3.31 \(\int \frac{(a+b \log (c x^n)) \log (d (\frac{1}{d}+f x^2))}{x^4} \, dx\)

Optimal. Leaf size=211 \[ \frac{1}{3} i b d^{3/2} f^{3/2} n \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )-\frac{1}{3} i b d^{3/2} f^{3/2} n \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )-\frac{2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac{\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{2}{9} b d^{3/2} f^{3/2} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )-\frac{b n \log \left (d f x^2+1\right )}{9 x^3}-\frac{8 b d f n}{9 x} \]

[Out]

(-8*b*d*f*n)/(9*x) - (2*b*d^(3/2)*f^(3/2)*n*ArcTan[Sqrt[d]*Sqrt[f]*x])/9 - (2*d*f*(a + b*Log[c*x^n]))/(3*x) -
(2*d^(3/2)*f^(3/2)*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]))/3 - (b*n*Log[1 + d*f*x^2])/(9*x^3) - ((a + b*
Log[c*x^n])*Log[1 + d*f*x^2])/(3*x^3) + (I/3)*b*d^(3/2)*f^(3/2)*n*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - (I/3)*b
*d^(3/2)*f^(3/2)*n*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]

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Rubi [A]  time = 0.139932, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {2455, 325, 205, 2376, 4848, 2391, 203} \[ \frac{1}{3} i b d^{3/2} f^{3/2} n \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )-\frac{1}{3} i b d^{3/2} f^{3/2} n \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )-\frac{2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac{\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{2}{9} b d^{3/2} f^{3/2} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )-\frac{b n \log \left (d f x^2+1\right )}{9 x^3}-\frac{8 b d f n}{9 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b*d*f*n)/(9*x) - (2*b*d^(3/2)*f^(3/2)*n*ArcTan[Sqrt[d]*Sqrt[f]*x])/9 - (2*d*f*(a + b*Log[c*x^n]))/(3*x) -
(2*d^(3/2)*f^(3/2)*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]))/3 - (b*n*Log[1 + d*f*x^2])/(9*x^3) - ((a + b*
Log[c*x^n])*Log[1 + d*f*x^2])/(3*x^3) + (I/3)*b*d^(3/2)*f^(3/2)*n*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - (I/3)*b
*d^(3/2)*f^(3/2)*n*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac{1}{d}+f x^2\right )\right )}{x^4} \, dx &=-\frac{2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac{2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{3 x^3}-(b n) \int \left (-\frac{2 d f}{3 x^2}-\frac{2 d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{3 x}-\frac{\log \left (1+d f x^2\right )}{3 x^4}\right ) \, dx\\ &=-\frac{2 b d f n}{3 x}-\frac{2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac{2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{3 x^3}+\frac{1}{3} (b n) \int \frac{\log \left (1+d f x^2\right )}{x^4} \, dx+\frac{1}{3} \left (2 b d^{3/2} f^{3/2} n\right ) \int \frac{\tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{x} \, dx\\ &=-\frac{2 b d f n}{3 x}-\frac{2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac{2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f x^2\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{3 x^3}+\frac{1}{9} (2 b d f n) \int \frac{1}{x^2 \left (1+d f x^2\right )} \, dx+\frac{1}{3} \left (i b d^{3/2} f^{3/2} n\right ) \int \frac{\log \left (1-i \sqrt{d} \sqrt{f} x\right )}{x} \, dx-\frac{1}{3} \left (i b d^{3/2} f^{3/2} n\right ) \int \frac{\log \left (1+i \sqrt{d} \sqrt{f} x\right )}{x} \, dx\\ &=-\frac{8 b d f n}{9 x}-\frac{2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac{2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f x^2\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{3 x^3}+\frac{1}{3} i b d^{3/2} f^{3/2} n \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )-\frac{1}{3} i b d^{3/2} f^{3/2} n \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )-\frac{1}{9} \left (2 b d^2 f^2 n\right ) \int \frac{1}{1+d f x^2} \, dx\\ &=-\frac{8 b d f n}{9 x}-\frac{2}{9} b d^{3/2} f^{3/2} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )-\frac{2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac{2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f x^2\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{3 x^3}+\frac{1}{3} i b d^{3/2} f^{3/2} n \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )-\frac{1}{3} i b d^{3/2} f^{3/2} n \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )\\ \end{align*}

Mathematica [C]  time = 0.181683, size = 285, normalized size = 1.35 \[ \frac{2}{3} b d f n \left (\frac{1}{2} i \sqrt{d} \sqrt{f} \left (\text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+\log (x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )-\frac{1}{2} i \sqrt{d} \sqrt{f} \left (\text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\log (x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )\right )-\frac{1}{x}-\frac{\log (x)}{x}\right )-\frac{2 a d f \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-d f x^2\right )}{3 x}-\frac{a \log \left (d f x^2+1\right )}{3 x^3}-\frac{2}{9} b d^{3/2} f^{3/2} \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )+n\right ) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )-\frac{2 b \left (3 d f \left (\log \left (c x^n\right )-n \log (x)\right )+d f n\right )}{9 x}-\frac{b \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )+3 n \log (x)+n\right ) \log \left (d f x^2+1\right )}{9 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*d*f*Hypergeometric2F1[-1/2, 1, 1/2, -(d*f*x^2)])/(3*x) - (2*b*d^(3/2)*f^(3/2)*ArcTan[Sqrt[d]*Sqrt[f]*x]*
(n + 3*(-(n*Log[x]) + Log[c*x^n])))/9 - (2*b*(d*f*n + 3*d*f*(-(n*Log[x]) + Log[c*x^n])))/(9*x) - (a*Log[1 + d*
f*x^2])/(3*x^3) - (b*(n + 3*n*Log[x] + 3*(-(n*Log[x]) + Log[c*x^n]))*Log[1 + d*f*x^2])/(9*x^3) + (2*b*d*f*n*(-
x^(-1) - Log[x]/x + (I/2)*Sqrt[d]*Sqrt[f]*(Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, (-I)*Sqrt[d]*Sqrt[
f]*x]) - (I/2)*Sqrt[d]*Sqrt[f]*(Log[x]*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])))/3

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ({d}^{-1}+f{x}^{2} \right ) \right ) }{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^2))/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a \log \left (d f x^{2} + 1\right )}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^2))/x^4,x, algorithm="fricas")

[Out]

integral((b*log(d*f*x^2 + 1)*log(c*x^n) + a*log(d*f*x^2 + 1))/x^4, 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))*ln(d*(1/d+f*x**2))/x**4,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 )} \log \left ({\left (f x^{2} + \frac{1}{d}\right )} d\right )}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^2))/x^4,x, algorithm="giac")

[Out]

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