\(\int \text {Chi}(d (a+b \log (c x^n))) \, dx\) [102]

Optimal result

Integrand size = 13, antiderivative size = 119 \[ \int \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6688, 12, 5649, 2347, 2209} \[ \int \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{2} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]

[In]

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

[Out]

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

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 5649

Int[Cosh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*(((e_.) + Log[(g_.)*(x_)^(m_.)]*(f_.))*(h_.))^(q_.), x_S
ymbol] :> Dist[1/((c*x^n)^(b*d)*(2/x^(b*d*n)))/E^(a*d), Int[(h*(e + f*Log[g*x^m]))^q/x^(b*d*n), x], x] + Dist[
E^(a*d)*((c*x^n)^(b*d)/(2*x^(b*d*n))), Int[x^(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 6688

Int[CoshIntegral[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[x*CoshIntegral[d*(a + b*Log[c
*x^n])], x] - Dist[b*d*n, Int[Cosh[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x], x] /; FreeQ[{a, b, c, d,
n}, x]

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

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.78 \[ \int \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \left (\operatorname {ExpIntegralEi}\left (-\frac {(-1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(cosh_integral(b*d*log(c*x^n) + a*d), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(coshint(d*(a + b*log(c*x^n))),x)

[Out]

int(coshint(d*(a + b*log(c*x^n))), x)