) a^3\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a^3 (8 A+7 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left ((32 A+49 C) a^3+21 C \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a^3 (8 A+7 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (21 a^3 C \int \frac {\sqrt {\cos (c+d x) a+a}}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^4 (32 A+49 C) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (8 A+7 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (21 a^3 C \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^4 (32 A+49 C) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (8 A+7 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3253 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {2 a^4 (32 A+49 C) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-\frac {42 a^3 C \int \frac {1}{\sqrt {1-\frac {a \sin ^2(c+d x)}{\cos (c+d x) a+a}}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}\right )+\frac {2 a^3 (8 A+7 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {2 a^3 (8 A+7 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{3} \left (\frac {42 a^{7/2} C \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}+\frac {2 a^4 (32 A+49 C) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{7 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*(a + a*Cos[c + d*x])^(5/2)*Sin [c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*a^3*(8*A + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a^2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)) + ((42*a^(7/2)*C*ArcSin[( Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^4*(32*A + 49*C)* Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]))/3)/(7*a))
3.13.24.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[-2/f Subst[Int[1/Sqrt[1 - x^2/a], x], x, b*(Co s[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, d, e, f}, x] && E qQ[a^2 - b^2, 0] && EqQ[d, a/b]
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 = 0.47 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.10
\[\frac {2 a^{2} \left (\sec ^{\frac {9}{2}}\left (d x +c \right )\right ) \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (21 C \left (\cos ^{5}\left (d x +c \right )\right ) \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+21 C \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right )+46 A \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+56 C \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+23 A \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+7 C \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+12 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 A \cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{21 d \left (1+\cos \left (d x +c \right )\right )}\]
2/21*a^2/d*sec(d*x+c)^(9/2)*((1+cos(d*x+c))*a)^(1/2)/(1+cos(d*x+c))*(21*C* cos(d*x+c)^5*arctan((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan(d*x+c))*(cos(d*x +c)/(1+cos(d*x+c)))^(1/2)+21*C*cos(d*x+c)^4*(cos(d*x+c)/(1+cos(d*x+c)))^(1 /2)*arctan((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan(d*x+c))+46*A*cos(d*x+c)^4 *sin(d*x+c)+56*C*cos(d*x+c)^4*sin(d*x+c)+23*A*cos(d*x+c)^3*sin(d*x+c)+7*C* cos(d*x+c)^3*sin(d*x+c)+12*A*cos(d*x+c)^2*sin(d*x+c)+3*A*cos(d*x+c)*sin(d* x+c))
Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.77 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {2 \, {\left (21 \, {\left (C a^{2} \cos \left (d x + c\right )^{4} + C a^{2} \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - \frac {{\left (2 \, {\left (23 \, A + 28 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (23 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 12 \, A a^{2} \cos \left (d x + c\right ) + 3 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{21 \, {\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}} \]
-2/21*(21*(C*a^2*cos(d*x + c)^4 + C*a^2*cos(d*x + c)^3)*sqrt(a)*arctan(sqr t(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) - (2*(23* A + 28*C)*a^2*cos(d*x + c)^3 + (23*A + 7*C)*a^2*cos(d*x + c)^2 + 12*A*a^2* cos(d*x + c) + 3*A*a^2)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^4 + d*cos(d*x + c)^3)
Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2343 vs. \(2 (196) = 392\).
Time = 0.77 (sec) , antiderivative size = 2343, normalized size of antiderivative = 10.19 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\text {Too large to display} \]
1/210*(7*(6*(a^2*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + 25*(a^2*cos(2*d*x + 2*c)^2 + a^2*sin(2*d*x + 2*c)^2 + 2*a^2*cos(2*d*x + 2*c) + a^2)*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)^(3/4)*sqrt( a) + 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)*((15*a^2*sin(6*d*x + 6*c) + 50*a^2*sin(4*d*x + 4*c) + 58*a^2*sin(2*d *x + 2*c) - 20*(3*a^2*sin(6*d*x + 6*c) + 10*a^2*sin(4*d*x + 4*c) + 11*a^2* sin(2*d*x + 2*c))*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2 0*(3*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 11*a^2*cos(2*d*x + 2 *c) + 4*a^2)*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*cos(7/2 *arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - (15*a^2*cos(6*d*x + 6* c) + 50*a^2*cos(4*d*x + 4*c) + 58*a^2*cos(2*d*x + 2*c) + 23*a^2 + 20*(3*a^ 2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 11*a^2*cos(2*d*x + 2*c) + 4 *a^2)*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 20*(3*a^2*sin (6*d*x + 6*c) + 10*a^2*sin(4*d*x + 4*c) + 11*a^2*sin(2*d*x + 2*c))*sin(7/2 *arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + 25*(a^2*cos(2*d*x + 2*c)^2 + a^2*sin(2*d*x + 2*c)^2 + 2*a^2*cos(2*d*x + 2*c) + a^2)*sin(3/2*arctan2(sin(2*d*x + 2*c) , cos(2*d*x + 2*c) + 1)))*sqrt(a) + 15*((a^2*cos(2*d*x + 2*c)^4 + a^2*sin( 2*d*x + 2*c)^4 + 4*a^2*cos(2*d*x + 2*c)^3 + 6*a^2*cos(2*d*x + 2*c)^2 + ...
Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\text {Timed out} \]
Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]