\(\int x \operatorname {CosIntegral}(d (a+b \log (c x^n))) \, dx\) [101]

Optimal result

Integrand size = 15, antiderivative size = 133 \[ \int x \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} x^2 \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{4} e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{4} e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(2+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.77 \[ \int x \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{4} x^2 \left (2 \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \left (\operatorname {ExpIntegralEi}\left (\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (\frac {(2+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, 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.

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

\(\Big \downarrow \) 7081

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5001

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

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

Input:

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

Output:

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

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

Input:

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

Output:

int(x*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. 448 vs. \(2 (125) = 250\).

Time = 0.11 (sec) , antiderivative size = 448, normalized size of antiderivative = 3.37 \[ \int x \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {1}{4} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {2 \, \log \left (c\right )}{n} - \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{4} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {2 \, \log \left (c\right )}{n} - \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \frac {1}{4} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {2 \, \log \left (c\right )}{n} - \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{4} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {2 \, \log \left (c\right )}{n} - \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \frac {1}{2} \, x^{2} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right ) \] Input:

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

Output:

-1/4*pi*sqrt(b^2*d^2*n^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)) - 1/4*pi*sqrt(b^2*d^2*n^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)) + 1/4*I*pi*sqrt(b^2*d^2*n^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*log 
(c) + pi*a*b*d^2*n + 2*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - 1/4*I*pi*s 
qrt(b^2*d^2*n^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)) + 1/2*x^2*fresnel_cos(b*d*log(c*x^n) + 
a*d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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