3.75 \(\int \frac {x}{(a+b \log (c x^n))^2} \, dx\)

Optimal. Leaf size=76 \[ \frac {2 x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac {x^2}{b n \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2306, 2310, 2178} \[ \frac {2 x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac {x^2}{b n \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Log[c*x^n])^2,x]

[Out]

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

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 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]

Rubi steps

\begin {align*} \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac {x^2}{b n \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 n}\\ &=-\frac {x^2}{b n \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 n^2}\\ &=\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^2 n^2}-\frac {x^2}{b n \left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 70, normalized size = 0.92 \[ \frac {x^2 \left (2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {b n}{a+b \log \left (c x^n\right )}\right )}{b^2 n^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Log[c*x^n])^2,x]

[Out]

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

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fricas [A]  time = 0.42, size = 101, normalized size = 1.33 \[ -\frac {{\left (b n x^{2} e^{\left (\frac {2 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )} - 2 \, {\left (b n \log \relax (x) + b \log \relax (c) + a\right )} \operatorname {log\_integral}\left (x^{2} e^{\left (\frac {2 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )}}{b^{3} n^{3} \log \relax (x) + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.42, size = 261, normalized size = 3.43 \[ -\frac {b n x^{2}}{b^{3} n^{3} \log \relax (x) + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}} + \frac {2 \, b n {\rm Ei}\left (\frac {2 \, \log \relax (c)}{n} + \frac {2 \, a}{b n} + 2 \, \log \relax (x)\right ) e^{\left (-\frac {2 \, a}{b n}\right )} \log \relax (x)}{{\left (b^{3} n^{3} \log \relax (x) + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}\right )} c^{\frac {2}{n}}} + \frac {2 \, b {\rm Ei}\left (\frac {2 \, \log \relax (c)}{n} + \frac {2 \, a}{b n} + 2 \, \log \relax (x)\right ) e^{\left (-\frac {2 \, a}{b n}\right )} \log \relax (c)}{{\left (b^{3} n^{3} \log \relax (x) + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}\right )} c^{\frac {2}{n}}} + \frac {2 \, a {\rm Ei}\left (\frac {2 \, \log \relax (c)}{n} + \frac {2 \, a}{b n} + 2 \, \log \relax (x)\right ) e^{\left (-\frac {2 \, a}{b n}\right )}}{{\left (b^{3} n^{3} \log \relax (x) + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}\right )} c^{\frac {2}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*log(c*x^n))^2,x, algorithm="giac")

[Out]

-b*n*x^2/(b^3*n^3*log(x) + b^3*n^2*log(c) + a*b^2*n^2) + 2*b*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x))*e^(-2*a/(
b*n))*log(x)/((b^3*n^3*log(x) + b^3*n^2*log(c) + a*b^2*n^2)*c^(2/n)) + 2*b*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x
))*e^(-2*a/(b*n))*log(c)/((b^3*n^3*log(x) + b^3*n^2*log(c) + a*b^2*n^2)*c^(2/n)) + 2*a*Ei(2*log(c)/n + 2*a/(b*
n) + 2*log(x))*e^(-2*a/(b*n))/((b^3*n^3*log(x) + b^3*n^2*log(c) + a*b^2*n^2)*c^(2/n))

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maple [F]  time = 0.89, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*ln(c*x^n)+a)^2,x)

[Out]

int(x/(b*ln(c*x^n)+a)^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {x^{2}}{b^{2} n \log \relax (c) + b^{2} n \log \left (x^{n}\right ) + a b n} + 2 \, \int \frac {x}{b^{2} n \log \relax (c) + b^{2} n \log \left (x^{n}\right ) + a b n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*log(c*x^n))^2,x, algorithm="maxima")

[Out]

-x^2/(b^2*n*log(c) + b^2*n*log(x^n) + a*b*n) + 2*integrate(x/(b^2*n*log(c) + b^2*n*log(x^n) + a*b*n), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*log(c*x^n))^2,x)

[Out]

int(x/(a + b*log(c*x^n))^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*ln(c*x**n))**2,x)

[Out]

Integral(x/(a + b*log(c*x**n))**2, x)

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