3.193 \(\int (d+e x^2) \coth ^{-1}(a x) \log (c x^n) \, dx\)

Optimal. Leaf size=180 \[ \frac{n \left (3 a^2 d+e\right ) \text{PolyLog}\left (2,a^2 x^2\right )}{12 a^3}+\frac{\left (3 a^2 d+e\right ) \log \left (1-a^2 x^2\right ) \log \left (c x^n\right )}{6 a^3}-\frac{d n \log \left (1-a^2 x^2\right )}{2 a}-\frac{e n \log \left (1-a^2 x^2\right )}{18 a^3}+d x \coth ^{-1}(a x) \log \left (c x^n\right )+\frac{e x^2 \log \left (c x^n\right )}{6 a}+\frac{1}{3} e x^3 \coth ^{-1}(a x) \log \left (c x^n\right )-d n x \coth ^{-1}(a x)-\frac{5 e n x^2}{36 a}-\frac{1}{9} e n x^3 \coth ^{-1}(a x) \]

[Out]

(-5*e*n*x^2)/(36*a) - d*n*x*ArcCoth[a*x] - (e*n*x^3*ArcCoth[a*x])/9 + (e*x^2*Log[c*x^n])/(6*a) + d*x*ArcCoth[a
*x]*Log[c*x^n] + (e*x^3*ArcCoth[a*x]*Log[c*x^n])/3 - (d*n*Log[1 - a^2*x^2])/(2*a) - (e*n*Log[1 - a^2*x^2])/(18
*a^3) + ((3*a^2*d + e)*Log[c*x^n]*Log[1 - a^2*x^2])/(6*a^3) + ((3*a^2*d + e)*n*PolyLog[2, a^2*x^2])/(12*a^3)

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Rubi [A]  time = 0.156183, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {5977, 1593, 444, 43, 2388, 5911, 260, 5917, 266, 2391} \[ \frac{n \left (3 a^2 d+e\right ) \text{PolyLog}\left (2,a^2 x^2\right )}{12 a^3}+\frac{\left (3 a^2 d+e\right ) \log \left (1-a^2 x^2\right ) \log \left (c x^n\right )}{6 a^3}-\frac{d n \log \left (1-a^2 x^2\right )}{2 a}-\frac{e n \log \left (1-a^2 x^2\right )}{18 a^3}+d x \coth ^{-1}(a x) \log \left (c x^n\right )+\frac{e x^2 \log \left (c x^n\right )}{6 a}+\frac{1}{3} e x^3 \coth ^{-1}(a x) \log \left (c x^n\right )-d n x \coth ^{-1}(a x)-\frac{5 e n x^2}{36 a}-\frac{1}{9} e n x^3 \coth ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*ArcCoth[a*x]*Log[c*x^n],x]

[Out]

(-5*e*n*x^2)/(36*a) - d*n*x*ArcCoth[a*x] - (e*n*x^3*ArcCoth[a*x])/9 + (e*x^2*Log[c*x^n])/(6*a) + d*x*ArcCoth[a
*x]*Log[c*x^n] + (e*x^3*ArcCoth[a*x]*Log[c*x^n])/3 - (d*n*Log[1 - a^2*x^2])/(2*a) - (e*n*Log[1 - a^2*x^2])/(18
*a^3) + ((3*a^2*d + e)*Log[c*x^n]*Log[1 - a^2*x^2])/(6*a^3) + ((3*a^2*d + e)*n*PolyLog[2, a^2*x^2])/(12*a^3)

Rule 5977

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^
2)^q, x]}, Dist[a + b*ArcCoth[c*x], u, x] - Dist[b*c, Int[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x
] && (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2388

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(Px_.)*(F_)[(d_.)*((e_.) + (f_.)*(x_))], x_Symbol] :> With[{u = IntH
ide[Px*F[d*(e + f*x)], 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, n}, x] && PolynomialQ[Px, x] && MemberQ[{ArcTan, ArcCot, ArcTanh, ArcCoth}, F]

Rule 5911

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcCoth[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 5917

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
oth[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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]

Rubi steps

\begin{align*} \int \left (d+e x^2\right ) \coth ^{-1}(a x) \log \left (c x^n\right ) \, dx &=\frac{e x^2 \log \left (c x^n\right )}{6 a}+d x \coth ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \coth ^{-1}(a x) \log \left (c x^n\right )+\frac{\left (3 a^2 d+e\right ) \log \left (c x^n\right ) \log \left (1-a^2 x^2\right )}{6 a^3}-n \int \left (\frac{e x}{6 a}+d \coth ^{-1}(a x)+\frac{1}{3} e x^2 \coth ^{-1}(a x)+\frac{\left (3 a^2 d+e\right ) \log \left (1-a^2 x^2\right )}{6 a^3 x}\right ) \, dx\\ &=-\frac{e n x^2}{12 a}+\frac{e x^2 \log \left (c x^n\right )}{6 a}+d x \coth ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \coth ^{-1}(a x) \log \left (c x^n\right )+\frac{\left (3 a^2 d+e\right ) \log \left (c x^n\right ) \log \left (1-a^2 x^2\right )}{6 a^3}-(d n) \int \coth ^{-1}(a x) \, dx-\frac{1}{3} (e n) \int x^2 \coth ^{-1}(a x) \, dx-\frac{\left (\left (3 a^2 d+e\right ) n\right ) \int \frac{\log \left (1-a^2 x^2\right )}{x} \, dx}{6 a^3}\\ &=-\frac{e n x^2}{12 a}-d n x \coth ^{-1}(a x)-\frac{1}{9} e n x^3 \coth ^{-1}(a x)+\frac{e x^2 \log \left (c x^n\right )}{6 a}+d x \coth ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \coth ^{-1}(a x) \log \left (c x^n\right )+\frac{\left (3 a^2 d+e\right ) \log \left (c x^n\right ) \log \left (1-a^2 x^2\right )}{6 a^3}+\frac{\left (3 a^2 d+e\right ) n \text{Li}_2\left (a^2 x^2\right )}{12 a^3}+(a d n) \int \frac{x}{1-a^2 x^2} \, dx+\frac{1}{9} (a e n) \int \frac{x^3}{1-a^2 x^2} \, dx\\ &=-\frac{e n x^2}{12 a}-d n x \coth ^{-1}(a x)-\frac{1}{9} e n x^3 \coth ^{-1}(a x)+\frac{e x^2 \log \left (c x^n\right )}{6 a}+d x \coth ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \coth ^{-1}(a x) \log \left (c x^n\right )-\frac{d n \log \left (1-a^2 x^2\right )}{2 a}+\frac{\left (3 a^2 d+e\right ) \log \left (c x^n\right ) \log \left (1-a^2 x^2\right )}{6 a^3}+\frac{\left (3 a^2 d+e\right ) n \text{Li}_2\left (a^2 x^2\right )}{12 a^3}+\frac{1}{18} (a e n) \operatorname{Subst}\left (\int \frac{x}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac{e n x^2}{12 a}-d n x \coth ^{-1}(a x)-\frac{1}{9} e n x^3 \coth ^{-1}(a x)+\frac{e x^2 \log \left (c x^n\right )}{6 a}+d x \coth ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \coth ^{-1}(a x) \log \left (c x^n\right )-\frac{d n \log \left (1-a^2 x^2\right )}{2 a}+\frac{\left (3 a^2 d+e\right ) \log \left (c x^n\right ) \log \left (1-a^2 x^2\right )}{6 a^3}+\frac{\left (3 a^2 d+e\right ) n \text{Li}_2\left (a^2 x^2\right )}{12 a^3}+\frac{1}{18} (a e n) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2}-\frac{1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{5 e n x^2}{36 a}-d n x \coth ^{-1}(a x)-\frac{1}{9} e n x^3 \coth ^{-1}(a x)+\frac{e x^2 \log \left (c x^n\right )}{6 a}+d x \coth ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \coth ^{-1}(a x) \log \left (c x^n\right )-\frac{d n \log \left (1-a^2 x^2\right )}{2 a}-\frac{e n \log \left (1-a^2 x^2\right )}{18 a^3}+\frac{\left (3 a^2 d+e\right ) \log \left (c x^n\right ) \log \left (1-a^2 x^2\right )}{6 a^3}+\frac{\left (3 a^2 d+e\right ) n \text{Li}_2\left (a^2 x^2\right )}{12 a^3}\\ \end{align*}

Mathematica [A]  time = 0.129761, size = 178, normalized size = 0.99 \[ \frac{3 n \left (3 a^2 d+e\right ) \text{PolyLog}\left (2,a^2 x^2\right )-4 a^3 x \coth ^{-1}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+18 a^2 d \log \left (1-a^2 x^2\right ) \log \left (c x^n\right )+6 a^2 e x^2 \log \left (c x^n\right )+6 e \log \left (1-a^2 x^2\right ) \log \left (c x^n\right )+36 a^2 d n \log \left (\frac{1}{a x \sqrt{1-\frac{1}{a^2 x^2}}}\right )-5 a^2 e n x^2-2 e n \log \left (a^2 x^2-1\right )}{36 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)*ArcCoth[a*x]*Log[c*x^n],x]

[Out]

(-5*a^2*e*n*x^2 + 36*a^2*d*n*Log[1/(a*Sqrt[1 - 1/(a^2*x^2)]*x)] + 6*a^2*e*x^2*Log[c*x^n] - 4*a^3*x*ArcCoth[a*x
]*(n*(9*d + e*x^2) - 3*(3*d + e*x^2)*Log[c*x^n]) + 18*a^2*d*Log[c*x^n]*Log[1 - a^2*x^2] + 6*e*Log[c*x^n]*Log[1
- a^2*x^2] - 2*e*n*Log[-1 + a^2*x^2] + 3*(3*a^2*d + e)*n*PolyLog[2, a^2*x^2])/(36*a^3)

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Maple [F]  time = 5.356, size = 0, normalized size = 0. \begin{align*} \int \left ( e{x}^{2}+d \right ){\rm arccoth} \left (ax\right )\ln \left ( c{x}^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*arccoth(a*x)*ln(c*x^n),x)

[Out]

int((e*x^2+d)*arccoth(a*x)*ln(c*x^n),x)

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Maxima [A]  time = 1.4611, size = 431, normalized size = 2.39 \begin{align*} -\frac{1}{36} \, n{\left (\frac{6 \,{\left (3 \, a^{2} d + e\right )}{\left (\log \left (a x - 1\right ) \log \left (a x\right ) +{\rm Li}_2\left (-a x + 1\right )\right )}}{a^{3}} + \frac{6 \,{\left (3 \, a^{2} d + e\right )}{\left (\log \left (a x + 1\right ) \log \left (-a x\right ) +{\rm Li}_2\left (a x + 1\right )\right )}}{a^{3}} + \frac{2 \,{\left (9 \, a^{2} d + e\right )} \log \left (a x + 1\right )}{a^{3}} + \frac{5 \, a^{2} e x^{2} + 2 \,{\left (a^{3} e x^{3} + 9 \, a^{3} d x\right )} \log \left (a x + 1\right ) - 2 \,{\left (a^{3} e x^{3} + 9 \, a^{3} d x - 9 \, a^{2} d - e\right )} \log \left (a x - 1\right )}{a^{3}}\right )} + \frac{1}{12} \,{\left (6 \,{\left (x \log \left (\frac{1}{a x} + 1\right ) + \frac{\log \left (a x + 1\right )}{a}\right )} d - 6 \,{\left (x \log \left (-\frac{1}{a x} + 1\right ) - \frac{\log \left (a x - 1\right )}{a}\right )} d +{\left (2 \, x^{3} \log \left (\frac{1}{a x} + 1\right ) + \frac{\frac{a x^{2} - 2 \, x}{a} + \frac{2 \, \log \left (a x + 1\right )}{a^{2}}}{a}\right )} e -{\left (2 \, x^{3} \log \left (-\frac{1}{a x} + 1\right ) - \frac{\frac{a x^{2} + 2 \, x}{a} + \frac{2 \, \log \left (a x - 1\right )}{a^{2}}}{a}\right )} e\right )} \log \left (c x^{n}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*arccoth(a*x)*log(c*x^n),x, algorithm="maxima")

[Out]

-1/36*n*(6*(3*a^2*d + e)*(log(a*x - 1)*log(a*x) + dilog(-a*x + 1))/a^3 + 6*(3*a^2*d + e)*(log(a*x + 1)*log(-a*
x) + dilog(a*x + 1))/a^3 + 2*(9*a^2*d + e)*log(a*x + 1)/a^3 + (5*a^2*e*x^2 + 2*(a^3*e*x^3 + 9*a^3*d*x)*log(a*x
+ 1) - 2*(a^3*e*x^3 + 9*a^3*d*x - 9*a^2*d - e)*log(a*x - 1))/a^3) + 1/12*(6*(x*log(1/(a*x) + 1) + log(a*x + 1
)/a)*d - 6*(x*log(-1/(a*x) + 1) - log(a*x - 1)/a)*d + (2*x^3*log(1/(a*x) + 1) + ((a*x^2 - 2*x)/a + 2*log(a*x +
1)/a^2)/a)*e - (2*x^3*log(-1/(a*x) + 1) - ((a*x^2 + 2*x)/a + 2*log(a*x - 1)/a^2)/a)*e)*log(c*x^n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*arccoth(a*x)*log(c*x^n),x, algorithm="fricas")

[Out]

integral((e*x^2 + d)*arccoth(a*x)*log(c*x^n), 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((e*x**2+d)*acoth(a*x)*ln(c*x**n),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*arccoth(a*x)*log(c*x^n),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)*arccoth(a*x)*log(c*x^n), x)