3.2.1 \(\int \frac {(a+b \log (c x^n))^2 \log (d (e+f x^2)^m)}{x} \, dx\) [101]

Optimal. Leaf size=147 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{3 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^2}{e}\right )}{3 b n}-\frac {1}{2} m \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )+\frac {1}{2} b m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x^2}{e}\right )-\frac {1}{4} b^2 m n^2 \text {Li}_4\left (-\frac {f x^2}{e}\right ) \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2422, 2375, 2421, 2430, 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 )^2+\frac {1}{2} b m n \text {PolyLog}\left (3,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b^2 m n^2 \text {PolyLog}\left (4,-\frac {f x^2}{e}\right )+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{3 b n}-\frac {m \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/(3*b*n) - (m*(a + b*Log[c*x^n])^3*Log[1 + (f*x^2)/e])/(3*b*n) - (m
*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*x^2)/e)])/2 + (b*m*n*(a + b*Log[c*x^n])*PolyLog[3, -((f*x^2)/e)])/2 - (b
^2*m*n^2*PolyLog[4, -((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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

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 )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{3 b n}-\frac {(2 f m) \int \frac {x \left (a+b \log \left (c x^n\right )\right )^3}{e+f x^2} \, dx}{3 b n}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{3 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^2}{e}\right )}{3 b n}+m \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{3 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^2}{e}\right )}{3 b n}-\frac {1}{2} m \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )+(b m n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{3 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^2}{e}\right )}{3 b n}-\frac {1}{2} m \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )+\frac {1}{2} b m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x^2}{e}\right )-\frac {1}{2} \left (b^2 m n^2\right ) \int \frac {\text {Li}_3\left (-\frac {f x^2}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{3 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^2}{e}\right )}{3 b n}-\frac {1}{2} m \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )+\frac {1}{2} b m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x^2}{e}\right )-\frac {1}{4} b^2 m n^2 \text {Li}_4\left (-\frac {f x^2}{e}\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.13, size = 736, normalized size = 5.01 \begin {gather*} -a^2 m \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+a b m n \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-\frac {1}{3} b^2 m n^2 \log ^3(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 a b m \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+b^2 m n \log ^2(x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-b^2 m \log (x) \log ^2\left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-a^2 m \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+a b m n \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-\frac {1}{3} b^2 m n^2 \log ^3(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 a b m \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+b^2 m n \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-b^2 m \log (x) \log ^2\left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+a^2 \log (x) \log \left (d \left (e+f x^2\right )^m\right )-a b n \log ^2(x) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} b^2 n^2 \log ^3(x) \log \left (d \left (e+f x^2\right )^m\right )+2 a b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-m \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-m \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 a b m n \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 b^2 m n \log \left (c x^n\right ) \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 a b m n \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 b^2 m n \log \left (c x^n\right ) \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b^2 m n^2 \text {Li}_4\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b^2 m n^2 \text {Li}_4\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*m*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]]) + a*b*m*n*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (b^2*m*n^2*
Log[x]^3*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]])/3 - 2*a*b*m*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + b^2*m*
n*Log[x]^2*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - b^2*m*Log[x]*Log[c*x^n]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]
] - a^2*m*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + a*b*m*n*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (b^2*m*n^2
*Log[x]^3*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]])/3 - 2*a*b*m*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + b^2*m
*n*Log[x]^2*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - b^2*m*Log[x]*Log[c*x^n]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e
]] + a^2*Log[x]*Log[d*(e + f*x^2)^m] - a*b*n*Log[x]^2*Log[d*(e + f*x^2)^m] + (b^2*n^2*Log[x]^3*Log[d*(e + f*x^
2)^m])/3 + 2*a*b*Log[x]*Log[c*x^n]*Log[d*(e + f*x^2)^m] - b^2*n*Log[x]^2*Log[c*x^n]*Log[d*(e + f*x^2)^m] + b^2
*Log[x]*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] - m*(a + b*Log[c*x^n])^2*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] - m*(a
+ b*Log[c*x^n])^2*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]] + 2*a*b*m*n*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] + 2*b^2*
m*n*Log[c*x^n]*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] + 2*a*b*m*n*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] + 2*b^2*m*n*
Log[c*x^n]*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] - 2*b^2*m*n^2*PolyLog[4, ((-I)*Sqrt[f]*x)/Sqrt[e]] - 2*b^2*m*n^2*
PolyLog[4, (I*Sqrt[f]*x)/Sqrt[e]]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.25, size = 27214, normalized size = 185.13

method	result	size
risch	\(\text {Expression too large to display}\)	\(27214\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

1/3*(b^2*m*n^2*log(x)^3 + 3*b^2*m*log(x)*log(x^n)^2 - 3*(b^2*m*n*log(c) + a*b*m*n)*log(x)^2 - 3*(b^2*m*n*log(x
)^2 - 2*(b^2*m*log(c) + a*b*m)*log(x))*log(x^n) + 3*(b^2*m*log(c)^2 + 2*a*b*m*log(c) + a^2*m)*log(x))*log(f*x^
2 + e) - integrate(1/3*(2*b^2*f*m*n^2*x^2*log(x)^3 - 6*(b^2*f*m*n*log(c) + a*b*f*m*n)*x^2*log(x)^2 + 6*(b^2*f*
m*log(c)^2 + 2*a*b*f*m*log(c) + a^2*f*m)*x^2*log(x) - 3*(b^2*f*log(c)^2*log(d) + 2*a*b*f*log(c)*log(d) + a^2*f
*log(d))*x^2 + 3*(2*b^2*f*m*x^2*log(x) - b^2*f*x^2*log(d) - b^2*e*log(d))*log(x^n)^2 - 3*(b^2*log(c)^2*log(d)
+ 2*a*b*log(c)*log(d) + a^2*log(d))*e - 6*(b^2*f*m*n*x^2*log(x)^2 - 2*(b^2*f*m*log(c) + a*b*f*m)*x^2*log(x) +
(b^2*f*log(c)*log(d) + a*b*f*log(d))*x^2 + (b^2*log(c)*log(d) + a*b*log(d))*e)*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))^2*log(d*(f*x^2+e)^m)/x,x, algorithm="fricas")

[Out]

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

[Out]

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

[Out]

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

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