3.409 \(\int \frac{1}{(a+b \log (c (d (e+f x)^m)^n))^2} \, dx\)

Optimal. Leaf size=123 \[ \frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b^2 f m^2 n^2}-\frac{e+f x}{b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.167533, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2389, 2297, 2300, 2178, 2445} \[ \frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b^2 f m^2 n^2}-\frac{e+f x}{b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-2),x]

[Out]

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

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2300

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, 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

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

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

Mathematica [A]  time = 0.110355, size = 163, normalized size = 1.33 \[ -\frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \left (b m n e^{\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{\frac{1}{m n}}-\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right ) \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )\right )}{b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.087, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{m} \right ) ^{n} \right ) \right ) ^{-2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*ln(c*(d*(f*x+e)^m)^n))^2,x)

[Out]

int(1/(a+b*ln(c*(d*(f*x+e)^m)^n))^2,x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.25319, size = 425, normalized size = 3.46 \begin{align*} -\frac{{\left ({\left (b f m n x + b e m n\right )} e^{\left (\frac{b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )} -{\left (b m n \log \left (f x + e\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a\right )} \logintegral \left ({\left (f x + e\right )} e^{\left (\frac{b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}\right )\right )} e^{\left (-\frac{b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}}{b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*ln(c*(d*(f*x+e)**m)**n))**2,x)

[Out]

Integral((a + b*log(c*(d*(e + f*x)**m)**n))**(-2), x)

________________________________________________________________________________________

Giac [B]  time = 1.24344, size = 801, normalized size = 6.51 \begin{align*} -\frac{{\left (f x + e\right )} b m n}{b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}} + \frac{b m n{\rm Ei}\left (\frac{\log \left (d\right )}{m} + \frac{\log \left (c\right )}{m n} + \frac{a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac{a}{b m n}\right )} \log \left (f x + e\right )}{{\left (b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}\right )} c^{\frac{1}{m n}} d^{\left (\frac{1}{m}\right )}} + \frac{b n{\rm Ei}\left (\frac{\log \left (d\right )}{m} + \frac{\log \left (c\right )}{m n} + \frac{a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac{a}{b m n}\right )} \log \left (d\right )}{{\left (b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}\right )} c^{\frac{1}{m n}} d^{\left (\frac{1}{m}\right )}} + \frac{b{\rm Ei}\left (\frac{\log \left (d\right )}{m} + \frac{\log \left (c\right )}{m n} + \frac{a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac{a}{b m n}\right )} \log \left (c\right )}{{\left (b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}\right )} c^{\frac{1}{m n}} d^{\left (\frac{1}{m}\right )}} + \frac{a{\rm Ei}\left (\frac{\log \left (d\right )}{m} + \frac{\log \left (c\right )}{m n} + \frac{a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac{a}{b m n}\right )}}{{\left (b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}\right )} c^{\frac{1}{m n}} d^{\left (\frac{1}{m}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^2,x, algorithm="giac")

[Out]

-(f*x + e)*b*m*n/(b^3*f*m^3*n^3*log(f*x + e) + b^3*f*m^2*n^3*log(d) + b^3*f*m^2*n^2*log(c) + a*b^2*f*m^2*n^2)
+ b*m*n*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(f*x + e)/((b^3*f*m^3*n^3*log
(f*x + e) + b^3*f*m^2*n^3*log(d) + b^3*f*m^2*n^2*log(c) + a*b^2*f*m^2*n^2)*c^(1/(m*n))*d^(1/m)) + b*n*Ei(log(d
)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(d)/((b^3*f*m^3*n^3*log(f*x + e) + b^3*f*m^2*
n^3*log(d) + b^3*f*m^2*n^2*log(c) + a*b^2*f*m^2*n^2)*c^(1/(m*n))*d^(1/m)) + b*Ei(log(d)/m + log(c)/(m*n) + a/(
b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(c)/((b^3*f*m^3*n^3*log(f*x + e) + b^3*f*m^2*n^3*log(d) + b^3*f*m^2*n
^2*log(c) + a*b^2*f*m^2*n^2)*c^(1/(m*n))*d^(1/m)) + a*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e
^(-a/(b*m*n))/((b^3*f*m^3*n^3*log(f*x + e) + b^3*f*m^2*n^3*log(d) + b^3*f*m^2*n^2*log(c) + a*b^2*f*m^2*n^2)*c^
(1/(m*n))*d^(1/m))