3.42 \(\int \frac{(a+b \log (c x^n))^3 \log (d (\frac{1}{d}+f x^2))}{x} \, dx\)

Optimal. Leaf size=101 \[ -\frac{3}{4} b^2 n^2 \text{PolyLog}\left (4,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{3}{4} b n \text{PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac{3}{8} b^3 n^3 \text{PolyLog}\left (5,-d f x^2\right ) \]

[Out]

-((a + b*Log[c*x^n])^3*PolyLog[2, -(d*f*x^2)])/2 + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[3, -(d*f*x^2)])/4 - (3*
b^2*n^2*(a + b*Log[c*x^n])*PolyLog[4, -(d*f*x^2)])/4 + (3*b^3*n^3*PolyLog[5, -(d*f*x^2)])/8

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Rubi [A]  time = 0.0999935, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {2374, 2383, 6589} \[ -\frac{3}{4} b^2 n^2 \text{PolyLog}\left (4,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{3}{4} b n \text{PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac{3}{8} b^3 n^3 \text{PolyLog}\left (5,-d f x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*Log[c*x^n])^3*PolyLog[2, -(d*f*x^2)])/2 + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[3, -(d*f*x^2)])/4 - (3*
b^2*n^2*(a + b*Log[c*x^n])*PolyLog[4, -(d*f*x^2)])/4 + (3*b^3*n^3*PolyLog[5, -(d*f*x^2)])/8

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[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 6589

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 )^3 \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 )^3 \text{Li}_2\left (-d f x^2\right )+\frac{1}{2} (3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^2\right )}{x} \, dx\\ &=-\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-d f x^2\right )+\frac{3}{4} b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-d f x^2\right )-\frac{1}{2} \left (3 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f x^2\right )}{x} \, dx\\ &=-\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-d f x^2\right )+\frac{3}{4} b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-d f x^2\right )-\frac{3}{4} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_4\left (-d f x^2\right )+\frac{1}{4} \left (3 b^3 n^3\right ) \int \frac{\text{Li}_4\left (-d f x^2\right )}{x} \, dx\\ &=-\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-d f x^2\right )+\frac{3}{4} b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-d f x^2\right )-\frac{3}{4} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_4\left (-d f x^2\right )+\frac{3}{8} b^3 n^3 \text{Li}_5\left (-d f x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.312354, size = 754, normalized size = 7.47 \[ \frac{1}{4} \left (4 b^2 n^2 \left (6 \text{PolyLog}\left (4,-i \sqrt{d} \sqrt{f} x\right )+6 \text{PolyLog}\left (4,i \sqrt{d} \sqrt{f} x\right )+3 \log ^2(x) \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+3 \log ^2(x) \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )-6 \log (x) \text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right )-6 \log (x) \text{PolyLog}\left (3,i \sqrt{d} \sqrt{f} x\right )+\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 )\right ) \left (-a-b \log \left (c x^n\right )+b n \log (x)\right )-6 b n \left (-2 \text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right )-2 \text{PolyLog}\left (3,i \sqrt{d} \sqrt{f} x\right )+2 \log (x) \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+2 \log (x) \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\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 )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2-4 \left (\text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+\text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\log (x) \left (\log \left (1-i \sqrt{d} \sqrt{f} x\right )+\log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3-b^3 n^3 \left (-24 \text{PolyLog}\left (5,-i \sqrt{d} \sqrt{f} x\right )-24 \text{PolyLog}\left (5,i \sqrt{d} \sqrt{f} x\right )+4 \log ^3(x) \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+4 \log ^3(x) \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )-12 \log ^2(x) \text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right )-12 \log ^2(x) \text{PolyLog}\left (3,i \sqrt{d} \sqrt{f} x\right )+24 \log (x) \text{PolyLog}\left (4,-i \sqrt{d} \sqrt{f} x\right )+24 \log (x) \text{PolyLog}\left (4,i \sqrt{d} \sqrt{f} x\right )+\log ^4(x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )+\log ^4(x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )-\log (x) \log \left (d f x^2+1\right ) \left (-4 b^2 n^2 \log ^2(x) \left (a+b \log \left (c x^n\right )\right )+6 b n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-4 \left (a+b \log \left (c x^n\right )\right )^3+b^3 n^3 \log ^3(x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Log[x]*(b^3*n^3*Log[x]^3 - 4*b^2*n^2*Log[x]^2*(a + b*Log[c*x^n]) + 6*b*n*Log[x]*(a + b*Log[c*x^n])^2 - 4*(a
+ b*Log[c*x^n])^3)*Log[1 + d*f*x^2]) - 4*(a - b*n*Log[x] + b*Log[c*x^n])^3*(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]) -
6*b*n*(a - b*n*Log[x] + b*Log[c*x^n])^2*(Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + Log[x]^2*Log[1 + I*Sqrt[d]*Sq
rt[f]*x] + 2*Log[x]*PolyLog[2, (-I)*Sqrt[d]*Sqrt[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*Sqrt[d]*Sqrt[f]*x]) + 4*b^2*n^2*(-a + b*n*Log[x] - b*Log[c*x^n])*(
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)*Sqr
t[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]) - b^3*n^3*(Log[x]^4*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + Log[x]^4*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + 4*Log[x]^3*Po
lyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] + 4*Log[x]^3*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] - 12*Log[x]^2*PolyLog[3, (-I)*Sq
rt[d]*Sqrt[f]*x] - 12*Log[x]^2*PolyLog[3, I*Sqrt[d]*Sqrt[f]*x] + 24*Log[x]*PolyLog[4, (-I)*Sqrt[d]*Sqrt[f]*x]
+ 24*Log[x]*PolyLog[4, I*Sqrt[d]*Sqrt[f]*x] - 24*PolyLog[5, (-I)*Sqrt[d]*Sqrt[f]*x] - 24*PolyLog[5, I*Sqrt[d]*
Sqrt[f]*x]))/4

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Maple [F]  time = 0.113, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ({d}^{-1}+f{x}^{2} \right ) \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b^3*n^3*log(x)^4 - 4*b^3*log(x)*log(x^n)^3 - 4*(b^3*n^2*log(c) + a*b^2*n^2)*log(x)^3 + 6*(b^3*n*log(c)^2
+ 2*a*b^2*n*log(c) + a^2*b*n)*log(x)^2 + 6*(b^3*n*log(x)^2 - 2*(b^3*log(c) + a*b^2)*log(x))*log(x^n)^2 - 4*(b
^3*n^2*log(x)^3 - 3*(b^3*n*log(c) + a*b^2*n)*log(x)^2 + 3*(b^3*log(c)^2 + 2*a*b^2*log(c) + a^2*b)*log(x))*log(
x^n) - 4*(b^3*log(c)^3 + 3*a*b^2*log(c)^2 + 3*a^2*b*log(c) + a^3)*log(x))*log(d*f*x^2 + 1) - integrate(-1/2*(b
^3*d*f*n^3*x*log(x)^4 - 4*b^3*d*f*x*log(x)*log(x^n)^3 - 4*(b^3*d*f*n^2*log(c) + a*b^2*d*f*n^2)*x*log(x)^3 + 6*
(b^3*d*f*n*log(c)^2 + 2*a*b^2*d*f*n*log(c) + a^2*b*d*f*n)*x*log(x)^2 - 4*(b^3*d*f*log(c)^3 + 3*a*b^2*d*f*log(c
)^2 + 3*a^2*b*d*f*log(c) + a^3*d*f)*x*log(x) + 6*(b^3*d*f*n*x*log(x)^2 - 2*(b^3*d*f*log(c) + a*b^2*d*f)*x*log(
x))*log(x^n)^2 - 4*(b^3*d*f*n^2*x*log(x)^3 - 3*(b^3*d*f*n*log(c) + a*b^2*d*f*n)*x*log(x)^2 + 3*(b^3*d*f*log(c)
^2 + 2*a*b^2*d*f*log(c) + a^2*b*d*f)*x*log(x))*log(x^n))/(d*f*x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{3} \log \left (d f x^{2} + 1\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^3*log(d*f*x^2 + 1)*log(c*x^n)^3 + 3*a*b^2*log(d*f*x^2 + 1)*log(c*x^n)^2 + 3*a^2*b*log(d*f*x^2 + 1)
*log(c*x^n) + a^3*log(d*f*x^2 + 1))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**3*ln(d*(1/d+f*x**2))/x,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac{1}{d}\right )} d\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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