Integrand size = 20, antiderivative size = 83 \[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\frac {e^{-\frac {a}{b p q}} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b f p q} \] Output:
(f*x+e)*Ei((a+b*ln(c*(d*(f*x+e)^p)^q))/b/p/q)/b/exp(a/b/p/q)/f/p/q/((c*(d* (f*x+e)^p)^q)^(1/p/q))
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\frac {e^{-\frac {a}{b p q}} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b f p q} \] Input:
Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])^(-1),x]
Output:
((e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(b*E^( a/(b*p*q))*f*p*q*(c*(d*(e + f*x)^p)^q)^(1/(p*q)))
Time = 0.75 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2895, 2836, 2737, 2609}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx\) |
\(\Big \downarrow \) 2895 |
\(\displaystyle \int \frac {1}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx\) |
\(\Big \downarrow \) 2836 |
\(\displaystyle \frac {\int \frac {1}{a+b \log \left (c d^q (e+f x)^{p q}\right )}d(e+f x)}{f}\) |
\(\Big \downarrow \) 2737 |
\(\displaystyle \frac {(e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \int \frac {\left (c d^q (e+f x)^{p q}\right )^{\frac {1}{p q}}}{a+b \log \left (c d^q (e+f x)^{p q}\right )}d\log \left (c d^q (e+f x)^{p q}\right )}{f p q}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {(e+f x) e^{-\frac {a}{b p q}} \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b f p q}\) |
Input:
Int[(a + b*Log[c*(d*(e + f*x)^p)^q])^(-1),x]
Output:
((e + f*x)*ExpIntegralEi[(a + b*Log[c*d^q*(e + f*x)^(p*q)])/(b*p*q)])/(b*E ^(a/(b*p*q))*f*p*q*(c*d^q*(e + f*x)^(p*q))^(1/(p*q)))
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : > Simp[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]
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]]
\[\int \frac {1}{a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}d x\]
Input:
int(1/(a+b*ln(c*(d*(f*x+e)^p)^q)),x)
Output:
int(1/(a+b*ln(c*(d*(f*x+e)^p)^q)),x)
Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.78 \[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\frac {e^{\left (-\frac {b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\right )} \operatorname {log\_integral}\left ({\left (f x + e\right )} e^{\left (\frac {b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\right )}\right )}{b f p q} \] Input:
integrate(1/(a+b*log(c*(d*(f*x+e)^p)^q)),x, algorithm="fricas")
Output:
e^(-(b*q*log(d) + b*log(c) + a)/(b*p*q))*log_integral((f*x + e)*e^((b*q*lo g(d) + b*log(c) + a)/(b*p*q)))/(b*f*p*q)
\[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int \frac {1}{a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}\, dx \] Input:
integrate(1/(a+b*ln(c*(d*(f*x+e)**p)**q)),x)
Output:
Integral(1/(a + b*log(c*(d*(e + f*x)**p)**q)), x)
\[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int { \frac {1}{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a} \,d x } \] Input:
integrate(1/(a+b*log(c*(d*(f*x+e)^p)^q)),x, algorithm="maxima")
Output:
integrate(1/(b*log(((f*x + e)^p*d)^q*c) + a), x)
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94 \[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\frac {{\rm Ei}\left (\frac {\log \left (d\right )}{p} + \frac {\log \left (c\right )}{p q} + \frac {a}{b p q} + \log \left (f x + e\right )\right ) e^{\left (-\frac {a}{b p q}\right )}}{b c^{\frac {1}{p q}} d^{\left (\frac {1}{p}\right )} f p q} \] Input:
integrate(1/(a+b*log(c*(d*(f*x+e)^p)^q)),x, algorithm="giac")
Output:
Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))/(b*c ^(1/(p*q))*d^(1/p)*f*p*q)
Timed out. \[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int \frac {1}{a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )} \,d x \] Input:
int(1/(a + b*log(c*(d*(e + f*x)^p)^q)),x)
Output:
int(1/(a + b*log(c*(d*(e + f*x)^p)^q)), x)
\[ \int \frac {1}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\frac {\left (\int \frac {x}{\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right ) b e +\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right ) b f x +a e +a f x}d x \right ) b \,f^{2} p q +\mathrm {log}\left (\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right ) b +a \right ) e}{b f p q} \] Input:
int(1/(a+b*log(c*(d*(f*x+e)^p)^q)),x)
Output:
(int(x/(log(d**q*(e + f*x)**(p*q)*c)*b*e + log(d**q*(e + f*x)**(p*q)*c)*b* f*x + a*e + a*f*x),x)*b*f**2*p*q + log(log(d**q*(e + f*x)**(p*q)*c)*b + a) *e)/(b*f*p*q)