3.5.10 \(\int \frac {1}{(a+b \log (c (d (e+f x)^m)^n))^3} \, dx\) [410]

Optimal. Leaf size=169 \[ \frac {e^{-\frac {a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{2 b^3 f m^3 n^3}-\frac {e+f x}{2 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}-\frac {e+f x}{2 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2436, 2334, 2337, 2209, 2495} \begin {gather*} \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}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{2 b^3 f m^3 n^3}-\frac {e+f x}{2 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )}-\frac {e+f x}{2 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-3),x]

[Out]

((e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n)])/(2*b^3*E^(a/(b*m*n))*f*m^3*n^3*(c*(d*(e +
f*x)^m)^n)^(1/(m*n))) - (e + f*x)/(2*b*f*m*n*(a + b*Log[c*(d*(e + f*x)^m)^n])^2) - (e + f*x)/(2*b^2*f*m^2*n^2*
(a + b*Log[c*(d*(e + f*x)^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 2334

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

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Mathematica [A]
time = 0.13, size = 189, normalized size = 1.12 \begin {gather*} -\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}} \left (-\text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right ) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2+b e^{\frac {a}{b m n}} m n \left (c \left (d (e+f x)^m\right )^n\right )^{\frac {1}{m n}} \left (a+b m n+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )\right )}{2 b^3 f m^3 n^3 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-3),x]

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*ln(c*(d*(f*x+e)^m)^n))^3,x)

[Out]

int(1/(a+b*ln(c*(d*(f*x+e)^m)^n))^3,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^3,x, algorithm="maxima")

[Out]

-1/2*(((f*m*n + f*n*log(d) + f*log(c))*b + a*f)*x + ((m*n + n*log(d) + log(c))*b + a)*e + (b*f*x + b*e)*log(((
f*x + e)^m)^n))/(b^4*f*m^2*n^2*log(((f*x + e)^m)^n)^2 + a^2*b^2*f*m^2*n^2 + 2*(f*m^2*n^3*log(d) + f*m^2*n^2*lo
g(c))*a*b^3 + (f*m^2*n^4*log(d)^2 + 2*f*m^2*n^3*log(c)*log(d) + f*m^2*n^2*log(c)^2)*b^4 + 2*(a*b^3*f*m^2*n^2 +
(f*m^2*n^3*log(d) + f*m^2*n^2*log(c))*b^4)*log(((f*x + e)^m)^n)) + integrate(1/2/(b^3*m^2*n^2*log(((f*x + e)^
m)^n) + a*b^2*m^2*n^2 + (m^2*n^3*log(d) + m^2*n^2*log(c))*b^3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 455 vs. \(2 (170) = 340\).
time = 0.37, size = 455, normalized size = 2.69 \begin {gather*} -\frac {{\left ({\left ({\left (b^{2} f m^{2} n^{2} + a b f m n\right )} x + {\left (b^{2} m^{2} n^{2} + a b m n\right )} e + {\left (b^{2} f m^{2} n^{2} x + b^{2} m^{2} n^{2} e\right )} \log \left (f x + e\right ) + {\left (b^{2} f m n x + b^{2} m n e\right )} \log \left (c\right ) + {\left (b^{2} f m n^{2} x + b^{2} m n^{2} e\right )} \log \left (d\right )\right )} e^{\left (\frac {b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )} - {\left (b^{2} m^{2} n^{2} \log \left (f x + e\right )^{2} + b^{2} n^{2} \log \left (d\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \, {\left (b^{2} m n^{2} \log \left (d\right ) + b^{2} m n \log \left (c\right ) + a b m n\right )} \log \left (f x + e\right ) + 2 \, {\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (d\right )\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 )\right )} e^{\left (-\frac {b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}}{2 \, {\left (b^{5} f m^{5} n^{5} \log \left (f x + e\right )^{2} + b^{5} f m^{3} n^{5} \log \left (d\right )^{2} + b^{5} f m^{3} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} f m^{3} n^{3} \log \left (c\right ) + a^{2} b^{3} f m^{3} n^{3} + 2 \, {\left (b^{5} f m^{4} n^{5} \log \left (d\right ) + b^{5} f m^{4} n^{4} \log \left (c\right ) + a b^{4} f m^{4} n^{4}\right )} \log \left (f x + e\right ) + 2 \, {\left (b^{5} f m^{3} n^{4} \log \left (c\right ) + a b^{4} f m^{3} n^{4}\right )} \log \left (d\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^3,x, algorithm="fricas")

[Out]

-1/2*(((b^2*f*m^2*n^2 + a*b*f*m*n)*x + (b^2*m^2*n^2 + a*b*m*n)*e + (b^2*f*m^2*n^2*x + b^2*m^2*n^2*e)*log(f*x +
e) + (b^2*f*m*n*x + b^2*m*n*e)*log(c) + (b^2*f*m*n^2*x + b^2*m*n^2*e)*log(d))*e^((b*n*log(d) + b*log(c) + a)/
(b*m*n)) - (b^2*m^2*n^2*log(f*x + e)^2 + b^2*n^2*log(d)^2 + b^2*log(c)^2 + 2*a*b*log(c) + a^2 + 2*(b^2*m*n^2*l
og(d) + b^2*m*n*log(c) + a*b*m*n)*log(f*x + e) + 2*(b^2*n*log(c) + a*b*n)*log(d))*log_integral((f*x + e)*e^((b
*n*log(d) + b*log(c) + a)/(b*m*n))))*e^(-(b*n*log(d) + b*log(c) + a)/(b*m*n))/(b^5*f*m^5*n^5*log(f*x + e)^2 +
b^5*f*m^3*n^5*log(d)^2 + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3 + 2*(b^5*f*m^4*
n^5*log(d) + b^5*f*m^4*n^4*log(c) + a*b^4*f*m^4*n^4)*log(f*x + e) + 2*(b^5*f*m^3*n^4*log(c) + a*b^4*f*m^3*n^4)
*log(d))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*ln(c*(d*(f*x+e)**m)**n))**3,x)

[Out]

Integral((a + b*log(c*(d*(e + f*x)**m)**n))**(-3), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3481 vs. \(2 (170) = 340\).
time = 6.21, size = 3481, normalized size = 20.60 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^3,x, algorithm="giac")

[Out]

-1/2*(f*x + e)*b^2*m^2*n^2*log(f*x + e)/(b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) +
2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^
4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3
*n^3) + 1/2*b^2*m^2*n^2*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(f*x + e)^2/(
(b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^
5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2
+ 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) - 1/2*(f*x +
e)*b^2*m^2*n^2/(b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x +
e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m
^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3) - 1/2*(f*x + e)*b^2
*m*n^2*log(d)/(b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x +
e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^
3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3) + b^2*m*n^2*Ei(log(d
)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(f*x + e)*log(d)/((b^5*f*m^5*n^5*log(f*x + e)
^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*
b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d)
+ 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) - 1/2*(f*x + e)*b^2*m*n*log(c)/(b^5*f*m^
5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n
^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^
4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3) + b^2*m*n*Ei(log(d)/m + log(c)/(m*n) + a/(b
*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(f*x + e)*log(c)/((b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(
f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e
) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(
c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) + 1/2*b^2*n^2*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x +
e))*e^(-a/(b*m*n))*log(d)^2/((b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4
*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*lo
g(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/
(m*n))*d^(1/m)) - 1/2*(f*x + e)*a*b*m*n/(b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) +
2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^
4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3
*n^3) + a*b*m*n*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(f*x + e)/((b^5*f*m^5
*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^
5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4
*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) + b^2*n*Ei(log(d)/m + l
og(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(c)*log(d)/((b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*
m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4
*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*
m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) + 1/2*b^2*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + lo
g(f*x + e))*e^(-a/(b*m*n))*log(c)^2/((b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b
^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*l
og(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^
3)*c^(1/(m*n))*d^(1/m)) + a*b*n*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(d)/(
(b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^
5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2
+ 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) + a*b*Ei(log(
d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*log(c*(d*(e + f*x)^m)^n))^3,x)

[Out]

int(1/(a + b*log(c*(d*(e + f*x)^m)^n))^3, x)

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