Integrand size = 24, antiderivative size = 132 \[ \int \frac {\cos ^5(a x)}{x^3 (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\cos (a x)}{x^2}+\frac {\cos ^3(a x)}{a^2 x^4}-\frac {3 \cos ^3(a x)}{2 x^2}-\frac {1}{8} a^2 \operatorname {CosIntegral}(a x)-\frac {27}{8} a^2 \operatorname {CosIntegral}(3 a x)-\frac {a \sin (a x)}{x}-\frac {\cos ^2(a x) \sin (a x)}{a x^3}+\frac {9 a \cos ^2(a x) \sin (a x)}{2 x}-\frac {\cos ^4(a x)}{a^2 x^4 (\cos (a x)+a x \sin (a x))} \]
-1/8*a^2*Ci(a*x)-27/8*a^2*Ci(3*a*x)+cos(a*x)/x^2+cos(a*x)^3/a^2/x^4-3/2*co s(a*x)^3/x^2-a*sin(a*x)/x-cos(a*x)^2*sin(a*x)/a/x^3+9/2*a*cos(a*x)^2*sin(a *x)/x-cos(a*x)^4/a^2/x^4/(cos(a*x)+a*x*sin(a*x))
Time = 0.76 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.03 \[ \int \frac {\cos ^5(a x)}{x^3 (\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {3-a^2 x^2+4 \cos (2 a x)-8 a^2 x^2 \cos (2 a x)+\cos (4 a x)+9 a^2 x^2 \cos (4 a x)+2 a^2 x^2 \operatorname {CosIntegral}(a x) (\cos (a x)+a x \sin (a x))+54 a^2 x^2 \operatorname {CosIntegral}(3 a x) (\cos (a x)+a x \sin (a x))-12 a x \sin (2 a x)-6 a x \sin (4 a x)}{16 x^2 (\cos (a x)+a x \sin (a x))} \]
-1/16*(3 - a^2*x^2 + 4*Cos[2*a*x] - 8*a^2*x^2*Cos[2*a*x] + Cos[4*a*x] + 9* a^2*x^2*Cos[4*a*x] + 2*a^2*x^2*CosIntegral[a*x]*(Cos[a*x] + a*x*Sin[a*x]) + 54*a^2*x^2*CosIntegral[3*a*x]*(Cos[a*x] + a*x*Sin[a*x]) - 12*a*x*Sin[2*a *x] - 6*a*x*Sin[4*a*x])/(x^2*(Cos[a*x] + a*x*Sin[a*x]))
Time = 1.03 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.36, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5110, 3042, 3795, 3042, 3778, 25, 3042, 3778, 3042, 3783, 3795, 3042, 3783, 3793, 2009}
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.
| \(\displaystyle \int \frac {\cos ^5(a x)}{x^3 (a x \sin (a x)+\cos (a x))^2} \, dx\) |
| \(\Big \downarrow \) 5110 |
| \(\displaystyle -\frac {4 \int \frac {\cos ^3(a x)}{x^5}dx}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^3}{x^5}dx}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle -\frac {4 \left (-\frac {3}{4} a^2 \int \frac {\cos ^3(a x)}{x^3}dx+\frac {1}{2} a^2 \int \frac {\cos (a x)}{x^3}dx-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )}{x^3}dx-\frac {3}{4} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^3}{x^3}dx-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle -\frac {4 \left (-\frac {3}{4} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^3}{x^3}dx+\frac {1}{2} a^2 \left (\frac {1}{2} a \int -\frac {\sin (a x)}{x^2}dx-\frac {\cos (a x)}{2 x^2}\right )-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 \left (-\frac {3}{4} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^3}{x^3}dx+\frac {1}{2} a^2 \left (-\frac {1}{2} a \int \frac {\sin (a x)}{x^2}dx-\frac {\cos (a x)}{2 x^2}\right )-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (-\frac {3}{4} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^3}{x^3}dx+\frac {1}{2} a^2 \left (-\frac {1}{2} a \int \frac {\sin (a x)}{x^2}dx-\frac {\cos (a x)}{2 x^2}\right )-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle -\frac {4 \left (-\frac {3}{4} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^3}{x^3}dx+\frac {1}{2} a^2 \left (-\frac {1}{2} a \left (a \int \frac {\cos (a x)}{x}dx-\frac {\sin (a x)}{x}\right )-\frac {\cos (a x)}{2 x^2}\right )-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (-\frac {3}{4} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^3}{x^3}dx+\frac {1}{2} a^2 \left (-\frac {1}{2} a \left (a \int \frac {\sin \left (a x+\frac {\pi }{2}\right )}{x}dx-\frac {\sin (a x)}{x}\right )-\frac {\cos (a x)}{2 x^2}\right )-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle -\frac {4 \left (-\frac {3}{4} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^3}{x^3}dx+\frac {1}{2} a^2 \left (-\frac {1}{2} a \left (a \operatorname {CosIntegral}(a x)-\frac {\sin (a x)}{x}\right )-\frac {\cos (a x)}{2 x^2}\right )-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle -\frac {4 \left (-\frac {3}{4} a^2 \left (-\frac {9}{2} a^2 \int \frac {\cos ^3(a x)}{x}dx+3 a^2 \int \frac {\cos (a x)}{x}dx-\frac {\cos ^3(a x)}{2 x^2}+\frac {3 a \sin (a x) \cos ^2(a x)}{2 x}\right )+\frac {1}{2} a^2 \left (-\frac {1}{2} a \left (a \operatorname {CosIntegral}(a x)-\frac {\sin (a x)}{x}\right )-\frac {\cos (a x)}{2 x^2}\right )-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (-\frac {3}{4} a^2 \left (3 a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )}{x}dx-\frac {9}{2} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^3}{x}dx-\frac {\cos ^3(a x)}{2 x^2}+\frac {3 a \sin (a x) \cos ^2(a x)}{2 x}\right )+\frac {1}{2} a^2 \left (-\frac {1}{2} a \left (a \operatorname {CosIntegral}(a x)-\frac {\sin (a x)}{x}\right )-\frac {\cos (a x)}{2 x^2}\right )-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle -\frac {4 \left (-\frac {3}{4} a^2 \left (-\frac {9}{2} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^3}{x}dx+3 a^2 \operatorname {CosIntegral}(a x)-\frac {\cos ^3(a x)}{2 x^2}+\frac {3 a \sin (a x) \cos ^2(a x)}{2 x}\right )+\frac {1}{2} a^2 \left (-\frac {1}{2} a \left (a \operatorname {CosIntegral}(a x)-\frac {\sin (a x)}{x}\right )-\frac {\cos (a x)}{2 x^2}\right )-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle -\frac {4 \left (-\frac {3}{4} a^2 \left (-\frac {9}{2} a^2 \int \left (\frac {3 \cos (a x)}{4 x}+\frac {\cos (3 a x)}{4 x}\right )dx+3 a^2 \operatorname {CosIntegral}(a x)-\frac {\cos ^3(a x)}{2 x^2}+\frac {3 a \sin (a x) \cos ^2(a x)}{2 x}\right )+\frac {1}{2} a^2 \left (-\frac {1}{2} a \left (a \operatorname {CosIntegral}(a x)-\frac {\sin (a x)}{x}\right )-\frac {\cos (a x)}{2 x^2}\right )-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \left (-\frac {3}{4} a^2 \left (3 a^2 \operatorname {CosIntegral}(a x)-\frac {9}{2} a^2 \left (\frac {3 \operatorname {CosIntegral}(a x)}{4}+\frac {\operatorname {CosIntegral}(3 a x)}{4}\right )-\frac {\cos ^3(a x)}{2 x^2}+\frac {3 a \sin (a x) \cos ^2(a x)}{2 x}\right )+\frac {1}{2} a^2 \left (-\frac {1}{2} a \left (a \operatorname {CosIntegral}(a x)-\frac {\sin (a x)}{x}\right )-\frac {\cos (a x)}{2 x^2}\right )-\frac {\cos ^3(a x)}{4 x^4}+\frac {a \sin (a x) \cos ^2(a x)}{4 x^3}\right )}{a^2}-\frac {\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}\) |
-(Cos[a*x]^4/(a^2*x^4*(Cos[a*x] + a*x*Sin[a*x]))) - (4*(-1/4*Cos[a*x]^3/x^ 4 + (a*Cos[a*x]^2*Sin[a*x])/(4*x^3) - (3*a^2*(-1/2*Cos[a*x]^3/x^2 + 3*a^2* CosIntegral[a*x] - (9*a^2*((3*CosIntegral[a*x])/4 + CosIntegral[3*a*x]/4)) /2 + (3*a*Cos[a*x]^2*Sin[a*x])/(2*x)))/4 + (a^2*(-1/2*Cos[a*x]/x^2 - (a*(a *CosIntegral[a*x] - Sin[a*x]/x))/2))/2))/a^2
3.6.95.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) *(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* (m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
Int[(Cos[(a_.)*(x_)]^(n_)*((b_.)*(x_))^(m_))/(Cos[(a_.)*(x_)]*(c_.) + (d_.) *(x_)*Sin[(a_.)*(x_)])^2, x_Symbol] :> Simp[(-b)*(b*x)^(m - 1)*(Cos[a*x]^(n - 1)/(a*d*(c*Cos[a*x] + d*x*Sin[a*x]))), x] - Simp[b^2*((n - 1)/d^2) Int [(b*x)^(m - 2)*Cos[a*x]^(n - 2), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c - d, 0] && EqQ[m, 2 - n]
Timed out.
\[\int \frac {\cos \left (a x \right )^{5}}{x^{3} \left (\cos \left (a x \right )+a x \sin \left (a x \right )\right )^{2}}d x\]
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^5(a x)}{x^3 (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {44 \, a^{2} x^{2} \cos \left (a x\right )^{2} - 4 \, {\left (9 \, a^{2} x^{2} + 1\right )} \cos \left (a x\right )^{4} - 8 \, a^{2} x^{2} - {\left (27 \, a^{2} x^{2} \operatorname {Ci}\left (3 \, a x\right ) + a^{2} x^{2} \operatorname {Ci}\left (a x\right )\right )} \cos \left (a x\right ) - {\left (27 \, a^{3} x^{3} \operatorname {Ci}\left (3 \, a x\right ) + a^{3} x^{3} \operatorname {Ci}\left (a x\right ) - 24 \, a x \cos \left (a x\right )^{3}\right )} \sin \left (a x\right )}{8 \, {\left (a x^{3} \sin \left (a x\right ) + x^{2} \cos \left (a x\right )\right )}} \]
1/8*(44*a^2*x^2*cos(a*x)^2 - 4*(9*a^2*x^2 + 1)*cos(a*x)^4 - 8*a^2*x^2 - (2 7*a^2*x^2*cos_integral(3*a*x) + a^2*x^2*cos_integral(a*x))*cos(a*x) - (27* a^3*x^3*cos_integral(3*a*x) + a^3*x^3*cos_integral(a*x) - 24*a*x*cos(a*x)^ 3)*sin(a*x))/(a*x^3*sin(a*x) + x^2*cos(a*x))
Timed out. \[ \int \frac {\cos ^5(a x)}{x^3 (\cos (a x)+a x \sin (a x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^5(a x)}{x^3 (\cos (a x)+a x \sin (a x))^2} \, dx=\text {Exception raised: RuntimeError} \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.48 (sec) , antiderivative size = 3130, normalized size of antiderivative = 23.71 \[ \int \frac {\cos ^5(a x)}{x^3 (\cos (a x)+a x \sin (a x))^2} \, dx=\text {Too large to display} \]
-1/16*(54*a^7*x^7*real_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a* x)^3 + 2*a^7*x^7*real_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^ 3 + 2*a^7*x^7*real_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 54*a^7*x^7*real_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - 27*a^6*x^6*real_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - a^6*x^6*real_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - a^ 6*x^6*real_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 27*a^6 *x^6*real_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + 54*a^ 7*x^7*real_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 2*a^7*x ^7*real_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 2*a^7*x^7*re al_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 54*a^7*x^7*real_ part(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 54*a^7*x^7*real_p art(cos_integral(3*a*x))*tan(1/2*a*x)^3 + 2*a^7*x^7*real_part(cos_integral (a*x))*tan(1/2*a*x)^3 + 2*a^7*x^7*real_part(cos_integral(-a*x))*tan(1/2*a* x)^3 + 54*a^7*x^7*real_part(cos_integral(-3*a*x))*tan(1/2*a*x)^3 - 27*a^6* x^6*real_part(cos_integral(3*a*x))*tan(1/2*a*x)^4 - a^6*x^6*real_part(cos_ integral(a*x))*tan(1/2*a*x)^4 - a^6*x^6*real_part(cos_integral(-a*x))*tan( 1/2*a*x)^4 - 27*a^6*x^6*real_part(cos_integral(-3*a*x))*tan(1/2*a*x)^4 + 5 4*a^7*x^7*real_part(cos_integral(3*a*x))*tan(1/2*a*x) + 2*a^7*x^7*real_par t(cos_integral(a*x))*tan(1/2*a*x) + 2*a^7*x^7*real_part(cos_integral(-a...
Timed out. \[ \int \frac {\cos ^5(a x)}{x^3 (\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {{\cos \left (a\,x\right )}^5}{x^3\,{\left (\cos \left (a\,x\right )+a\,x\,\sin \left (a\,x\right )\right )}^2} \,d x \]