\(\int x \text {Si}(d (a+b \log (c x^n))) \, dx\) [33]

Optimal result

Integrand size = 15, antiderivative size = 137 \[ \int x \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {1}{4} i e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {1}{4} i e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(2+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {1}{2} x^2 \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.77 \[ \int x \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{4} x^2 \left (-i e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \left (\operatorname {ExpIntegralEi}\left (\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (\frac {(2+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+2 \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 156, 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 = {7080, 27, 5000, 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.

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

\(\Big \downarrow \) 7080

\(\displaystyle \frac {1}{2} x^2 \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} b d n \int \frac {x \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d \left (a+b \log \left (c x^n\right )\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} x^2 \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} b n \int \frac {x \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )}dx\)

\(\Big \downarrow \) 5000

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

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

Input:

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

Output:

-1/2*(b*n*(((I/2)*x^2*ExpIntegralEi[((2 - I*b*d*n)*(a + b*Log[c*x^n]))/(b* 
n)])/(b*E^((2*a)/(b*n))*n*(c*x^n)^(2/n)) - ((I/2)*x^2*ExpIntegralEi[((2 + 
I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(b*E^((2*a)/(b*n))*n*(c*x^n)^(2/n)))) 
 + (x^2*SinIntegral[d*(a + b*Log[c*x^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 5000

Int[(((e_.) + Log[(g_.)*(x_)^(m_.)]*(f_.))*(h_.))^(q_.)*((i_.)*(x_))^(r_.)* 
Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[(I*(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[I*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 7080

Int[((e_.)*(x_))^(m_.)*SinIntegral[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d 
_.)], x_Symbol] :> Simp[(e*x)^(m + 1)*(SinIntegral[d*(a + b*Log[c*x^n])]/(e 
*(m + 1))), x] - Simp[b*d*(n/(m + 1))   Int[(e*x)^m*(Sin[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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.02 \[ \int x \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \frac {1}{4} \, {\left (i \, {\rm Ei}\left (\frac {i \, a b d n + {\left (i \, b^{2} d n + 2 \, b\right )} \log \left (c\right ) + {\left (i \, b^{2} d n^{2} + 2 \, b n\right )} \log \left (x\right ) + 2 \, a}{b n}\right ) - i \, {\rm Ei}\left (\frac {-i \, a b d n + {\left (-i \, b^{2} d n + 2 \, b\right )} \log \left (c\right ) + {\left (-i \, b^{2} d n^{2} + 2 \, b n\right )} \log \left (x\right ) + 2 \, a}{b n}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \] Input:

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

Output:

1/2*x^2*sin_integral(b*d*log(c*x^n) + a*d) + 1/4*(I*Ei((I*a*b*d*n + (I*b^2 
*d*n + 2*b)*log(c) + (I*b^2*d*n^2 + 2*b*n)*log(x) + 2*a)/(b*n)) - I*Ei((-I 
*a*b*d*n + (-I*b^2*d*n + 2*b)*log(c) + (-I*b^2*d*n^2 + 2*b*n)*log(x) + 2*a 
)/(b*n)))*e^(-2*(b*log(c) + a)/(b*n))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

Timed out. \[ \int x \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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