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

Optimal. Leaf size=100 \[ \frac{(m+1) (d x)^{m+1} e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 d n^2}-\frac{(d x)^{m+1}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

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Rubi [A]  time = 0.0931316, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2306, 2310, 2178} \[ \frac{(m+1) (d x)^{m+1} e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 d n^2}-\frac{(d x)^{m+1}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m/(a + b*Log[c*x^n])^2,x]

[Out]

((1 + m)*(d*x)^(1 + m)*ExpIntegralEi[((1 + m)*(a + b*Log[c*x^n]))/(b*n)])/(b^2*d*E^((a*(1 + m))/(b*n))*n^2*(c*
x^n)^((1 + m)/n)) - (d*x)^(1 + m)/(b*d*n*(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{(d x)^m}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac{(d x)^{1+m}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac{(1+m) \int \frac{(d x)^m}{a+b \log \left (c x^n\right )} \, dx}{b n}\\ &=-\frac{(d x)^{1+m}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac{\left ((1+m) (d x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(1+m) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b d n^2}\\ &=\frac{e^{-\frac{a (1+m)}{b n}} (1+m) (d x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \text{Ei}\left (\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 d n^2}-\frac{(d x)^{1+m}}{b d n \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}

Mathematica [A]  time = 0.233248, size = 89, normalized size = 0.89 \[ \frac{(d x)^m \left ((m+1) x^{-m} \exp \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \text{Ei}\left (\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{b n x}{a+b \log \left (c x^n\right )}\right )}{b^2 n^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m/(a + b*Log[c*x^n])^2,x]

[Out]

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

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Maple [F]  time = 1.303, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d^{m}{\left (m + 1\right )} \int \frac{x^{m}}{b^{2} n \log \left (c\right ) + b^{2} n \log \left (x^{n}\right ) + a b n}\,{d x} - \frac{d^{m} x x^{m}}{b^{2} n \log \left (c\right ) + b^{2} n \log \left (x^{n}\right ) + a b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^m*(m + 1)*integrate(x^m/(b^2*n*log(c) + b^2*n*log(x^n) + a*b*n), x) - d^m*x*x^m/(b^2*n*log(c) + b^2*n*log(x^
n) + a*b*n)

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Fricas [A]  time = 1.00347, size = 332, normalized size = 3.32 \begin{align*} -\frac{b n x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} -{\left ({\left (b m + b\right )} n \log \left (x\right ) + a m +{\left (b m + b\right )} \log \left (c\right ) + a\right )}{\rm Ei}\left (\frac{{\left (b m + b\right )} n \log \left (x\right ) + a m +{\left (b m + b\right )} \log \left (c\right ) + a}{b n}\right ) e^{\left (\frac{b m n \log \left (d\right ) - a m -{\left (b m + b\right )} \log \left (c\right ) - a}{b n}\right )}}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*n*x*e^(m*log(d) + m*log(x)) - ((b*m + b)*n*log(x) + a*m + (b*m + b)*log(c) + a)*Ei(((b*m + b)*n*log(x) + a
*m + (b*m + b)*log(c) + a)/(b*n))*e^((b*m*n*log(d) - a*m - (b*m + 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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{m}}{\left (a + b \log{\left (c x^{n} \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d*x)**m/(a + b*log(c*x**n))**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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