Integrand size = 12, antiderivative size = 66 \[ \int \frac {\cos ^4(a+b x)}{x^2} \, dx=-\frac {\cos ^4(a+b x)}{x}-b \operatorname {CosIntegral}(2 b x) \sin (2 a)-\frac {1}{2} b \operatorname {CosIntegral}(4 b x) \sin (4 a)-b \cos (2 a) \text {Si}(2 b x)-\frac {1}{2} b \cos (4 a) \text {Si}(4 b x) \]
-cos(b*x+a)^4/x-b*cos(2*a)*Si(2*b*x)-1/2*b*cos(4*a)*Si(4*b*x)-b*Ci(2*b*x)* sin(2*a)-1/2*b*Ci(4*b*x)*sin(4*a)
Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.20 \[ \int \frac {\cos ^4(a+b x)}{x^2} \, dx=-\frac {3+4 \cos (2 (a+b x))+\cos (4 (a+b x))+8 b x \operatorname {CosIntegral}(2 b x) \sin (2 a)+4 b x \operatorname {CosIntegral}(4 b x) \sin (4 a)+8 b x \cos (2 a) \text {Si}(2 b x)+4 b x \cos (4 a) \text {Si}(4 b x)}{8 x} \]
-1/8*(3 + 4*Cos[2*(a + b*x)] + Cos[4*(a + b*x)] + 8*b*x*CosIntegral[2*b*x] *Sin[2*a] + 4*b*x*CosIntegral[4*b*x]*Sin[4*a] + 8*b*x*Cos[2*a]*SinIntegral [2*b*x] + 4*b*x*Cos[4*a]*SinIntegral[4*b*x])/x
Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3794, 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.
| \(\displaystyle \int \frac {\cos ^4(a+b x)}{x^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )^4}{x^2}dx\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle 4 b \int \left (-\frac {\sin (2 a+2 b x)}{4 x}-\frac {\sin (4 a+4 b x)}{8 x}\right )dx-\frac {\cos ^4(a+b x)}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 b \left (-\frac {1}{4} \sin (2 a) \operatorname {CosIntegral}(2 b x)-\frac {1}{8} \sin (4 a) \operatorname {CosIntegral}(4 b x)-\frac {1}{4} \cos (2 a) \text {Si}(2 b x)-\frac {1}{8} \cos (4 a) \text {Si}(4 b x)\right )-\frac {\cos ^4(a+b x)}{x}\) |
-(Cos[a + b*x]^4/x) + 4*b*(-1/4*(CosIntegral[2*b*x]*Sin[2*a]) - (CosIntegr al[4*b*x]*Sin[4*a])/8 - (Cos[2*a]*SinIntegral[2*b*x])/4 - (Cos[4*a]*SinInt egral[4*b*x])/8)
3.1.27.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 ))) Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & & LtQ[m, -1]
Time = 0.74 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.36
| method | result | size |
| derivativedivides | \(b \left (-\frac {\cos \left (4 b x +4 a \right )}{8 b x}-\frac {\operatorname {Si}\left (4 b x \right ) \cos \left (4 a \right )}{2}-\frac {\operatorname {Ci}\left (4 b x \right ) \sin \left (4 a \right )}{2}-\frac {\cos \left (2 b x +2 a \right )}{2 b x}-\operatorname {Si}\left (2 b x \right ) \cos \left (2 a \right )-\operatorname {Ci}\left (2 b x \right ) \sin \left (2 a \right )-\frac {3}{8 b x}\right )\) | \(90\) |
| default | \(b \left (-\frac {\cos \left (4 b x +4 a \right )}{8 b x}-\frac {\operatorname {Si}\left (4 b x \right ) \cos \left (4 a \right )}{2}-\frac {\operatorname {Ci}\left (4 b x \right ) \sin \left (4 a \right )}{2}-\frac {\cos \left (2 b x +2 a \right )}{2 b x}-\operatorname {Si}\left (2 b x \right ) \cos \left (2 a \right )-\operatorname {Ci}\left (2 b x \right ) \sin \left (2 a \right )-\frac {3}{8 b x}\right )\) | \(90\) |
| risch | \(\frac {4 \,{\mathrm e}^{-2 i a} \pi \,\operatorname {csgn}\left (b x \right ) b x +4 i {\mathrm e}^{-2 i a} \operatorname {Ei}_{1}\left (-2 i x b \right ) b x -2 i b \,\operatorname {Ei}_{1}\left (-4 i x b \right ) {\mathrm e}^{4 i a} x +2 i \operatorname {Ei}_{1}\left (-4 i x b \right ) {\mathrm e}^{-4 i a} b x +2 \,{\mathrm e}^{-4 i a} \pi \,\operatorname {csgn}\left (b x \right ) b x -4 i b \,\operatorname {Ei}_{1}\left (-2 i x b \right ) {\mathrm e}^{2 i a} x -8 \,{\mathrm e}^{-2 i a} \operatorname {Si}\left (2 b x \right ) b x -4 \,{\mathrm e}^{-4 i a} \operatorname {Si}\left (4 b x \right ) b x -\cos \left (4 b x +4 a \right )-4 \cos \left (2 b x +2 a \right )-3}{8 x}\) | \(154\) |
b*(-1/8*cos(4*b*x+4*a)/b/x-1/2*Si(4*b*x)*cos(4*a)-1/2*Ci(4*b*x)*sin(4*a)-1 /2*cos(2*b*x+2*a)/b/x-Si(2*b*x)*cos(2*a)-Ci(2*b*x)*sin(2*a)-3/8/b/x)
Time = 0.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^4(a+b x)}{x^2} \, dx=-\frac {2 \, \cos \left (b x + a\right )^{4} + b x \operatorname {Ci}\left (4 \, b x\right ) \sin \left (4 \, a\right ) + 2 \, b x \operatorname {Ci}\left (2 \, b x\right ) \sin \left (2 \, a\right ) + b x \cos \left (4 \, a\right ) \operatorname {Si}\left (4 \, b x\right ) + 2 \, b x \cos \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x\right )}{2 \, x} \]
-1/2*(2*cos(b*x + a)^4 + b*x*cos_integral(4*b*x)*sin(4*a) + 2*b*x*cos_inte gral(2*b*x)*sin(2*a) + b*x*cos(4*a)*sin_integral(4*b*x) + 2*b*x*cos(2*a)*s in_integral(2*b*x))/x
\[ \int \frac {\cos ^4(a+b x)}{x^2} \, dx=\int \frac {\cos ^{4}{\left (a + b x \right )}}{x^{2}}\, dx \]
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 717, normalized size of antiderivative = 10.86 \[ \int \frac {\cos ^4(a+b x)}{x^2} \, dx=\text {Too large to display} \]
1/32*(((exp_integral_e(2, 4*I*b*x) + exp_integral_e(2, -4*I*b*x))*cos(2*a) ^2 + (exp_integral_e(2, 4*I*b*x) + exp_integral_e(2, -4*I*b*x))*sin(2*a)^2 )*cos(4*a)^3 + ((-I*exp_integral_e(2, 4*I*b*x) + I*exp_integral_e(2, -4*I* b*x))*cos(2*a)^2 + (-I*exp_integral_e(2, 4*I*b*x) + I*exp_integral_e(2, -4 *I*b*x))*sin(2*a)^2)*sin(4*a)^3 + 4*((exp_integral_e(2, 2*I*b*x) + exp_int egral_e(2, -2*I*b*x))*cos(2*a)^3 + (-I*exp_integral_e(2, 2*I*b*x) + I*exp_ integral_e(2, -2*I*b*x))*sin(2*a)^3 + ((exp_integral_e(2, 2*I*b*x) + exp_i ntegral_e(2, -2*I*b*x))*cos(2*a) + 3)*sin(2*a)^2 + (exp_integral_e(2, 2*I* b*x) + exp_integral_e(2, -2*I*b*x))*cos(2*a) + 3*cos(2*a)^2 + ((-I*exp_int egral_e(2, 2*I*b*x) + I*exp_integral_e(2, -2*I*b*x))*cos(2*a)^2 - I*exp_in tegral_e(2, 2*I*b*x) + I*exp_integral_e(2, -2*I*b*x))*sin(2*a))*cos(4*a)^2 + (4*(exp_integral_e(2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*cos(2*a)^ 3 + 4*(-I*exp_integral_e(2, 2*I*b*x) + I*exp_integral_e(2, -2*I*b*x))*sin( 2*a)^3 + 4*((exp_integral_e(2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*cos (2*a) + 3)*sin(2*a)^2 + ((exp_integral_e(2, 4*I*b*x) + exp_integral_e(2, - 4*I*b*x))*cos(2*a)^2 + (exp_integral_e(2, 4*I*b*x) + exp_integral_e(2, -4* I*b*x))*sin(2*a)^2)*cos(4*a) + 4*(exp_integral_e(2, 2*I*b*x) + exp_integra l_e(2, -2*I*b*x))*cos(2*a) + 12*cos(2*a)^2 + 4*((-I*exp_integral_e(2, 2*I* b*x) + I*exp_integral_e(2, -2*I*b*x))*cos(2*a)^2 - I*exp_integral_e(2, 2*I *b*x) + I*exp_integral_e(2, -2*I*b*x))*sin(2*a))*sin(4*a)^2 + ((exp_int...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.38 (sec) , antiderivative size = 3220, normalized size of antiderivative = 48.79 \[ \int \frac {\cos ^4(a+b x)}{x^2} \, dx=\text {Too large to display} \]
1/4*(b*x*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 *tan(a)^2 + 2*b*x*imag_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*t an(2*a)^2*tan(a)^2 - 2*b*x*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*ta n(b*x)^2*tan(2*a)^2*tan(a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(2*b *x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 2*b*x*sin_integral(4*b*x)*tan(2*b*x )^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b*x*sin_integral(2*b*x)*tan(2*b*x)^ 2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 - 4*b*x*real_part(cos_integral(2*b*x))*ta n(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a) - 4*b*x*real_part(cos_integral(-2* b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a) - 2*b*x*real_part(cos_inte gral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)*tan(a)^2 - 2*b*x*real_part(c os_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)*tan(a)^2 + b*x*imag_ part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - 2*b*x*imag_ part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 + 2*b*x*imag_ part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - b*x*imag_p art(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 + 2*b*x*sin_i ntegral(4*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - 4*b*x*sin_integral(2*b *x)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - b*x*imag_part(cos_integral(4*b*x) )*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 + 2*b*x*imag_part(cos_integral(2*b*x))* tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 - 2*b*x*imag_part(cos_integral(-2*b*x))*t an(2*b*x)^2*tan(b*x)^2*tan(a)^2 + b*x*imag_part(cos_integral(-4*b*x))*t...
Timed out. \[ \int \frac {\cos ^4(a+b x)}{x^2} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^4}{x^2} \,d x \]