3.102 \(\int \text{CosIntegral}(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=124 \[ x \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{ExpIntegralEi}\left (\frac{(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{1}{2} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{ExpIntegralEi}\left (\frac{(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]

[Out]

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

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Rubi [A]  time = 0.228058, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {6524, 12, 4496, 2310, 2178} \[ x \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{1}{2} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{(i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6524

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

Rule 12

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

Rule 4496

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

Rubi steps

\begin{align*} \int \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=x \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-(b d n) \int \frac{\cos \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{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-(b n) \int \frac{\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )} \, dx\\ &=x \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} \left (b e^{-i a d} n x^{i b d n} \left (c x^n\right )^{-i b d}\right ) \int \frac{x^{-i b d n}}{a+b \log \left (c x^n\right )} \, dx-\frac{1}{2} \left (b e^{i a d} n x^{-i b d n} \left (c x^n\right )^{i b d}\right ) \int \frac{x^{i b d n}}{a+b \log \left (c x^n\right )} \, dx\\ &=x \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} \left (b e^{-i a d} x \left (c x^n\right )^{-i b d-\frac{1-i b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(1-i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )-\frac{1}{2} \left (b e^{i a d} x \left (c x^n\right )^{i b d-\frac{1+i b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(1+i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )\\ &=x \text{Ci}\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} \text{Ei}\left (\frac{(1-i 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} \text{Ei}\left (\frac{(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\\ \end{align*}

Mathematica [A]  time = 1.65783, size = 98, normalized size = 0.79 \[ x \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \left (\text{ExpIntegralEi}\left (\frac{(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\text{ExpIntegralEi}\left (\frac{(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.073, size = 0, normalized size = 0. \begin{align*} \int{\it Ci} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Ci}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{Ci}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{Ci}{\left (d \left (a + b \log{\left (c x^{n} \right )}\right ) \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out