3.1.34 \(\int \frac {(a+b \log (c x^n))^2 \log (d (\frac {1}{d}+f x^2))}{x} \, dx\) [34]

Optimal. Leaf size=70 \[ -\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )+\frac {1}{2} b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )-\frac {1}{4} b^2 n^2 \text {Li}_4\left (-d f x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2421, 2430, 6724} \begin {gather*} \frac {1}{2} b n \text {PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} \text {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{4} b^2 n^2 \text {PolyLog}\left (4,-d f x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.13, size = 484, normalized size = 6.91 \begin {gather*} \frac {1}{3} \left (\log (x) \left (b^2 n^2 \log ^2(x)-3 b n \log (x) \left (a+b \log \left (c x^n\right )\right )+3 \left (a+b \log \left (c x^n\right )\right )^2\right ) \log \left (1+d f x^2\right )-3 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\log (x) \left (\log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )+\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\right )+3 b n \left (-a+b n \log (x)-b \log \left (c x^n\right )\right ) \left (\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-2 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )-2 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )\right )-b^2 n^2 \left (\log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_4\left (-i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_4\left (i \sqrt {d} \sqrt {f} x\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[x]*(b^2*n^2*Log[x]^2 - 3*b*n*Log[x]*(a + b*Log[c*x^n]) + 3*(a + b*Log[c*x^n])^2)*Log[1 + d*f*x^2] - 3*(a
- b*n*Log[x] + b*Log[c*x^n])^2*(Log[x]*(Log[1 - I*Sqrt[d]*Sqrt[f]*x] + Log[1 + I*Sqrt[d]*Sqrt[f]*x]) + PolyLog
[2, (-I)*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]) + 3*b*n*(-a + b*n*Log[x] - b*Log[c*x^n])*(Log[x
]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + Log[x]^2*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + 2*Log[x]*PolyLog[2, (-I)*Sqrt[d]*Sq
rt[f]*x] + 2*Log[x]*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] - 2*PolyLog[3, (-I)*Sqrt[d]*Sqrt[f]*x] - 2*PolyLog[3, I*Sq
rt[d]*Sqrt[f]*x]) - b^2*n^2*(Log[x]^3*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + Log[x]^3*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + 3
*Log[x]^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] + 3*Log[x]^2*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] - 6*Log[x]*PolyLog[3
, (-I)*Sqrt[d]*Sqrt[f]*x] - 6*Log[x]*PolyLog[3, I*Sqrt[d]*Sqrt[f]*x] + 6*PolyLog[4, (-I)*Sqrt[d]*Sqrt[f]*x] +
6*PolyLog[4, I*Sqrt[d]*Sqrt[f]*x]))/3

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.22, size = 6180, normalized size = 88.29

method	result	size
risch	\(\text {Expression too large to display}\)	\(6180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^2*ln(d*(1/d+f*x^2))/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*(1/d+f*x^2))/x,x, algorithm="maxima")

[Out]

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

[Out]

integral((b^2*log(d*f*x^2 + 1)*log(c*x^n)^2 + 2*a*b*log(d*f*x^2 + 1)*log(c*x^n) + a^2*log(d*f*x^2 + 1))/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*(1/d+f*x**2))/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*(1/d+f*x^2))/x,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^2*log((f*x^2 + 1/d)*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+\frac {1}{d}\right )\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*(f*x^2 + 1/d))*(a + b*log(c*x^n))^2)/x,x)

[Out]

int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^2)/x, x)

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