Integrand size = 19, antiderivative size = 48 \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\frac {b p \left (d x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {\log \left (d x^n\right )}{n}\right )}{x}-\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \] Output:
b*p*(d*x^n)^(1/n)*Ei(-ln(d*x^n)/n)/x-(a+b*ln(c*ln(d*x^n)^p))/x
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=-\frac {a-b p \left (d x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {\log \left (d x^n\right )}{n}\right )+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \] Input:
Integrate[(a + b*Log[c*Log[d*x^n]^p])/x^2,x]
Output:
-((a - b*p*(d*x^n)^n^(-1)*ExpIntegralEi[-(Log[d*x^n]/n)] + b*Log[c*Log[d*x ^n]^p])/x)
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3002, 2747, 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.
\(\displaystyle \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 3002 |
\(\displaystyle b n p \int \frac {1}{x^2 \log \left (d x^n\right )}dx-\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x}\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle \frac {b p \left (d x^n\right )^{\frac {1}{n}} \int \frac {\left (d x^n\right )^{-1/n}}{\log \left (d x^n\right )}d\log \left (d x^n\right )}{x}-\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {b p \left (d x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {\log \left (d x^n\right )}{n}\right )}{x}-\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x}\) |
Input:
Int[(a + b*Log[c*Log[d*x^n]^p])/x^2,x]
Output:
(b*p*(d*x^n)^n^(-1)*ExpIntegralEi[-(Log[d*x^n]/n)])/x - (a + b*Log[c*Log[d *x^n]^p])/x
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_)*((d_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n)) Subst[Int[E^(((m + 1)/n )*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
Int[((a_.) + Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)]*(b_.))*((e_.)*(x_))^(m_ .), x_Symbol] :> Simp[(e*x)^(m + 1)*((a + b*Log[c*Log[d*x^n]^p])/(e*(m + 1) )), x] - Simp[b*n*(p/(m + 1)) Int[(e*x)^m/Log[d*x^n], x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[m, -1]
\[\int \frac {a +b \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right )}{x^{2}}d x\]
Input:
int((a+b*ln(c*ln(d*x^n)^p))/x^2,x)
Output:
int((a+b*ln(c*ln(d*x^n)^p))/x^2,x)
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\frac {b d^{\left (\frac {1}{n}\right )} p x \operatorname {log\_integral}\left (\frac {1}{d^{\left (\frac {1}{n}\right )} x}\right ) - b p \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - b \log \left (c\right ) - a}{x} \] Input:
integrate((a+b*log(c*log(d*x^n)^p))/x^2,x, algorithm="fricas")
Output:
(b*d^(1/n)*p*x*log_integral(1/(d^(1/n)*x)) - b*p*log(n*log(x) + log(d)) - b*log(c) - a)/x
\[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\int \frac {a + b \log {\left (c \log {\left (d x^{n} \right )}^{p} \right )}}{x^{2}}\, dx \] Input:
integrate((a+b*ln(c*ln(d*x**n)**p))/x**2,x)
Output:
Integral((a + b*log(c*log(d*x**n)**p))/x**2, x)
\[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\int { \frac {b \log \left (c \log \left (d x^{n}\right )^{p}\right ) + a}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*log(d*x^n)^p))/x^2,x, algorithm="maxima")
Output:
(n*p*integrate(1/(x^2*log(d) + x^2*log(x^n)), x) - (log(c) + log((log(d) + log(x^n))^p))/x)*b - a/x
\[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\int { \frac {b \log \left (c \log \left (d x^{n}\right )^{p}\right ) + a}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*log(d*x^n)^p))/x^2,x, algorithm="giac")
Output:
integrate((b*log(c*log(d*x^n)^p) + a)/x^2, x)
Timed out. \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\int \frac {a+b\,\ln \left (c\,{\ln \left (d\,x^n\right )}^p\right )}{x^2} \,d x \] Input:
int((a + b*log(c*log(d*x^n)^p))/x^2,x)
Output:
int((a + b*log(c*log(d*x^n)^p))/x^2, x)
\[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\frac {\left (\int \frac {1}{\mathrm {log}\left (x^{n} d \right ) x^{2}}d x \right ) b n p x -\mathrm {log}\left (\mathrm {log}\left (x^{n} d \right )^{p} c \right ) b -a}{x} \] Input:
int((a+b*log(c*log(d*x^n)^p))/x^2,x)
Output:
(int(1/(log(x**n*d)*x**2),x)*b*n*p*x - log(log(x**n*d)**p*c)*b - a)/x