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

Optimal. Leaf size=89 \[ \frac {x \left (c x^n\right )^{-1/n} e^{-\frac {d}{e n}} (a e-b d+b e n) \text {Ei}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )}{e^3 n^2}+\frac {x (b d-a e)}{e^2 n \left (e \log \left (c x^n\right )+d\right )} \]

[Out]

(b*e*n+a*e-b*d)*x*Ei((d+e*ln(c*x^n))/e/n)/e^3/exp(d/e/n)/n^2/((c*x^n)^(1/n))+(-a*e+b*d)*x/e^2/n/(d+e*ln(c*x^n)
)

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Rubi [A]  time = 0.14, antiderivative size = 135, normalized size of antiderivative = 1.52, number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2360, 2297, 2300, 2178} \[ -\frac {x \left (c x^n\right )^{-1/n} e^{-\frac {d}{e n}} (b d-a e) \text {Ei}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )}{e^3 n^2}+\frac {x (b d-a e)}{e^2 n \left (e \log \left (c x^n\right )+d\right )}+\frac {b x \left (c x^n\right )^{-1/n} e^{-\frac {d}{e n}} \text {Ei}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )}{e^2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b*d - a*e)*x*ExpIntegralEi[(d + e*Log[c*x^n])/(e*n)])/(e^3*E^(d/(e*n))*n^2*(c*x^n)^n^(-1))) + (b*x*ExpInte
gralEi[(d + e*Log[c*x^n])/(e*n)])/(e^2*E^(d/(e*n))*n*(c*x^n)^n^(-1)) + ((b*d - a*e)*x)/(e^2*n*(d + e*Log[c*x^n
]))

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 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 2360

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]*(e_.) + (d_))^(q_.), x_Symbol] :> Int[E
xpandIntegrand[(a + b*Log[c*x^n])^p*(d + e*Log[c*x^n])^q, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[p
] && IntegerQ[q]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx &=\int \left (\frac {-b d+a e}{e \left (d+e \log \left (c x^n\right )\right )^2}+\frac {b}{e \left (d+e \log \left (c x^n\right )\right )}\right ) \, dx\\ &=\frac {b \int \frac {1}{d+e \log \left (c x^n\right )} \, dx}{e}+\frac {(-b d+a e) \int \frac {1}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx}{e}\\ &=\frac {(b d-a e) x}{e^2 n \left (d+e \log \left (c x^n\right )\right )}-\frac {(b d-a e) \int \frac {1}{d+e \log \left (c x^n\right )} \, dx}{e^2 n}+\frac {\left (b x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{n}}}{d+e x} \, dx,x,\log \left (c x^n\right )\right )}{e n}\\ &=\frac {b e^{-\frac {d}{e n}} x \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )}{e^2 n}+\frac {(b d-a e) x}{e^2 n \left (d+e \log \left (c x^n\right )\right )}-\frac {\left ((b d-a e) x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{n}}}{d+e x} \, dx,x,\log \left (c x^n\right )\right )}{e^2 n^2}\\ &=-\frac {(b d-a e) e^{-\frac {d}{e n}} x \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )}{e^3 n^2}+\frac {b e^{-\frac {d}{e n}} x \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )}{e^2 n}+\frac {(b d-a e) x}{e^2 n \left (d+e \log \left (c x^n\right )\right )}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 87, normalized size = 0.98 \[ \frac {x \left (c x^n\right )^{-1/n} e^{-\frac {d}{e n}} (a e-b d+b e n) \text {Ei}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )-\frac {e n x (a e-b d)}{e \log \left (c x^n\right )+d}}{e^3 n^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-(b*d) + a*e + b*e*n)*x*ExpIntegralEi[(d + e*Log[c*x^n])/(e*n)])/(E^(d/(e*n))*(c*x^n)^n^(-1)) - (e*(-(b*d)
+ a*e)*n*x)/(d + e*Log[c*x^n]))/(e^3*n^2)

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fricas [A]  time = 0.70, size = 154, normalized size = 1.73 \[ \frac {{\left ({\left (b d e - a e^{2}\right )} n x e^{\left (\frac {e \log \relax (c) + d}{e n}\right )} + {\left (b d e n - b d^{2} + a d e + {\left (b e^{2} n - b d e + a e^{2}\right )} \log \relax (c) + {\left (b e^{2} n^{2} - {\left (b d e - a e^{2}\right )} n\right )} \log \relax (x)\right )} \operatorname {log\_integral}\left (x e^{\left (\frac {e \log \relax (c) + d}{e n}\right )}\right )\right )} e^{\left (-\frac {e \log \relax (c) + d}{e n}\right )}}{e^{4} n^{3} \log \relax (x) + e^{4} n^{2} \log \relax (c) + d e^{3} n^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*d*e - a*e^2)*n*x*e^((e*log(c) + d)/(e*n)) + (b*d*e*n - b*d^2 + a*d*e + (b*e^2*n - b*d*e + a*e^2)*log(c) +
(b*e^2*n^2 - (b*d*e - a*e^2)*n)*log(x))*log_integral(x*e^((e*log(c) + d)/(e*n))))*e^(-(e*log(c) + d)/(e*n))/(e
^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2)

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giac [B]  time = 0.52, size = 661, normalized size = 7.43 \[ \frac {b d n x e}{n^{3} e^{4} \log \relax (x) + d n^{2} e^{3} + n^{2} e^{4} \log \relax (c)} + \frac {b n^{2} {\rm Ei}\left (\frac {d e^{\left (-1\right )}}{n} + \frac {\log \relax (c)}{n} + \log \relax (x)\right ) e^{\left (-\frac {d e^{\left (-1\right )}}{n} + 2\right )} \log \relax (x)}{{\left (n^{3} e^{4} \log \relax (x) + d n^{2} e^{3} + n^{2} e^{4} \log \relax (c)\right )} c^{\left (\frac {1}{n}\right )}} - \frac {b d n {\rm Ei}\left (\frac {d e^{\left (-1\right )}}{n} + \frac {\log \relax (c)}{n} + \log \relax (x)\right ) e^{\left (-\frac {d e^{\left (-1\right )}}{n} + 1\right )} \log \relax (x)}{{\left (n^{3} e^{4} \log \relax (x) + d n^{2} e^{3} + n^{2} e^{4} \log \relax (c)\right )} c^{\left (\frac {1}{n}\right )}} - \frac {a n x e^{2}}{n^{3} e^{4} \log \relax (x) + d n^{2} e^{3} + n^{2} e^{4} \log \relax (c)} + \frac {b d n {\rm Ei}\left (\frac {d e^{\left (-1\right )}}{n} + \frac {\log \relax (c)}{n} + \log \relax (x)\right ) e^{\left (-\frac {d e^{\left (-1\right )}}{n} + 1\right )}}{{\left (n^{3} e^{4} \log \relax (x) + d n^{2} e^{3} + n^{2} e^{4} \log \relax (c)\right )} c^{\left (\frac {1}{n}\right )}} - \frac {b d^{2} {\rm Ei}\left (\frac {d e^{\left (-1\right )}}{n} + \frac {\log \relax (c)}{n} + \log \relax (x)\right ) e^{\left (-\frac {d e^{\left (-1\right )}}{n}\right )}}{{\left (n^{3} e^{4} \log \relax (x) + d n^{2} e^{3} + n^{2} e^{4} \log \relax (c)\right )} c^{\left (\frac {1}{n}\right )}} + \frac {b n {\rm Ei}\left (\frac {d e^{\left (-1\right )}}{n} + \frac {\log \relax (c)}{n} + \log \relax (x)\right ) e^{\left (-\frac {d e^{\left (-1\right )}}{n} + 2\right )} \log \relax (c)}{{\left (n^{3} e^{4} \log \relax (x) + d n^{2} e^{3} + n^{2} e^{4} \log \relax (c)\right )} c^{\left (\frac {1}{n}\right )}} - \frac {b d {\rm Ei}\left (\frac {d e^{\left (-1\right )}}{n} + \frac {\log \relax (c)}{n} + \log \relax (x)\right ) e^{\left (-\frac {d e^{\left (-1\right )}}{n} + 1\right )} \log \relax (c)}{{\left (n^{3} e^{4} \log \relax (x) + d n^{2} e^{3} + n^{2} e^{4} \log \relax (c)\right )} c^{\left (\frac {1}{n}\right )}} + \frac {a n {\rm Ei}\left (\frac {d e^{\left (-1\right )}}{n} + \frac {\log \relax (c)}{n} + \log \relax (x)\right ) e^{\left (-\frac {d e^{\left (-1\right )}}{n} + 2\right )} \log \relax (x)}{{\left (n^{3} e^{4} \log \relax (x) + d n^{2} e^{3} + n^{2} e^{4} \log \relax (c)\right )} c^{\left (\frac {1}{n}\right )}} + \frac {a d {\rm Ei}\left (\frac {d e^{\left (-1\right )}}{n} + \frac {\log \relax (c)}{n} + \log \relax (x)\right ) e^{\left (-\frac {d e^{\left (-1\right )}}{n} + 1\right )}}{{\left (n^{3} e^{4} \log \relax (x) + d n^{2} e^{3} + n^{2} e^{4} \log \relax (c)\right )} c^{\left (\frac {1}{n}\right )}} + \frac {a {\rm Ei}\left (\frac {d e^{\left (-1\right )}}{n} + \frac {\log \relax (c)}{n} + \log \relax (x)\right ) e^{\left (-\frac {d e^{\left (-1\right )}}{n} + 2\right )} \log \relax (c)}{{\left (n^{3} e^{4} \log \relax (x) + d n^{2} e^{3} + n^{2} e^{4} \log \relax (c)\right )} c^{\left (\frac {1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*d*n*x*e/(n^3*e^4*log(x) + d*n^2*e^3 + n^2*e^4*log(c)) + b*n^2*Ei(d*e^(-1)/n + log(c)/n + log(x))*e^(-d*e^(-1
)/n + 2)*log(x)/((n^3*e^4*log(x) + d*n^2*e^3 + n^2*e^4*log(c))*c^(1/n)) - b*d*n*Ei(d*e^(-1)/n + log(c)/n + log
(x))*e^(-d*e^(-1)/n + 1)*log(x)/((n^3*e^4*log(x) + d*n^2*e^3 + n^2*e^4*log(c))*c^(1/n)) - a*n*x*e^2/(n^3*e^4*l
og(x) + d*n^2*e^3 + n^2*e^4*log(c)) + b*d*n*Ei(d*e^(-1)/n + log(c)/n + log(x))*e^(-d*e^(-1)/n + 1)/((n^3*e^4*l
og(x) + d*n^2*e^3 + n^2*e^4*log(c))*c^(1/n)) - b*d^2*Ei(d*e^(-1)/n + log(c)/n + log(x))*e^(-d*e^(-1)/n)/((n^3*
e^4*log(x) + d*n^2*e^3 + n^2*e^4*log(c))*c^(1/n)) + b*n*Ei(d*e^(-1)/n + log(c)/n + log(x))*e^(-d*e^(-1)/n + 2)
*log(c)/((n^3*e^4*log(x) + d*n^2*e^3 + n^2*e^4*log(c))*c^(1/n)) - b*d*Ei(d*e^(-1)/n + log(c)/n + log(x))*e^(-d
*e^(-1)/n + 1)*log(c)/((n^3*e^4*log(x) + d*n^2*e^3 + n^2*e^4*log(c))*c^(1/n)) + a*n*Ei(d*e^(-1)/n + log(c)/n +
log(x))*e^(-d*e^(-1)/n + 2)*log(x)/((n^3*e^4*log(x) + d*n^2*e^3 + n^2*e^4*log(c))*c^(1/n)) + a*d*Ei(d*e^(-1)/
n + log(c)/n + log(x))*e^(-d*e^(-1)/n + 1)/((n^3*e^4*log(x) + d*n^2*e^3 + n^2*e^4*log(c))*c^(1/n)) + a*Ei(d*e^
(-1)/n + log(c)/n + log(x))*e^(-d*e^(-1)/n + 2)*log(c)/((n^3*e^4*log(x) + d*n^2*e^3 + n^2*e^4*log(c))*c^(1/n))

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maple [C]  time = 0.65, size = 370, normalized size = 4.16 \[ -\frac {\left (b e n +a e -b d \right ) x \,c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} \Ei \left (1, -\ln \relax (x )-\frac {-i \pi e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi e \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 e \ln \relax (c )+2 d +2 \left (-n \ln \relax (x )+\ln \left (x^{n}\right )\right ) e}{2 e n}\right ) {\mathrm e}^{-\frac {-i \pi e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi e \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 d}{2 e n}}}{e^{3} n^{2}}-\frac {2 \left (a e -b d \right ) x}{\left (-i \pi e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi e \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 e \ln \relax (c )+2 e \ln \left (x^{n}\right )+2 d \right ) e^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ {\left ({\left (e n - d\right )} b + a e\right )} \int \frac {1}{e^{3} n \log \relax (c) + e^{3} n \log \left (x^{n}\right ) + d e^{2} n}\,{d x} + \frac {{\left (b d - a e\right )} x}{e^{3} n \log \relax (c) + e^{3} n \log \left (x^{n}\right ) + d e^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((e*n - d)*b + a*e)*integrate(1/(e^3*n*log(c) + e^3*n*log(x^n) + d*e^2*n), x) + (b*d - a*e)*x/(e^3*n*log(c) +
e^3*n*log(x^n) + d*e^2*n)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (d+e\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(d + e*log(c*x^n))^2,x)

[Out]

int((a + b*log(c*x^n))/(d + e*log(c*x^n))^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \log {\left (c x^{n} \right )}}{\left (d + e \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*log(c*x**n))/(d + e*log(c*x**n))**2, x)

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