\(\int \text {Si}(d (a+b \log (c x^n))) \, dx\) [34]

Optimal result

Integrand size = 13, antiderivative size = 128 \[ \int \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {1}{2} i e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {1}{2} i e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+x \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80 \[ \int \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {1}{2} i e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \left (\operatorname {ExpIntegralEi}\left (\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (\frac {(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+x \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {7077, 27, 4998, 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.

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

\(\Big \downarrow \) 7077

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 4998

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

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

Input:

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

Output:

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

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 4998

Int[(((e_.) + Log[(g_.)*(x_)^(m_.)]*(f_.))*(h_.))^(q_.)*Sin[((a_.) + Log[(c 
_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[(I*(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 
] - Simp[I*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 7077

Int[SinIntegral[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> 
Simp[x*SinIntegral[d*(a + b*Log[c*x^n])], x] - Simp[b*d*n   Int[Sin[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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.99 \[ \int \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} \, {\left (i \, {\rm Ei}\left (\frac {i \, a b d n + {\left (i \, b^{2} d n + b\right )} \log \left (c\right ) + {\left (i \, b^{2} d n^{2} + b n\right )} \log \left (x\right ) + a}{b n}\right ) - i \, {\rm Ei}\left (\frac {-i \, a b d n + {\left (-i \, b^{2} d n + b\right )} \log \left (c\right ) + {\left (-i \, b^{2} d n^{2} + b n\right )} \log \left (x\right ) + a}{b n}\right )\right )} e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )} + x \operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right ) \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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