\(\int \frac {\text {Si}(d (a+b \log (c x^n)))}{x^3} \, dx\) [37]

Optimal result

Integrand size = 17, antiderivative size = 139 \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {i e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}+\frac {i e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.80 \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {i \left (e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \left (\operatorname {ExpIntegralEi}\left (-\frac {i (-2 i+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (\frac {i (2 i+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+2 i \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{4 x^2} \] Input:

Integrate[SinIntegral[d*(a + b*Log[c*x^n])]/x^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, 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.

\(\displaystyle \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\)

\(\Big \downarrow \) 7080

\(\displaystyle \frac {1}{2} b d n \int \frac {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d x^3 \left (a+b \log \left (c x^n\right )\right )}dx-\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b n \int \frac {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3 \left (a+b \log \left (c x^n\right )\right )}dx-\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

\(\Big \downarrow \) 5000

\(\displaystyle -\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} b n \left (\frac {1}{2} i e^{-i a d} x^{i b d n} \left (c x^n\right )^{-i b d} \int \frac {x^{-i b d n-3}}{a+b \log \left (c x^n\right )}dx-\frac {1}{2} i e^{i a d} x^{-i b d n} \left (c x^n\right )^{i b d} \int \frac {x^{i b d n-3}}{a+b \log \left (c x^n\right )}dx\right )\)

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

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

Input:

Int[SinIntegral[d*(a + b*Log[c*x^n])]/x^3,x]
 

Output:

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

\[\int \frac {\operatorname {Si}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}d x\]

Input:

int(Si(d*(a+b*ln(c*x^n)))/x^3,x)
 

Output:

int(Si(d*(a+b*ln(c*x^n)))/x^3,x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.06 \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {{\left (-i \, x^{2} {\rm Ei}\left (\frac {i \, a b d n + {\left (i \, b^{2} d n - 2 \, b\right )} \log \left (c\right ) + {\left (i \, b^{2} d n^{2} - 2 \, b n\right )} \log \left (x\right ) - 2 \, a}{b n}\right ) + i \, x^{2} {\rm Ei}\left (\frac {-i \, a b d n + {\left (-i \, b^{2} d n - 2 \, b\right )} \log \left (c\right ) + {\left (-i \, b^{2} d n^{2} - 2 \, b n\right )} \log \left (x\right ) - 2 \, a}{b n}\right )\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 2 \, \operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right )}{4 \, x^{2}} \] Input:

integrate(sin_integral(d*(a+b*log(c*x^n)))/x^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\operatorname {Si}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \] Input:

integrate(Si(d*(a+b*ln(c*x**n)))/x**3,x)
 

Output:

Integral(Si(a*d + b*d*log(c*x**n))/x**3, x)
 

Maxima [F]

\[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\operatorname {Si}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \] Input:

integrate(sin_integral(d*(a+b*log(c*x^n)))/x^3,x, algorithm="maxima")
 

Output:

integrate(sin_integral((b*log(c*x^n) + a)*d)/x^3, x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\text {Timed out} \] Input:

integrate(sin_integral(d*(a+b*log(c*x^n)))/x^3,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathrm {sinint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \] Input:

int(sinint(d*(a + b*log(c*x^n)))/x^3,x)
 

Output:

int(sinint(d*(a + b*log(c*x^n)))/x^3, x)
 

Reduce [F]

\[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathit {si} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}{x^{3}}d x \] Input:

int(Si(d*(a+b*log(c*x^n)))/x^3,x)
 

Output:

int(si(log(x**n*c)*b*d + a*d)/x**3,x)