\(\int \frac {x}{(a+b \log (c x^n))^2} \, dx\) [75]

Optimal result

Integrand size = 14, antiderivative size = 76 \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {2 e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac {x^2}{b n \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

2*x^2*Ei(2*(a+b*ln(c*x^n))/b/n)/b^2/exp(2*a/b/n)/n^2/((c*x^n)^(2/n))-x^2/b/n/(a+b*ln(c*x^n))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2343, 2347, 2209} \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {2 x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac {x^2}{b n \left (a+b \log \left (c x^n\right )\right )} \]

[In]

Int[x/(a + b*Log[c*x^n])^2,x]

[Out]

(2*x^2*ExpIntegralEi[(2*(a + b*Log[c*x^n]))/(b*n)])/(b^2*E^((2*a)/(b*n))*n^2*(c*x^n)^(2/n)) - x^2/(b*n*(a + b*
Log[c*x^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 2343

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 2347

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)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^2}{b n \left (a+b \log \left (c x^n\right )\right )}+\frac {2 \int \frac {x}{a+b \log \left (c x^n\right )} \, dx}{b n} \\ & = -\frac {x^2}{b n \left (a+b \log \left (c x^n\right )\right )}+\frac {\left (2 x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b n^2} \\ & = \frac {2 e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac {x^2}{b n \left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {x^2 \left (2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {b n}{a+b \log \left (c x^n\right )}\right )}{b^2 n^2} \]

[In]

Integrate[x/(a + b*Log[c*x^n])^2,x]

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.34 (sec) , antiderivative size = 350, normalized size of antiderivative = 4.61

method	result	size
risch	\(-\frac {2 i x^{2}}{b n \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 i b \ln \left (c \right )+2 i b \ln \left (x^{n}\right )+2 i a \right )}-\frac {2 x^{2} c^{-\frac {2}{n}} \left (x^{n}\right )^{-\frac {2}{n}} {\mathrm e}^{-\frac {-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 a}{b n}} \operatorname {Ei}_{1}\left (-2 \ln \left (x \right )+\frac {i \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 i b \ln \left (c \right )+2 i b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 i a \right )}{b n}\right )}{b^{2} n^{2}}\)	\(350\)

[In]

int(x/(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)

[Out]

-2*I*x^2/b/n/(b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-b*Pi*csgn(I*x^n)*csgn(I*
c*x^n)^2+b*Pi*csgn(I*c*x^n)^3+2*I*b*ln(c)+2*I*b*ln(x^n)+2*I*a)-2/b^2/n^2*x^2*c^(-2/n)*(x^n)^(-2/n)*exp(-(-I*b*
Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b
*Pi*csgn(I*c*x^n)^3+2*a)/b/n)*Ei(1,-2*ln(x)+I*(b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-b*Pi*csgn(I*c)*csgn(I*
c*x^n)^2-b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+b*Pi*csgn(I*c*x^n)^3+2*I*b*ln(c)+2*I*b*(ln(x^n)-n*ln(x))+2*I*a)/b/n)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.33 \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {{\left (b n x^{2} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 2 \, {\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \operatorname {log\_integral}\left (x^{2} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {x}{\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

Integral(x/(a + b*log(c*x**n))**2, x)

Maxima [F]

\[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {x}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (77) = 154\).

Time = 0.36 (sec) , antiderivative size = 261, normalized size of antiderivative = 3.43 \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {b n x^{2}}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} + \frac {2 \, b n {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + 2 \, \log \left (x\right )\right ) e^{\left (-\frac {2 \, a}{b n}\right )} \log \left (x\right )}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\frac {2}{n}}} + \frac {2 \, b {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + 2 \, \log \left (x\right )\right ) e^{\left (-\frac {2 \, a}{b n}\right )} \log \left (c\right )}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\frac {2}{n}}} + \frac {2 \, a {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + 2 \, \log \left (x\right )\right ) e^{\left (-\frac {2 \, a}{b n}\right )}}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\frac {2}{n}}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {x}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]

[In]

int(x/(a + b*log(c*x^n))^2,x)

[Out]

int(x/(a + b*log(c*x^n))^2, x)