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

Optimal. Leaf size=139 \[ -\frac {i e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \text {Ei}\left (-\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}+\frac {i e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \text {Ei}\left (-\frac {(i b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]

[Out]

-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

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Rubi [A]  time = 0.25, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6526, 12, 4497, 2310, 2178} \[ -\frac {i e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \text {Ei}\left (-\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}+\frac {i e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \text {Ei}\left (-\frac {(i b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/4)*E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/x^2 + ((I/4)
*E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/x^2 - SinIntegral[d
*(a + b*Log[c*x^n])]/(2*x^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 2178

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

Rule 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 4497

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)/(E^(I*a*d)*(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] - 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 6526

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*} \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=-\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\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}+\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}+\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^{-3-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^{-3+i b d n}}{a+b \log \left (c x^n\right )} \, dx\\ &=-\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {\left (i b e^{-i a d} \left (c x^n\right )^{-i b d-\frac {-2-i b d n}{n}}\right ) \operatorname {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 )}{4 x^2}-\frac {\left (i b e^{i a d} \left (c x^n\right )^{i b d-\frac {-2+i b d n}{n}}\right ) \operatorname {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 )}{4 x^2}\\ &=-\frac {i e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \text {Ei}\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} \text {Ei}\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}\\ \end {align*}

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Mathematica [A]  time = 1.93, size = 111, normalized size = 0.80 \[ \frac {i \left (e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \left (\text {Ei}\left (-\frac {i (b d n-2 i) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\text {Ei}\left (\frac {i (b d n+2 i) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+2 i \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((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)] - ExpInte
gralEi[(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

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\Si \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Si}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {sinint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Si}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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