\(\int \operatorname {CosIntegral}(d (a+b \log (c x^n))) \, dx\) [102]

Optimal result

Integrand size = 13, antiderivative size = 124 \[ \int \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.79 \[ \int \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \left (\operatorname {ExpIntegralEi}\left (\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (\frac {(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {7078, 27, 4999, 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.

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

\(\Big \downarrow \) 7078

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 4999

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

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

Input:

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

Output:

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

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 4999

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

rule 7078

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

Maple [F]

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

Input:

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

Output:

int(Ci(d*(a+b*ln(c*x^n))),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. 445 vs. \(2 (114) = 228\).

Time = 0.12 (sec) , antiderivative size = 445, normalized size of antiderivative = 3.59 \[ \int \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {1}{2} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{2} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \frac {1}{2} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{2} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + x \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right ) \] Input:

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

Output:

-1/2*pi*sqrt(b^2*d^2*n^2)*e^(-log(c)/n - a/(b*n) - 1/2*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 + 
 I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - 1/2*pi*sqrt(b^2*d^2*n^2)*e^(-log 
(c)/n - a/(b*n) + 1/2*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 - I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2 
*n^2)) + 1/2*I*pi*sqrt(b^2*d^2*n^2)*e^(-log(c)/n - a/(b*n) - 1/2*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*log(c) + pi*a 
*b*d^2*n + I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - 1/2*I*pi*sqrt(b^2*d^2* 
n^2)*e^(-log(c)/n - a/(b*n) + 1/2*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*log(c) + pi*a*b*d^2*n - I)*sqrt(b^2*d^2*n^2) 
/(pi*b^2*d^2*n^2)) + x*fresnel_cos(b*d*log(c*x^n) + a*d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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