\(\int \frac {\operatorname {CosIntegral}(d (a+b \log (c x^n)))}{x^3} \, dx\) [105]

Optimal result

Integrand size = 17, antiderivative size = 135 \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2} \] Output:

-1/2*Ci(d*(a+b*ln(c*x^n)))/x^2+1/4*exp(2*a/b/n)*(c*x^n)^(2/n)*Ei(-(2-I*b*d 
*n)*(a+b*ln(c*x^n))/b/n)/x^2+1/4*exp(2*a/b/n)*(c*x^n)^(2/n)*Ei(-(2+I*b*d*n 
)*(a+b*ln(c*x^n))/b/n)/x^2
 

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.78 \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {-2 \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \left (\operatorname {ExpIntegralEi}\left (-\frac {i (-2 i+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (\frac {i (2 i+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )}{4 x^2} \] Input:

Integrate[CosIntegral[d*(a + b*Log[c*x^n])]/x^3,x]
 

Output:

(-2*CosIntegral[d*(a + b*Log[c*x^n])] + E^((2*a)/(b*n))*(c*x^n)^(2/n)*(Exp 
IntegralEi[((-I)*(-2*I + b*d*n)*(a + b*Log[c*x^n]))/(b*n)] + ExpIntegralEi 
[(I*(2*I + b*d*n)*(a + b*Log[c*x^n]))/(b*n)]))/(4*x^2)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {7081, 27, 5001, 2747, 2609}

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.

\(\displaystyle \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\)

\(\Big \downarrow \) 7081

\(\displaystyle \frac {1}{2} b d n \int \frac {\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d x^3 \left (a+b \log \left (c x^n\right )\right )}dx-\frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b n \int \frac {\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3 \left (a+b \log \left (c x^n\right )\right )}dx-\frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

\(\Big \downarrow \) 5001

\(\displaystyle -\frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} b n \left (\frac {1}{2} e^{-i a d} x^{i b d n} \left (c x^n\right )^{-i b d} \int \frac {x^{-i b d n-3}}{a+b \log \left (c x^n\right )}dx+\frac {1}{2} e^{i a d} x^{-i b d n} \left (c x^n\right )^{i b d} \int \frac {x^{i b d n-3}}{a+b \log \left (c x^n\right )}dx\right )\)

\(\Big \downarrow \) 2747

\(\displaystyle -\frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} b n \left (\frac {e^{i a d} \left (c x^n\right )^{2/n} \int \frac {\left (c x^n\right )^{-\frac {2-i b d n}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n x^2}+\frac {e^{-i a d} \left (c x^n\right )^{2/n} \int \frac {\left (c x^n\right )^{-\frac {i b d n+2}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n x^2}\right )\)

\(\Big \downarrow \) 2609

\(\displaystyle -\frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} b n \left (\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(i b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n x^2}\right )\)

Input:

Int[CosIntegral[d*(a + b*Log[c*x^n])]/x^3,x]
 

Output:

-1/2*CosIntegral[d*(a + b*Log[c*x^n])]/x^2 + (b*n*((E^((2*a)/(b*n))*(c*x^n 
)^(2/n)*ExpIntegralEi[-(((2 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(2*b*n 
*x^2) + (E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 + I*b*d*n)*(a + 
 b*Log[c*x^n]))/(b*n))])/(2*b*n*x^2)))/2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2609

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si 
mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F 
reeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2747

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n))   Subst[Int[E^(((m + 1)/n 
)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
 

rule 5001

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*(((e_.) + Log[(g_.)*(x 
_)^(m_.)]*(f_.))*(h_.))^(q_.)*((i_.)*(x_))^(r_.), x_Symbol] :> Simp[((i*x)^ 
r*(1/((c*x^n)^(I*b*d)*(2*x^(r - I*b*d*n)))))/E^(I*a*d)   Int[x^(r - I*b*d*n 
)*(h*(e + f*Log[g*x^m]))^q, x], x] + Simp[E^(I*a*d)*(i*x)^r*((c*x^n)^(I*b*d 
)/(2*x^(r + I*b*d*n)))   Int[x^(r + I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], 
x] /; FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]
 

rule 7081

Int[CosIntegral[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^( 
m_.), x_Symbol] :> Simp[(e*x)^(m + 1)*(CosIntegral[d*(a + b*Log[c*x^n])]/(e 
*(m + 1))), x] - Simp[b*d*(n/(m + 1))   Int[(e*x)^m*(Cos[d*(a + b*Log[c*x^n 
])]/(d*(a + b*Log[c*x^n]))), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && N 
eQ[m, -1]
 

Maple [F]

\[\int \frac {\operatorname {Ci}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}d x\]

Input:

int(Ci(d*(a+b*ln(c*x^n)))/x^3,x)
 

Output:

int(Ci(d*(a+b*ln(c*x^n)))/x^3,x)
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (121) = 242\).

Time = 0.10 (sec) , antiderivative size = 460, normalized size of antiderivative = 3.41 \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {\pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + i \, \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - i \, \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - 2 \, \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right )}{4 \, x^{2}} \] Input:

integrate(fresnel_cos(d*(a+b*log(c*x^n)))/x^3,x, algorithm="fricas")
 

Output:

1/4*(pi*sqrt(b^2*d^2*n^2)*x^2*e^(2*log(c)/n + 2*a/(b*n) + 2*I/(pi*b^2*d^2* 
n^2))*fresnel_cos((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^ 
2*n + 2*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) + pi*sqrt(b^2*d^2*n^2)*x^2* 
e^(2*log(c)/n + 2*a/(b*n) - 2*I/(pi*b^2*d^2*n^2))*fresnel_cos((pi*b^2*d^2* 
n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n - 2*I)*sqrt(b^2*d^2*n^2)/( 
pi*b^2*d^2*n^2)) + I*pi*sqrt(b^2*d^2*n^2)*x^2*e^(2*log(c)/n + 2*a/(b*n) + 
2*I/(pi*b^2*d^2*n^2))*fresnel_sin((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*lo 
g(c) + pi*a*b*d^2*n + 2*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - I*pi*sqrt 
(b^2*d^2*n^2)*x^2*e^(2*log(c)/n + 2*a/(b*n) - 2*I/(pi*b^2*d^2*n^2))*fresne 
l_sin((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n - 2*I)*s 
qrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - 2*fresnel_cos(b*d*log(c*x^n) + a*d))/ 
x^2
 

Sympy [F]

\[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\operatorname {Ci}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \] Input:

integrate(Ci(d*(a+b*ln(c*x**n)))/x**3,x)
 

Output:

Integral(Ci(a*d + b*d*log(c*x**n))/x**3, x)
 

Maxima [F]

\[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \] Input:

integrate(fresnel_cos(d*(a+b*log(c*x^n)))/x^3,x, algorithm="maxima")
 

Output:

integrate(fresnel_cos((b*log(c*x^n) + a)*d)/x^3, x)
 

Giac [F]

\[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \] Input:

integrate(fresnel_cos(d*(a+b*log(c*x^n)))/x^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate(fresnel_cos((b*log(c*x^n) + a)*d)/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathrm {cosint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \] Input:

int(cosint(d*(a + b*log(c*x^n)))/x^3,x)
 

Output:

int(cosint(d*(a + b*log(c*x^n)))/x^3, x)
 

Reduce [F]

\[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathit {ci} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}{x^{3}}d x \] Input:

int(Ci(d*(a+b*log(c*x^n)))/x^3,x)
 

Output:

int(ci(log(x**n*c)*b*d + a*d)/x**3,x)