\(\int \frac {1}{a+b \log (c (d (e+f x)^m)^n)} \, dx\) [408]

Optimal result

Integrand size = 20, antiderivative size = 83 \[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\frac {e^{-\frac {a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b f m n} \]

[Out]

(f*x+e)*Ei((a+b*ln(c*(d*(f*x+e)^m)^n))/b/m/n)/b/exp(a/b/m/n)/f/m/n/((c*(d*(f*x+e)^m)^n)^(1/m/n))

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2436, 2337, 2209, 2495} \[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\frac {(e+f x) e^{-\frac {a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b f m n} \]

[In]

Int[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-1),x]

[Out]

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

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2337

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 2436

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 2495

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\frac {e^{-\frac {a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b f m n} \]

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-1),x]

[Out]

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

Maple [F]

\[\int \frac {1}{a +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.78 \[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\frac {e^{\left (-\frac {b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )} \operatorname {log\_integral}\left ({\left (f x + e\right )} e^{\left (\frac {b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}\right )}{b f m n} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\int \frac {1}{a + b \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}\, dx \]

[In]

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

[Out]

Integral(1/(a + b*log(c*(d*(e + f*x)**m)**n)), x)

Maxima [F]

\[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\int { \frac {1}{b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94 \[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\frac {{\rm Ei}\left (\frac {\log \left (d\right )}{m} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac {a}{b m n}\right )}}{b c^{\frac {1}{m n}} d^{\left (\frac {1}{m}\right )} f m n} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\int \frac {1}{a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )} \,d x \]

[In]

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

[Out]

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