Integrand size = 25, antiderivative size = 217 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx=\frac {35 \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{1024 \sqrt {2} a^{9/2} d}-\frac {\cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac {19 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2}}-\frac {187 \sqrt {\cos (c+d x)} \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac {853 \sqrt {\cos (c+d x)} \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2}} \] Output:
35/2048*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))*2^(1/2)/a^(9/2)/d-1/8*cos(d*x+c)^(5/2)*sin(d*x+c)/d/(a+a*cos( d*x+c))^(9/2)-19/96*cos(d*x+c)^(3/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^(7/2) -187/768*cos(d*x+c)^(1/2)*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^(5/2)+853/3072 *cos(d*x+c)^(1/2)*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^(3/2)
Leaf count is larger than twice the leaf count of optimal. \(474\) vs. \(2(217)=434\).
Time = 6.08 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.18 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx=\frac {35 \arcsin \left (\frac {\sin \left (\frac {c}{2}+\frac {d x}{2}\right )}{\sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )}}\right ) \cos ^9\left (\frac {c}{2}+\frac {d x}{2}\right )}{64 d (a (1+\cos (c+d x)))^{9/2}}+\frac {93 \cos ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) \sin \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}{64 d \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )} (a (1+\cos (c+d x)))^{9/2}}-\frac {163 \cos ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) \sin ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}{96 d \cos ^2\left (\frac {1}{2} (c+d x)\right )^{3/2} (a (1+\cos (c+d x)))^{9/2}}+\frac {25 \cos ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) \sin ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}{24 d \cos ^2\left (\frac {1}{2} (c+d x)\right )^{5/2} (a (1+\cos (c+d x)))^{9/2}}-\frac {\cos ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) \sin ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}{4 d \cos ^2\left (\frac {1}{2} (c+d x)\right )^{7/2} (a (1+\cos (c+d x)))^{9/2}} \] Input:
Integrate[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(9/2),x]
Output:
(35*ArcSin[Sin[c/2 + (d*x)/2]/Sqrt[Cos[(c + d*x)/2]^2]]*Cos[c/2 + (d*x)/2] ^9)/(64*d*(a*(1 + Cos[c + d*x]))^(9/2)) + (93*Cos[c/2 + (d*x)/2]^9*Sin[c/2 + (d*x)/2]*Sqrt[1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2])/(64*d*Sqrt[ Cos[(c + d*x)/2]^2]*(a*(1 + Cos[c + d*x]))^(9/2)) - (163*Cos[c/2 + (d*x)/2 ]^9*Sin[c/2 + (d*x)/2]^3*Sqrt[1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2] )/(96*d*(Cos[(c + d*x)/2]^2)^(3/2)*(a*(1 + Cos[c + d*x]))^(9/2)) + (25*Cos [c/2 + (d*x)/2]^9*Sin[c/2 + (d*x)/2]^5*Sqrt[1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2])/(24*d*(Cos[(c + d*x)/2]^2)^(5/2)*(a*(1 + Cos[c + d*x]))^(9 /2)) - (Cos[c/2 + (d*x)/2]^9*Sin[c/2 + (d*x)/2]^7*Sqrt[1 - Sec[(c + d*x)/2 ]^2*Sin[c/2 + (d*x)/2]^2])/(4*d*(Cos[(c + d*x)/2]^2)^(7/2)*(a*(1 + Cos[c + d*x]))^(9/2))
Time = 1.18 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.10, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3244, 27, 3042, 3456, 27, 3042, 3456, 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.
| \(\displaystyle \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a \cos (c+d x)+a)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) (5 a-14 a \cos (c+d x))}{2 (\cos (c+d x) a+a)^{7/2}}dx}{8 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) (5 a-14 a \cos (c+d x))}{(\cos (c+d x) a+a)^{7/2}}dx}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (5 a-14 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{7/2}}dx}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\int \frac {\sqrt {\cos (c+d x)} \left (57 a^2-130 a^2 \cos (c+d x)\right )}{2 (\cos (c+d x) a+a)^{5/2}}dx}{6 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\sqrt {\cos (c+d x)} \left (57 a^2-130 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^{5/2}}dx}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (57 a^2-130 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}}dx}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {187 a^3-666 a^3 \cos (c+d x)}{2 \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^{3/2}}dx}{4 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {187 a^3-666 a^3 \cos (c+d x)}{\sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^{3/2}}dx}{8 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {187 a^3-666 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int -\frac {105 a^4}{2 \sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{2 a^2}-\frac {853 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {-\frac {105}{4} a^2 \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx-\frac {853 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {-\frac {105}{4} a^2 \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {853 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 3261 |
| \(\displaystyle -\frac {\frac {\frac {\frac {105 a^3 \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x) a^3}{\cos (c+d x) a+a}+2 a^2}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}\right )}{2 d}-\frac {853 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac {-\frac {105 a^{3/2} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} d}-\frac {853 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\) |
Input:
Int[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(9/2),x]
Output:
-1/8*(Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^(9/2)) - (( 19*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ( (187*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((-105*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*Sqrt[2]*d) - (853*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])
Time = 3.77 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(\frac {\left (\sin \left (d x +c \right ) \left (853 \cos \left (d x +c \right )^{3}+819 \cos \left (d x +c \right )^{2}+455 \cos \left (d x +c \right )+105\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (-105 \cos \left (d x +c \right )^{4}-420 \cos \left (d x +c \right )^{3}-630 \cos \left (d x +c \right )^{2}-420 \cos \left (d x +c \right )-105\right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right ) \sqrt {\cos \left (d x +c \right )}\, \sqrt {2}\, \sqrt {a \left (\cos \left (d x +c \right )+1\right )}}{6144 d \left (\cos \left (d x +c \right )^{5}+5 \cos \left (d x +c \right )^{4}+10 \cos \left (d x +c \right )^{3}+10 \cos \left (d x +c \right )^{2}+5 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, a^{5}}\) | \(218\) |
Input:
int(cos(d*x+c)^(7/2)/(a+a*cos(d*x+c))^(9/2),x,method=_RETURNVERBOSE)
Output:
1/6144/d*(sin(d*x+c)*(853*cos(d*x+c)^3+819*cos(d*x+c)^2+455*cos(d*x+c)+105 )*2^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+(-105*cos(d*x+c)^4-420*cos(d*x +c)^3-630*cos(d*x+c)^2-420*cos(d*x+c)-105)*arcsin(cot(d*x+c)-csc(d*x+c)))* cos(d*x+c)^(1/2)*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)/(cos(d*x+c)/(cos(d*x+ c)+1))^(1/2)/a^5
Time = 0.12 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.14 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx=\frac {105 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 10 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, {\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )}}\right ) + 2 \, {\left (853 \, \cos \left (d x + c\right )^{3} + 819 \, \cos \left (d x + c\right )^{2} + 455 \, \cos \left (d x + c\right ) + 105\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{6144 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \] Input:
integrate(cos(d*x+c)^(7/2)/(a+a*cos(d*x+c))^(9/2),x, algorithm="fricas")
Output:
1/6144*(105*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(1/2*sqrt(2)*sqrt (a*cos(d*x + c) + a)*sqrt(a)*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d*x + c)^2 + a*cos(d*x + c))) + 2*(853*cos(d*x + c)^3 + 819*cos(d*x + c)^2 + 455 *cos(d*x + c) + 105)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))*sin(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. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(7/2)/(a+a*cos(d*x+c))**(9/2),x)
Output:
Timed out
\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {7}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^(7/2)/(a+a*cos(d*x+c))^(9/2),x, algorithm="maxima")
Output:
integrate(cos(d*x + c)^(7/2)/(a*cos(d*x + c) + a)^(9/2), x)
Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^(7/2)/(a+a*cos(d*x+c))^(9/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{7/2}}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{9/2}} \,d x \] Input:
int(cos(c + d*x)^(7/2)/(a + a*cos(c + d*x))^(9/2),x)
Output:
int(cos(c + d*x)^(7/2)/(a + a*cos(c + d*x))^(9/2), x)
\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{5}+5 \cos \left (d x +c \right )^{4}+10 \cos \left (d x +c \right )^{3}+10 \cos \left (d x +c \right )^{2}+5 \cos \left (d x +c \right )+1}d x \right )}{a^{5}} \] Input:
int(cos(d*x+c)^(7/2)/(a+a*cos(d*x+c))^(9/2),x)
Output:
(sqrt(a)*int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x)**3)/( cos(c + d*x)**5 + 5*cos(c + d*x)**4 + 10*cos(c + d*x)**3 + 10*cos(c + d*x) **2 + 5*cos(c + d*x) + 1),x))/a**5