Integrand size = 8, antiderivative size = 46 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^3} \, dx=-\frac {\cos (b x)}{4 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)-\frac {\operatorname {CosIntegral}(b x)}{2 x^2}+\frac {b \sin (b x)}{4 x} \] Output:
-1/4*cos(b*x)/x^2-1/4*b^2*Ci(b*x)-1/2*Ci(b*x)/x^2+1/4*b*sin(b*x)/x
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^3} \, dx=-\frac {\cos (b x)}{4 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)-\frac {\operatorname {CosIntegral}(b x)}{2 x^2}+\frac {b \sin (b x)}{4 x} \] Input:
Integrate[CosIntegral[b*x]/x^3,x]
Output:
-1/4*Cos[b*x]/x^2 - (b^2*CosIntegral[b*x])/4 - CosIntegral[b*x]/(2*x^2) + (b*Sin[b*x])/(4*x)
Time = 0.39 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {7058, 27, 3042, 3778, 25, 3042, 3778, 3042, 3783}
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.
\(\displaystyle \int \frac {\operatorname {CosIntegral}(b x)}{x^3} \, dx\) |
\(\Big \downarrow \) 7058 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x)}{b x^3}dx-\frac {\operatorname {CosIntegral}(b x)}{2 x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {\cos (b x)}{x^3}dx-\frac {\operatorname {CosIntegral}(b x)}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \frac {\sin \left (b x+\frac {\pi }{2}\right )}{x^3}dx-\frac {\operatorname {CosIntegral}(b x)}{2 x^2}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} b \int -\frac {\sin (b x)}{x^2}dx-\frac {\cos (b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x)}{2 x^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} b \int \frac {\sin (b x)}{x^2}dx-\frac {\cos (b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x)}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} b \int \frac {\sin (b x)}{x^2}dx-\frac {\cos (b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x)}{2 x^2}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} b \left (b \int \frac {\cos (b x)}{x}dx-\frac {\sin (b x)}{x}\right )-\frac {\cos (b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x)}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} b \left (b \int \frac {\sin \left (b x+\frac {\pi }{2}\right )}{x}dx-\frac {\sin (b x)}{x}\right )-\frac {\cos (b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x)}{2 x^2}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} b \left (b \operatorname {CosIntegral}(b x)-\frac {\sin (b x)}{x}\right )-\frac {\cos (b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x)}{2 x^2}\) |
Input:
Int[CosIntegral[b*x]/x^3,x]
Output:
-1/2*CosIntegral[b*x]/x^2 + (-1/2*Cos[b*x]/x^2 - (b*(b*CosIntegral[b*x] - Sin[b*x]/x))/2)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[CosIntegral[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] : > Simp[(c + d*x)^(m + 1)*(CosIntegral[a + b*x]/(d*(m + 1))), x] - Simp[b/(d *(m + 1)) Int[(c + d*x)^(m + 1)*(Cos[a + b*x]/(a + b*x)), x], x] /; FreeQ [{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.77 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.02
method | result | size |
parts | \(-\frac {\operatorname {Ci}\left (b x \right )}{2 x^{2}}+\frac {b^{2} \left (-\frac {\cos \left (b x \right )}{2 b^{2} x^{2}}+\frac {\sin \left (b x \right )}{2 b x}-\frac {\operatorname {Ci}\left (b x \right )}{2}\right )}{2}\) | \(47\) |
derivativedivides | \(b^{2} \left (-\frac {\operatorname {Ci}\left (b x \right )}{2 b^{2} x^{2}}-\frac {\cos \left (b x \right )}{4 b^{2} x^{2}}+\frac {\sin \left (b x \right )}{4 b x}-\frac {\operatorname {Ci}\left (b x \right )}{4}\right )\) | \(48\) |
default | \(b^{2} \left (-\frac {\operatorname {Ci}\left (b x \right )}{2 b^{2} x^{2}}-\frac {\cos \left (b x \right )}{4 b^{2} x^{2}}+\frac {\sin \left (b x \right )}{4 b x}-\frac {\operatorname {Ci}\left (b x \right )}{4}\right )\) | \(48\) |
orering | \(\frac {\left (-\frac {1}{4} b^{2} x^{3}-\frac {7}{2} x \right ) \operatorname {Ci}\left (b x \right )}{x^{3}}-2 x^{2} \left (\frac {\cos \left (b x \right )}{x^{4}}-\frac {3 \,\operatorname {Ci}\left (b x \right )}{x^{4}}\right )-\frac {x^{3} \left (-\frac {b \sin \left (b x \right )}{x^{4}}-\frac {7 \cos \left (b x \right )}{x^{5}}+\frac {12 \,\operatorname {Ci}\left (b x \right )}{x^{5}}\right )}{4}\) | \(79\) |
meijerg | \(\frac {\sqrt {\pi }\, b^{2} \left (-\frac {4 \left (1+2 \gamma +2 \ln \left (x \right )+2 \ln \left (b \right )\right )}{\sqrt {\pi }\, x^{2} b^{2}}-\frac {2 \left (2 \gamma -4+2 \ln \left (x \right )+2 \ln \left (b \right )\right )}{\sqrt {\pi }}+\frac {-8 b^{2} x^{2}+4}{\sqrt {\pi }\, b^{2} x^{2}}+\frac {4 \left (3 b^{2} x^{2}+6\right ) \gamma }{3 \sqrt {\pi }\, b^{2} x^{2}}+\frac {4 \left (3 b^{2} x^{2}+6\right ) \ln \left (2\right )}{3 \sqrt {\pi }\, b^{2} x^{2}}+\frac {4 \left (3 b^{2} x^{2}+6\right ) \ln \left (\frac {b x}{2}\right )}{3 \sqrt {\pi }\, b^{2} x^{2}}-\frac {4 \cos \left (b x \right )}{\sqrt {\pi }\, b^{2} x^{2}}+\frac {4 \sin \left (b x \right )}{\sqrt {\pi }\, b x}-\frac {4 \left (3 b^{2} x^{2}+6\right ) \operatorname {Ci}\left (b x \right )}{3 \sqrt {\pi }\, b^{2} x^{2}}\right )}{16}\) | \(199\) |
Input:
int(Ci(b*x)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*Ci(b*x)/x^2+1/2*b^2*(-1/2*cos(b*x)/b^2/x^2+1/2*sin(b*x)/b/x-1/2*Ci(b* x))
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^3} \, dx=-\frac {\pi \sqrt {b^{2}} b x^{2} \operatorname {S}\left (\sqrt {b^{2}} x\right ) + b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \operatorname {C}\left (b x\right )}{2 \, x^{2}} \] Input:
integrate(fresnel_cos(b*x)/x^3,x, algorithm="fricas")
Output:
-1/2*(pi*sqrt(b^2)*b*x^2*fresnel_sin(sqrt(b^2)*x) + b*x*cos(1/2*pi*b^2*x^2 ) + fresnel_cos(b*x))/x^2
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (39) = 78\).
Time = 1.58 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.89 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^3} \, dx=\frac {b^{2} \log {\left (b x \right )}}{4} - \frac {b^{2} \log {\left (b^{2} x^{2} \right )}}{8} - \frac {b^{2} \operatorname {Ci}{\left (b x \right )}}{4} + \frac {b \sin {\left (b x \right )}}{4 x} + \frac {\log {\left (b x \right )}}{2 x^{2}} - \frac {\log {\left (b^{2} x^{2} \right )}}{4 x^{2}} - \frac {\cos {\left (b x \right )}}{4 x^{2}} - \frac {\operatorname {Ci}{\left (b x \right )}}{2 x^{2}} \] Input:
integrate(Ci(b*x)/x**3,x)
Output:
b**2*log(b*x)/4 - b**2*log(b**2*x**2)/8 - b**2*Ci(b*x)/4 + b*sin(b*x)/(4*x ) + log(b*x)/(2*x**2) - log(b**2*x**2)/(4*x**2) - cos(b*x)/(4*x**2) - Ci(b *x)/(2*x**2)
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.33 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^3} \, dx=-\frac {\sqrt {\frac {1}{2}} \sqrt {\pi x^{2}} {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{2}}{16 \, x} - \frac {\operatorname {C}\left (b x\right )}{2 \, x^{2}} \] Input:
integrate(fresnel_cos(b*x)/x^3,x, algorithm="maxima")
Output:
-1/16*sqrt(1/2)*sqrt(pi*x^2)*((I + 1)*sqrt(2)*gamma(-1/2, 1/2*I*pi*b^2*x^2 ) - (I - 1)*sqrt(2)*gamma(-1/2, -1/2*I*pi*b^2*x^2))*b^2/x - 1/2*fresnel_co s(b*x)/x^2
\[ \int \frac {\operatorname {CosIntegral}(b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right )}{x^{3}} \,d x } \] Input:
integrate(fresnel_cos(b*x)/x^3,x, algorithm="giac")
Output:
integrate(fresnel_cos(b*x)/x^3, x)
Timed out. \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^3} \, dx=-\frac {\frac {\cos \left (b\,x\right )}{2}-\frac {b\,x\,\sin \left (b\,x\right )}{2}}{2\,x^2}-\frac {b^2\,\mathrm {cosint}\left (b\,x\right )}{4}-\frac {\mathrm {cosint}\left (b\,x\right )}{2\,x^2} \] Input:
int(cosint(b*x)/x^3,x)
Output:
- (cos(b*x)/2 - (b*x*sin(b*x))/2)/(2*x^2) - (b^2*cosint(b*x))/4 - cosint(b *x)/(2*x^2)
Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^3} \, dx=\frac {-\mathit {ci} \left (b x \right ) b^{2} x^{2}-2 \mathit {ci} \left (b x \right )-\cos \left (b x \right )+\sin \left (b x \right ) b x}{4 x^{2}} \] Input:
int(Ci(b*x)/x^3,x)
Output:
( - ci(b*x)*b**2*x**2 - 2*ci(b*x) - cos(b*x) + sin(b*x)*b*x)/(4*x**2)