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

Optimal result

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

[Out]

-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)))

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6661, 12, 4585, 2347, 2209} \[ \int x \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {1}{4} i x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {1}{4} i x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(i b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {1}{2} x^2 \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]

[In]

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

[Out]

((-1/4*I)*x^2*ExpIntegralEi[((2 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*x^n)^(2/n)) + ((I/4
)*x^2*ExpIntegralEi[((2 + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*x^n)^(2/n)) + (x^2*SinInteg
ral[d*(a + b*Log[c*x^n])])/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2209

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

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(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 4585

Int[(((e_.) + Log[(g_.)*(x_)^(m_.)]*(f_.))*(h_.))^(q_.)*((i_.)*(x_))^(r_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*
(b_.))*(d_.)], x_Symbol] :> Dist[(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] - Dist[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 6661

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] - Dist[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] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \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 \\ & = \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 \\ & = \frac {1}{2} x^2 \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{4} \left (i b e^{-i a d} n x^{i b d n} \left (c x^n\right )^{-i b d}\right ) \int \frac {x^{1-i b d n}}{a+b \log \left (c x^n\right )} \, dx+\frac {1}{4} \left (i b e^{i a d} n x^{-i b d n} \left (c x^n\right )^{i b d}\right ) \int \frac {x^{1+i b d n}}{a+b \log \left (c x^n\right )} \, dx \\ & = \frac {1}{2} x^2 \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{4} \left (i b e^{-i a d} x^2 \left (c x^n\right )^{-i b d-\frac {2-i b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(2-i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )+\frac {1}{4} \left (i b e^{i a d} x^2 \left (c x^n\right )^{i b d-\frac {2+i b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(2+i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\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}{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 ) \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.02 \[ \int x \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \frac {1}{4} \, {\left (i \, {\rm Ei}\left (\frac {i \, a b d n + {\left (i \, b^{2} d n + 2 \, b\right )} \log \left (c\right ) + {\left (i \, b^{2} d n^{2} + 2 \, b n\right )} \log \left (x\right ) + 2 \, a}{b n}\right ) - i \, {\rm Ei}\left (\frac {-i \, a b d n + {\left (-i \, b^{2} d n + 2 \, b\right )} \log \left (c\right ) + {\left (-i \, b^{2} d n^{2} + 2 \, b n\right )} \log \left (x\right ) + 2 \, a}{b n}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \]

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

int(x*sinint(d*(a + b*log(c*x^n))),x)

[Out]

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