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

Optimal. Leaf size=172 \[ \frac{(e x)^{m+1} \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac{x (e x)^m e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{ExpIntegralEi}\left (\frac{(-i b d n+m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}-\frac{x (e x)^m e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{ExpIntegralEi}\left (\frac{(i b d n+m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.294702, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6527, 12, 4498, 2310, 2178} \[ \frac{(e x)^{m+1} \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac{x (e x)^m e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m-i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}-\frac{x (e x)^m e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6527

Int[CosIntegral[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[((e*x)^(m +
1)*CosIntegral[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(b*d*n)/(m + 1), Int[((e*x)^m*Cos[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]

Rule 12

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

Rule 4498

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*(((e_.) + Log[(g_.)*(x_)^(m_.)]*(f_.))*(h_.))^(q_.)*((i_.
)*(x_))^(r_.), x_Symbol] :> Dist[(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[(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 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 (e x)^m \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac{(e x)^{1+m} \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac{(b d n) \int \frac{(e x)^m \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}{1+m}\\ &=\frac{(e x)^{1+m} \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac{(b n) \int \frac{(e x)^m \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )} \, dx}{1+m}\\ &=\frac{(e x)^{1+m} \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac{\left (b e^{-i a d} n x^{-m+i b d n} (e x)^m \left (c x^n\right )^{-i b d}\right ) \int \frac{x^{m-i b d n}}{a+b \log \left (c x^n\right )} \, dx}{2 (1+m)}-\frac{\left (b e^{i a d} n x^{-m-i b d n} (e x)^m \left (c x^n\right )^{i b d}\right ) \int \frac{x^{m+i b d n}}{a+b \log \left (c x^n\right )} \, dx}{2 (1+m)}\\ &=\frac{(e x)^{1+m} \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac{\left (b e^{-i a d} x (e x)^m \left (c x^n\right )^{-i b d-\frac{1+m-i b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(1+m-i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}-\frac{\left (b e^{i a d} x (e x)^m \left (c x^n\right )^{i b d-\frac{1+m+i b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(1+m+i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}\\ &=\frac{(e x)^{1+m} \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac{e^{-\frac{a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac{1+m}{n}} \text{Ei}\left (\frac{(1+m-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)}-\frac{e^{-\frac{a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac{1+m}{n}} \text{Ei}\left (\frac{(1+m+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)}\\ \end{align*}

Mathematica [A]  time = 3.04162, size = 124, normalized size = 0.72 \[ \frac{(e x)^m \left (2 x \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x^{-m} \exp \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \left (\text{ExpIntegralEi}\left (\frac{(-i b d n+m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\text{ExpIntegralEi}\left (\frac{(i b d n+m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )\right )}{2 (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}{\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((e*x)^m*Ci(d*(a+b*ln(c*x^n))),x)

[Out]

int((e*x)^m*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 \left (e x\right )^{m}{\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((e*x)^m*Ci(d*(a+b*log(c*x^n))),x, algorithm="maxima")

[Out]

integrate((e*x)^m*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 (\left (e x\right )^{m} \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((e*x)^m*Ci(d*(a+b*log(c*x^n))),x, algorithm="fricas")

[Out]

integral((e*x)^m*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 \left (e x\right )^{m} \operatorname{Ci}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out