∫ (a+b log(cxn)) log(d(e+fx2)m)_____________________

3.1.92 \(\int \frac {(a+b \log (c x^n)) \log (d (e+f x^2)^m)}{x} \, dx\) [92]

Optimal. Leaf size=113 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{2 b n}-\frac {1}{2} m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )+\frac {1}{4} b m n \text {Li}_3\left (-\frac {f x^2}{e}\right ) \]

[Out]

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

lylog(2,-f*x^2/e)+1/4*b*m*n*polylog(3,-f*x^2/e)

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Rubi [A]
time = 0.08, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2422, 2375, 2421, 6724} \begin {gather*} -\frac {1}{2} m \text {PolyLog}\left (2,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} b m n \text {PolyLog}\left (3,-\frac {f x^2}{e}\right )+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac {m \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

*(a + b*Log[c*x^n])*PolyLog[2, -((f*x^2)/e)])/2 + (b*m*n*PolyLog[3, -((f*x^2)/e)])/4

Rule 2375

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si

mp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c*x^n])^p/(e*r)), x] - Dist[b*f^m*n*(p/(e*r)), Int[Log[1 + e*(x^r/d)]*((

a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &

& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp

[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L

og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2422

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :

> Simp[Log[d*(e + f*x^m)^r]*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Dist[f*m*(r/(b*n*(p + 1))), Int[x

^(m - 1)*((a + b*Log[c*x^n])^(p + 1)/(e + f*x^m)), x], x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p,

0] && NeQ[d*e, 1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac {(f m) \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx}{b n}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{2 b n}+m \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{2 b n}-\frac {1}{2} m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )+\frac {1}{2} (b m n) \int \frac {\text {Li}_2\left (-\frac {f x^2}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{2 b n}-\frac {1}{2} m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )+\frac {1}{4} b m n \text {Li}_3\left (-\frac {f x^2}{e}\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.06, size = 297, normalized size = 2.63 \begin {gather*} \frac {1}{2} \left (b m n \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b m \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+b m n \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b m \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-b n \log ^2(x) \log \left (d \left (e+f x^2\right )^m\right )+a \log \left (-\frac {f x^2}{e}\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 b m \log \left (c x^n\right ) \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b m \log \left (c x^n\right ) \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )+a m \text {Li}_2\left (1+\frac {f x^2}{e}\right )+2 b m n \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 b m n \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b*m*n*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 2*b*m*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + b*m*

n*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 2*b*m*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - b*n*Log[x

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

m] - 2*b*m*Log[c*x^n]*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] - 2*b*m*Log[c*x^n]*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]

] + a*m*PolyLog[2, 1 + (f*x^2)/e] + 2*b*m*n*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] + 2*b*m*n*PolyLog[3, (I*Sqrt[

f]*x)/Sqrt[e]])/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.53, size = 2842, normalized size = 25.15


method	result	size

risch	\(\text {Expression too large to display}\)	\(2842\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)*a*(ln((f*x^2+e)^m)-m*ln(f*x^2+e))-ln(x)*ln((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*m*(ln(x^n)-n*ln(x))+ln(x)*

ln(f*x^2+e)*b*m*(ln(x^n)-n*ln(x))+1/2*I*m*dilog((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*Pi*csgn(I*c*x^n)^3+1/2*I*m

*dilog((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*Pi*csgn(I*c*x^n)^3-1/4*I*Pi*csgn(I*d*(f*x^2+e)^m)^3*b/n*ln(x^n)^2+1/

2*I*Pi*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*a*ln(x)-1/2*I*m*dilog((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*Pi*csgn(I*c

)*csgn(I*c*x^n)^2-1/4*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*ln(x)*b*csgn(I*c)*csgn(I*c*x^n)^2-1/4*Pi^2*csgn(I

*d)*csgn(I*d*(f*x^2+e)^m)^2*ln(x)*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*ln(d)*b/n*ln(x^n)^2-ln(x)*ln((-f*x+(-e*f)^

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

ln(c)-1/2*I*m*dilog((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*m*dilog((f*x+(-e*f)^

(1/2))/(-e*f)^(1/2))*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*m*ln(x)*ln((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*Pi*

csgn(I*c*x^n)^3+1/2*I*m*ln(x)*ln((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*Pi*csgn(I*c*x^n)^3+1/4*Pi^2*csgn(I*d)*csgn

(I*d*(f*x^2+e)^m)^2*ln(x)*b*csgn(I*c*x^n)^3+1/4*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*ln(x)*b*csgn(

I*c*x^n)^3+1/4*Pi^2*csgn(I*d*(f*x^2+e)^m)^3*ln(x)*b*csgn(I*c)*csgn(I*c*x^n)^2+1/4*Pi^2*csgn(I*d*(f*x^2+e)^m)^3

*ln(x)*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*Pi^2*csgn(I*d*(f*x^2+e)^m)^3*ln(x)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x

^n)-1/4*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*ln(x)*b*csgn(I*c*x^n)^3+1/2*I*ln(x)*Pi*ln(f*x

^2+e)*b*m*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*m*ln(x)*ln((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*Pi*csgn(I*c)*csgn(I

*c*x^n)^2-1/2*I*ln(x)*Pi*ln(f*x^2+e)*b*m*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*m*ln(x)*ln((-f*x+(-e*f)^(1/

2))/(-e*f)^(1/2))*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-m*dilog((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*a-m*dilog

((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*a+1/4*I*Pi*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*b/n*ln(x^n)^2-1/2*I*P

i*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*a*ln(x)-m*ln(x)*ln((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*a-m

*ln(x)*ln((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*a+ln(x)*ln(f*x^2+e)*a*m-m*dilog((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b

*ln(c)-m*dilog((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*ln(c)+1/2*I*ln(d)*ln(x)*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2

*b*n*m*ln(x)^2*ln(f*x^2+e)-1/2*b*n*m*ln(x)^2*ln(1+f*x^2/e)-1/2*b*n*m*ln(x)*polylog(2,-f*x^2/e)-m*ln(x)*ln((-f*

x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*ln(c)-m*ln(x)*ln((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*ln(c)+ln(x)*ln(c)*ln(f*x^2

+e)*b*m-1/2*I*ln(x)*Pi*b*csgn(I*c*x^n)^3*(ln((f*x^2+e)^m)-m*ln(f*x^2+e))+1/2*I*m*ln(x)*ln((f*x+(-e*f)^(1/2))/(

-e*f)^(1/2))*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/4*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e

)^m)*ln(x)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*ln(d)*ln(x)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*Pi*csg

n(I*d)*csgn(I*d*(f*x^2+e)^m)^2*ln(x)*b*ln(c)+1/2*I*ln(x)*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2*(ln((f*x^2+e)^m)-m*l

n(f*x^2+e))+1/2*I*ln(x)*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2*(ln((f*x^2+e)^m)-m*ln(f*x^2+e))-1/2*I*ln(x)*Pi*ln(f*x^2

+e)*b*m*csgn(I*c*x^n)^3+1/4*I*Pi*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*b/n*ln(x^n)^2+1/2*I*Pi*csgn(I*(f*x^2+e)^m)*

csgn(I*d*(f*x^2+e)^m)^2*ln(x)*b*ln(c)+1/4*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*ln(x)*b*csgn(I*c)*c

sgn(I*x^n)*csgn(I*c*x^n)-1/2*I*ln(x)*Pi*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(ln((f*x^2+e)^m)-m*ln(f*x^2+e))+

1/4*b*m*n*polylog(3,-f*x^2/e)-1/2*I*ln(d)*ln(x)*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I*m*ln(x)*ln((-f*

x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*m*ln(x)*ln((f*x+(-e*f)^(1/2))/(-e*f)^(1/2

))*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/4*Pi^2*csgn(I*d*(f*x^2+e)^m)^3*ln(x)*b*csgn(I*c*x^n)^3-1/2*I*Pi*csgn(I*d*(

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

2+e)^m)*csgn(I*d*(f*x^2+e)^m)*ln(x)*b*ln(c)+1/2*I*Pi*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*a*ln(x)-1/2*I

*Pi*csgn(I*d*(f*x^2+e)^m)^3*ln(x)*b*ln(c)-1/4*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*ln(x)*b*csgn(I*

c)*csgn(I*c*x^n)^2-1/4*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*ln(x)*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/

4*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*ln(x)*b*csgn(I*c)*csgn(I*c*x^n)^2+1/4*Pi^2*csgn(I*d

)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*ln(x)*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*Pi^2*csgn(I*d)*csgn(I*d*(f

*x^2+e)^m)^2*ln(x)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/4*I*Pi*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^

2+e)^m)*b/n*ln(x^n)^2-1/2*I*m*dilog((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*m

*ln(x)*ln((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*m*dilog((-f*x+(-e*f)^(1/2))/

(-e*f)^(1/2))*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*m*dilog((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*Pi*csg

n(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-dilog((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b*m*(ln(x^n)-n*ln(x))-dilog((f*x+(-e*

f)^(1/2))/(-e*f)^(1/2))*b*m*(ln(x^n)-n*ln(x))+1...

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

[Out]

-1/2*(b*m*n*log(x)^2 - 2*b*m*log(x)*log(x^n) - 2*(b*m*log(c) + a*m)*log(x))*log(f*x^2 + e) - integrate(-(b*f*m

*n*x^2*log(x)^2 - 2*(b*f*m*log(c) + a*f*m)*x^2*log(x) + (b*f*log(c)*log(d) + a*f*log(d))*x^2 + (b*log(c)*log(d

) + a*log(d))*e - (2*b*f*m*x^2*log(x) - b*f*x^2*log(d) - b*e*log(d))*log(x^n))/(f*x^3 + x*e), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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