Integrand size = 25, antiderivative size = 237 \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {45 \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{1024 \sqrt {2} a^{9/2} d}-\frac {\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {5 \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2} \sqrt {\sec (c+d x)}}+\frac {33 \sin (c+d x)}{256 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}}+\frac {73 \sin (c+d x)}{1024 a^3 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}} \] Output:
45/2048*arctan(1/2*a^(1/2)*sin(d*x+c)*2^(1/2)/cos(d*x+c)^(1/2)/(a+a*cos(d* x+c))^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)*2^(1/2)/a^(9/2)/d-1/8*sin(d *x+c)/d/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(3/2)-5/32*sin(d*x+c)/a/d/(a+a*c os(d*x+c))^(7/2)/sec(d*x+c)^(1/2)+33/256*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c)) ^(5/2)/sec(d*x+c)^(1/2)+73/1024*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^(3/2)/se c(d*x+c)^(1/2)
Time = 4.77 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (2 (882+999 \cos (c+d x)+702 \cos (2 (c+d x))+73 \cos (3 (c+d x))) \sqrt {2-2 \sec (c+d x)}+5760 \text {arctanh}\left (\sqrt {-\sec (c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )}\right ) \cos ^8\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{65536 \sqrt {2} a^4 d \sqrt {a (1+\cos (c+d x))} \sqrt {-((-1+\sec (c+d x)) \sec (c+d x))}} \] Input:
Integrate[1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(5/2)),x]
Output:
(Sec[(c + d*x)/2]^6*(2*(882 + 999*Cos[c + d*x] + 702*Cos[2*(c + d*x)] + 73 *Cos[3*(c + d*x)])*Sqrt[2 - 2*Sec[c + d*x]] + 5760*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[(c + d*x)/2]^8*Sec[c + d*x])*Tan[(c + d*x)/ 2])/(65536*Sqrt[2]*a^4*d*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[-((-1 + Sec[c + d *x])*Sec[c + d*x])])
Time = 1.35 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.10, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3042, 4710, 3042, 3244, 27, 3042, 3456, 27, 3042, 3457, 27, 3042, 3457, 27, 3042, 3261, 218}
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.
| \(\displaystyle \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4710 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(\cos (c+d x) a+a)^{9/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\int \frac {3 \sqrt {\cos (c+d x)} (a-4 a \cos (c+d x))}{2 (\cos (c+d x) a+a)^{7/2}}dx}{8 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \int \frac {\sqrt {\cos (c+d x)} (a-4 a \cos (c+d x))}{(\cos (c+d x) a+a)^{7/2}}dx}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a-4 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{7/2}}dx}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \left (\frac {\int \frac {5 a^2-28 a^2 \cos (c+d x)}{2 \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^{5/2}}dx}{6 a^2}+\frac {5 a \sin (c+d x) \sqrt {\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}\right )}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \left (\frac {\int \frac {5 a^2-28 a^2 \cos (c+d x)}{\sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^{5/2}}dx}{12 a^2}+\frac {5 a \sin (c+d x) \sqrt {\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}\right )}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \left (\frac {\int \frac {5 a^2-28 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}}dx}{12 a^2}+\frac {5 a \sin (c+d x) \sqrt {\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}\right )}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \left (\frac {\frac {\int \frac {7 a^3-66 a^3 \cos (c+d x)}{2 \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^{3/2}}dx}{4 a^2}-\frac {33 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {5 a \sin (c+d x) \sqrt {\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}\right )}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \left (\frac {\frac {\int \frac {7 a^3-66 a^3 \cos (c+d x)}{\sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^{3/2}}dx}{8 a^2}-\frac {33 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {5 a \sin (c+d x) \sqrt {\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}\right )}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \left (\frac {\frac {\int \frac {7 a^3-66 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}-\frac {33 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {5 a \sin (c+d x) \sqrt {\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}\right )}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \left (\frac {\frac {\frac {\int -\frac {45 a^4}{2 \sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{2 a^2}-\frac {73 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {33 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {5 a \sin (c+d x) \sqrt {\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}\right )}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \left (\frac {\frac {-\frac {45}{4} a^2 \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx-\frac {73 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {33 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {5 a \sin (c+d x) \sqrt {\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}\right )}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \left (\frac {\frac {-\frac {45}{4} a^2 \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {73 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {33 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {5 a \sin (c+d x) \sqrt {\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}\right )}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 3261 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \left (\frac {\frac {\frac {45 a^3 \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x) a^3}{\cos (c+d x) a+a}+2 a^2}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}\right )}{2 d}-\frac {73 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {33 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {5 a \sin (c+d x) \sqrt {\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}\right )}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {3 \left (\frac {\frac {-\frac {45 a^{3/2} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} d}-\frac {73 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {33 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {5 a \sin (c+d x) \sqrt {\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}\right )}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\) |
Input:
Int[1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(5/2)),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1/8*(Cos[c + d*x]^(3/2)*Sin[c + d* x])/(d*(a + a*Cos[c + d*x])^(9/2)) - (3*((5*a*Sqrt[Cos[c + d*x]]*Sin[c + d *x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((-33*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((-45*a^(3/2)*ArcTan[(Sqrt[a]* Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*S qrt[2]*d) - (73*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d *x])^(3/2)))/(8*a^2))/(12*a^2)))/(16*a^2))
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[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Simp[1/(a*b* (2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], 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[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(a/f) Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*S in[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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{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[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In tegerQ[2*n] || EqQ[c, 0])
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*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[(csc[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m Int[ActivateTrig[u]/(c*Sin[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Time = 4.55 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.94
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \sqrt {a \left (\cos \left (d x +c \right )+1\right )}\, \left (\tan \left (d x +c \right ) \sec \left (d x +c \right ) \left (-73 \cos \left (d x +c \right )^{3}-351 \cos \left (d x +c \right )^{2}-195 \cos \left (d x +c \right )-45\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (45 \cos \left (d x +c \right )^{2}+180 \cos \left (d x +c \right )+270+180 \sec \left (d x +c \right )+45 \sec \left (d x +c \right )^{2}\right )\right )}{2048 d \left (\cos \left (d x +c \right )^{5}+5 \cos \left (d x +c \right )^{4}+10 \cos \left (d x +c \right )^{3}+10 \cos \left (d x +c \right )^{2}+5 \cos \left (d x +c \right )+1\right ) \sec \left (d x +c \right )^{\frac {5}{2}} \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, a^{5}}\) | \(222\) |
Input:
int(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/2048/d*2^(1/2)*(a*(cos(d*x+c)+1))^(1/2)/(cos(d*x+c)^5+5*cos(d*x+c)^4+10 *cos(d*x+c)^3+10*cos(d*x+c)^2+5*cos(d*x+c)+1)/sec(d*x+c)^(5/2)/(cos(d*x+c) /(cos(d*x+c)+1))^(1/2)*(tan(d*x+c)*sec(d*x+c)*(-73*cos(d*x+c)^3-351*cos(d* x+c)^2-195*cos(d*x+c)-45)*2^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+arcsin (cot(d*x+c)-csc(d*x+c))*(45*cos(d*x+c)^2+180*cos(d*x+c)+270+180*sec(d*x+c) +45*sec(d*x+c)^2))/a^5
Time = 0.14 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {45 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 10 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - \frac {2 \, {\left (73 \, \cos \left (d x + c\right )^{4} + 351 \, \cos \left (d x + c\right )^{3} + 195 \, \cos \left (d x + c\right )^{2} + 45 \, \cos \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{2048 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \] Input:
integrate(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(5/2),x, algorithm="fricas")
Output:
-1/2048*(45*sqrt(2)*(cos(d*x + c)^5 + 5*cos(d*x + c)^4 + 10*cos(d*x + c)^3 + 10*cos(d*x + c)^2 + 5*cos(d*x + c) + 1)*sqrt(a)*arctan(sqrt(2)*sqrt(a*c os(d*x + c) + a)*sqrt(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) - 2*(73*cos(d* x + c)^4 + 351*cos(d*x + c)^3 + 195*cos(d*x + c)^2 + 45*cos(d*x + c))*sqrt (a*cos(d*x + c) + a)*sin(d*x + c)/sqrt(cos(d*x + c)))/(a^5*d*cos(d*x + c)^ 5 + 5*a^5*d*cos(d*x + c)^4 + 10*a^5*d*cos(d*x + c)^3 + 10*a^5*d*cos(d*x + c)^2 + 5*a^5*d*cos(d*x + c) + a^5*d)
Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate(1/(a+a*cos(d*x+c))**(9/2)/sec(d*x+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {9}{2}} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(5/2),x, algorithm="maxima")
Output:
integrate(1/((a*cos(d*x + c) + a)^(9/2)*sec(d*x + c)^(5/2)), x)
Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(5/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {1}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{9/2}} \,d x \] Input:
int(1/((1/cos(c + d*x))^(5/2)*(a + a*cos(c + d*x))^(9/2)),x)
Output:
int(1/((1/cos(c + d*x))^(5/2)*(a + a*cos(c + d*x))^(9/2)), x)
\[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )+1}}{\cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{3}+5 \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{3}+10 \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3}+10 \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3}+5 \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3}+\sec \left (d x +c \right )^{3}}d x \right )}{a^{5}} \] Input:
int(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(5/2),x)
Output:
(sqrt(a)*int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x) + 1))/(cos(c + d*x)**5* sec(c + d*x)**3 + 5*cos(c + d*x)**4*sec(c + d*x)**3 + 10*cos(c + d*x)**3*s ec(c + d*x)**3 + 10*cos(c + d*x)**2*sec(c + d*x)**3 + 5*cos(c + d*x)*sec(c + d*x)**3 + sec(c + d*x)**3),x))/a**5