3.83 \(\int \frac{x}{(a+b \log (c x^n))^3} \, dx\)

Optimal. Leaf size=101 \[ \frac{2 x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3}-\frac{x^2}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

[Out]

(2*x^2*ExpIntegralEi[(2*(a + b*Log[c*x^n]))/(b*n)])/(b^3*E^((2*a)/(b*n))*n^3*(c*x^n)^(2/n)) - x^2/(2*b*n*(a +
b*Log[c*x^n])^2) - x^2/(b^2*n^2*(a + b*Log[c*x^n]))

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Rubi [A]  time = 0.0798244, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2306, 2310, 2178} \[ \frac{2 x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3}-\frac{x^2}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Log[c*x^n])^3,x]

[Out]

(2*x^2*ExpIntegralEi[(2*(a + b*Log[c*x^n]))/(b*n)])/(b^3*E^((2*a)/(b*n))*n^3*(c*x^n)^(2/n)) - x^2/(2*b*n*(a +
b*Log[c*x^n])^2) - x^2/(b^2*n^2*(a + b*Log[c*x^n]))

Rule 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

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

Mathematica [A]  time = 0.129612, size = 89, normalized size = 0.88 \[ \frac{x^2 \left (4 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{b n \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{\left (a+b \log \left (c x^n\right )\right )^2}\right )}{2 b^3 n^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Log[c*x^n])^3,x]

[Out]

(x^2*((4*ExpIntegralEi[(2*(a + b*Log[c*x^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*x^n)^(2/n)) - (b*n*(2*a + b*n + 2*b*
Log[c*x^n]))/(a + b*Log[c*x^n])^2))/(2*b^3*n^3)

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Maple [F]  time = 0.652, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, b x^{2} \log \left (x^{n}\right ) +{\left (b{\left (n + 2 \, \log \left (c\right )\right )} + 2 \, a\right )} x^{2}}{2 \,{\left (b^{4} n^{2} \log \left (c\right )^{2} + b^{4} n^{2} \log \left (x^{n}\right )^{2} + 2 \, a b^{3} n^{2} \log \left (c\right ) + a^{2} b^{2} n^{2} + 2 \,{\left (b^{4} n^{2} \log \left (c\right ) + a b^{3} n^{2}\right )} \log \left (x^{n}\right )\right )}} + 2 \, \int \frac{x}{b^{3} n^{2} \log \left (c\right ) + b^{3} n^{2} \log \left (x^{n}\right ) + a b^{2} n^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*x^2*log(x^n) + (b*(n + 2*log(c)) + 2*a)*x^2)/(b^4*n^2*log(c)^2 + b^4*n^2*log(x^n)^2 + 2*a*b^3*n^2*lo
g(c) + a^2*b^2*n^2 + 2*(b^4*n^2*log(c) + a*b^3*n^2)*log(x^n)) + 2*integrate(x/(b^3*n^2*log(c) + b^3*n^2*log(x^
n) + a*b^2*n^2), x)

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Fricas [B]  time = 0.782581, size = 518, normalized size = 5.13 \begin{align*} -\frac{{\left ({\left (2 \, b^{2} n^{2} x^{2} \log \left (x\right ) + 2 \, b^{2} n x^{2} \log \left (c\right ) +{\left (b^{2} n^{2} + 2 \, a b n\right )} x^{2}\right )} e^{\left (\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 4 \,{\left (b^{2} n^{2} \log \left (x\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \,{\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )\right )} \logintegral \left (x^{2} e^{\left (\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{2 \,{\left (b^{5} n^{5} \log \left (x\right )^{2} + b^{5} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} \log \left (c\right ) + a^{2} b^{3} n^{3} + 2 \,{\left (b^{5} n^{4} \log \left (c\right ) + a b^{4} n^{4}\right )} \log \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((2*b^2*n^2*x^2*log(x) + 2*b^2*n*x^2*log(c) + (b^2*n^2 + 2*a*b*n)*x^2)*e^(2*(b*log(c) + a)/(b*n)) - 4*(b^
2*n^2*log(x)^2 + b^2*log(c)^2 + 2*a*b*log(c) + a^2 + 2*(b^2*n*log(c) + a*b*n)*log(x))*log_integral(x^2*e^(2*(b
*log(c) + a)/(b*n))))*e^(-2*(b*log(c) + a)/(b*n))/(b^5*n^5*log(x)^2 + b^5*n^3*log(c)^2 + 2*a*b^4*n^3*log(c) +
a^2*b^3*n^3 + 2*(b^5*n^4*log(c) + a*b^4*n^4)*log(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (a + b \log{\left (c x^{n} \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(a + b*log(c*x**n))**3, x)

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Giac [B]  time = 1.80405, size = 1389, normalized size = 13.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^2*n^2*x^2*log(x)/(b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*
b^4*n^3*log(c) + a^2*b^3*n^3) + 2*b^2*n^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x))*e^(-2*a/(b*n))*log(x)^2/((b^5*
n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*
n^3)*c^(2/n)) - 1/2*b^2*n^2*x^2/(b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*l
og(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3) - b^2*n*x^2*log(c)/(b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5
*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3) + 4*b^2*n*Ei(2*log(c)/n + 2*a/(b*n) + 2
*log(x))*e^(-2*a/(b*n))*log(c)*log(x)/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^
4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(2/n)) - a*b*n*x^2/(b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log
(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3) + 2*b^2*Ei(2*log(c)/n + 2*a/(b
*n) + 2*log(x))*e^(-2*a/(b*n))*log(c)^2/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*
b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(2/n)) + 4*a*b*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x))*e^
(-2*a/(b*n))*log(x)/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a
*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(2/n)) + 4*a*b*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x))*e^(-2*a/(b*n))*log(c)/((
b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*
b^3*n^3)*c^(2/n)) + 2*a^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x))*e^(-2*a/(b*n))/((b^5*n^5*log(x)^2 + 2*b^5*n^4*
log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(2/n))