3.12 \(\int x^5 \coth ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=105 \[ \frac{x^4}{60 a^2}+\frac{4 x^2}{45 a^4}+\frac{23 \log \left (1-a^2 x^2\right )}{90 a^6}+\frac{x^3 \coth ^{-1}(a x)}{9 a^3}+\frac{x \coth ^{-1}(a x)}{3 a^5}-\frac{\coth ^{-1}(a x)^2}{6 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^2+\frac{x^5 \coth ^{-1}(a x)}{15 a} \]

[Out]

(4*x^2)/(45*a^4) + x^4/(60*a^2) + (x*ArcCoth[a*x])/(3*a^5) + (x^3*ArcCoth[a*x])/(9*a^3) + (x^5*ArcCoth[a*x])/(
15*a) - ArcCoth[a*x]^2/(6*a^6) + (x^6*ArcCoth[a*x]^2)/6 + (23*Log[1 - a^2*x^2])/(90*a^6)

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Rubi [A]  time = 0.245691, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5917, 5981, 266, 43, 5911, 260, 5949} \[ \frac{x^4}{60 a^2}+\frac{4 x^2}{45 a^4}+\frac{23 \log \left (1-a^2 x^2\right )}{90 a^6}+\frac{x^3 \coth ^{-1}(a x)}{9 a^3}+\frac{x \coth ^{-1}(a x)}{3 a^5}-\frac{\coth ^{-1}(a x)^2}{6 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^2+\frac{x^5 \coth ^{-1}(a x)}{15 a} \]

Antiderivative was successfully verified.

[In]

Int[x^5*ArcCoth[a*x]^2,x]

[Out]

(4*x^2)/(45*a^4) + x^4/(60*a^2) + (x*ArcCoth[a*x])/(3*a^5) + (x^3*ArcCoth[a*x])/(9*a^3) + (x^5*ArcCoth[a*x])/(
15*a) - ArcCoth[a*x]^2/(6*a^6) + (x^6*ArcCoth[a*x]^2)/6 + (23*Log[1 - a^2*x^2])/(90*a^6)

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 5981

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcCoth[c*x])
^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[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 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 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 5949

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^5 \coth ^{-1}(a x)^2 \, dx &=\frac{1}{6} x^6 \coth ^{-1}(a x)^2-\frac{1}{3} a \int \frac{x^6 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac{1}{6} x^6 \coth ^{-1}(a x)^2+\frac{\int x^4 \coth ^{-1}(a x) \, dx}{3 a}-\frac{\int \frac{x^4 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a}\\ &=\frac{x^5 \coth ^{-1}(a x)}{15 a}+\frac{1}{6} x^6 \coth ^{-1}(a x)^2-\frac{1}{15} \int \frac{x^5}{1-a^2 x^2} \, dx+\frac{\int x^2 \coth ^{-1}(a x) \, dx}{3 a^3}-\frac{\int \frac{x^2 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^3}\\ &=\frac{x^3 \coth ^{-1}(a x)}{9 a^3}+\frac{x^5 \coth ^{-1}(a x)}{15 a}+\frac{1}{6} x^6 \coth ^{-1}(a x)^2-\frac{1}{30} \operatorname{Subst}\left (\int \frac{x^2}{1-a^2 x} \, dx,x,x^2\right )+\frac{\int \coth ^{-1}(a x) \, dx}{3 a^5}-\frac{\int \frac{\coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^5}-\frac{\int \frac{x^3}{1-a^2 x^2} \, dx}{9 a^2}\\ &=\frac{x \coth ^{-1}(a x)}{3 a^5}+\frac{x^3 \coth ^{-1}(a x)}{9 a^3}+\frac{x^5 \coth ^{-1}(a x)}{15 a}-\frac{\coth ^{-1}(a x)^2}{6 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^2-\frac{1}{30} \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}-\frac{x}{a^2}-\frac{1}{a^4 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{\int \frac{x}{1-a^2 x^2} \, dx}{3 a^4}-\frac{\operatorname{Subst}\left (\int \frac{x}{1-a^2 x} \, dx,x,x^2\right )}{18 a^2}\\ &=\frac{x^2}{30 a^4}+\frac{x^4}{60 a^2}+\frac{x \coth ^{-1}(a x)}{3 a^5}+\frac{x^3 \coth ^{-1}(a x)}{9 a^3}+\frac{x^5 \coth ^{-1}(a x)}{15 a}-\frac{\coth ^{-1}(a x)^2}{6 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^2+\frac{\log \left (1-a^2 x^2\right )}{5 a^6}-\frac{\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 )}{18 a^2}\\ &=\frac{4 x^2}{45 a^4}+\frac{x^4}{60 a^2}+\frac{x \coth ^{-1}(a x)}{3 a^5}+\frac{x^3 \coth ^{-1}(a x)}{9 a^3}+\frac{x^5 \coth ^{-1}(a x)}{15 a}-\frac{\coth ^{-1}(a x)^2}{6 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^2+\frac{23 \log \left (1-a^2 x^2\right )}{90 a^6}\\ \end{align*}

Mathematica [A]  time = 0.0218823, size = 80, normalized size = 0.76 \[ \frac{3 a^4 x^4+16 a^2 x^2+46 \log \left (1-a^2 x^2\right )+4 a x \left (3 a^4 x^4+5 a^2 x^2+15\right ) \coth ^{-1}(a x)+30 \left (a^6 x^6-1\right ) \coth ^{-1}(a x)^2}{180 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*ArcCoth[a*x]^2,x]

[Out]

(16*a^2*x^2 + 3*a^4*x^4 + 4*a*x*(15 + 5*a^2*x^2 + 3*a^4*x^4)*ArcCoth[a*x] + 30*(-1 + a^6*x^6)*ArcCoth[a*x]^2 +
46*Log[1 - a^2*x^2])/(180*a^6)

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Maple [B]  time = 0.052, size = 196, normalized size = 1.9 \begin{align*}{\frac{{x}^{6} \left ({\rm arccoth} \left (ax\right ) \right ) ^{2}}{6}}+{\frac{{x}^{5}{\rm arccoth} \left (ax\right )}{15\,a}}+{\frac{{x}^{3}{\rm arccoth} \left (ax\right )}{9\,{a}^{3}}}+{\frac{x{\rm arccoth} \left (ax\right )}{3\,{a}^{5}}}+{\frac{{\rm arccoth} \left (ax\right )\ln \left ( ax-1 \right ) }{6\,{a}^{6}}}-{\frac{{\rm arccoth} \left (ax\right )\ln \left ( ax+1 \right ) }{6\,{a}^{6}}}+{\frac{ \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{24\,{a}^{6}}}-{\frac{\ln \left ( ax-1 \right ) }{12\,{a}^{6}}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{1}{12\,{a}^{6}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{\ln \left ( ax+1 \right ) }{12\,{a}^{6}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }+{\frac{ \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{24\,{a}^{6}}}+{\frac{{x}^{4}}{60\,{a}^{2}}}+{\frac{4\,{x}^{2}}{45\,{a}^{4}}}+{\frac{23\,\ln \left ( ax-1 \right ) }{90\,{a}^{6}}}+{\frac{23\,\ln \left ( ax+1 \right ) }{90\,{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*arccoth(a*x)^2,x)

[Out]

1/6*x^6*arccoth(a*x)^2+1/15*x^5*arccoth(a*x)/a+1/9*x^3*arccoth(a*x)/a^3+1/3*x*arccoth(a*x)/a^5+1/6/a^6*arccoth
(a*x)*ln(a*x-1)-1/6/a^6*arccoth(a*x)*ln(a*x+1)+1/24/a^6*ln(a*x-1)^2-1/12/a^6*ln(a*x-1)*ln(1/2+1/2*a*x)+1/12/a^
6*ln(-1/2*a*x+1/2)*ln(1/2+1/2*a*x)-1/12/a^6*ln(-1/2*a*x+1/2)*ln(a*x+1)+1/24/a^6*ln(a*x+1)^2+1/60*x^4/a^2+4/45*
x^2/a^4+23/90/a^6*ln(a*x-1)+23/90/a^6*ln(a*x+1)

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Maxima [A]  time = 0.970857, size = 182, normalized size = 1.73 \begin{align*} \frac{1}{6} \, x^{6} \operatorname{arcoth}\left (a x\right )^{2} + \frac{1}{90} \, a{\left (\frac{2 \,{\left (3 \, a^{4} x^{5} + 5 \, a^{2} x^{3} + 15 \, x\right )}}{a^{6}} - \frac{15 \, \log \left (a x + 1\right )}{a^{7}} + \frac{15 \, \log \left (a x - 1\right )}{a^{7}}\right )} \operatorname{arcoth}\left (a x\right ) + \frac{6 \, a^{4} x^{4} + 32 \, a^{2} x^{2} - 2 \,{\left (15 \, \log \left (a x - 1\right ) - 46\right )} \log \left (a x + 1\right ) + 15 \, \log \left (a x + 1\right )^{2} + 15 \, \log \left (a x - 1\right )^{2} + 92 \, \log \left (a x - 1\right )}{360 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccoth(a*x)^2,x, algorithm="maxima")

[Out]

1/6*x^6*arccoth(a*x)^2 + 1/90*a*(2*(3*a^4*x^5 + 5*a^2*x^3 + 15*x)/a^6 - 15*log(a*x + 1)/a^7 + 15*log(a*x - 1)/
a^7)*arccoth(a*x) + 1/360*(6*a^4*x^4 + 32*a^2*x^2 - 2*(15*log(a*x - 1) - 46)*log(a*x + 1) + 15*log(a*x + 1)^2
+ 15*log(a*x - 1)^2 + 92*log(a*x - 1))/a^6

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Fricas [A]  time = 1.57716, size = 224, normalized size = 2.13 \begin{align*} \frac{6 \, a^{4} x^{4} + 32 \, a^{2} x^{2} + 15 \,{\left (a^{6} x^{6} - 1\right )} \log \left (\frac{a x + 1}{a x - 1}\right )^{2} + 4 \,{\left (3 \, a^{5} x^{5} + 5 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (\frac{a x + 1}{a x - 1}\right ) + 92 \, \log \left (a^{2} x^{2} - 1\right )}{360 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccoth(a*x)^2,x, algorithm="fricas")

[Out]

1/360*(6*a^4*x^4 + 32*a^2*x^2 + 15*(a^6*x^6 - 1)*log((a*x + 1)/(a*x - 1))^2 + 4*(3*a^5*x^5 + 5*a^3*x^3 + 15*a*
x)*log((a*x + 1)/(a*x - 1)) + 92*log(a^2*x^2 - 1))/a^6

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Sympy [A]  time = 7.50491, size = 114, normalized size = 1.09 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{acoth}^{2}{\left (a x \right )}}{6} + \frac{x^{5} \operatorname{acoth}{\left (a x \right )}}{15 a} + \frac{x^{4}}{60 a^{2}} + \frac{x^{3} \operatorname{acoth}{\left (a x \right )}}{9 a^{3}} + \frac{4 x^{2}}{45 a^{4}} + \frac{x \operatorname{acoth}{\left (a x \right )}}{3 a^{5}} + \frac{23 \log{\left (a x + 1 \right )}}{45 a^{6}} - \frac{\operatorname{acoth}^{2}{\left (a x \right )}}{6 a^{6}} - \frac{23 \operatorname{acoth}{\left (a x \right )}}{45 a^{6}} & \text{for}\: a \neq 0 \\- \frac{\pi ^{2} x^{6}}{24} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*acoth(a*x)**2,x)

[Out]

Piecewise((x**6*acoth(a*x)**2/6 + x**5*acoth(a*x)/(15*a) + x**4/(60*a**2) + x**3*acoth(a*x)/(9*a**3) + 4*x**2/
(45*a**4) + x*acoth(a*x)/(3*a**5) + 23*log(a*x + 1)/(45*a**6) - acoth(a*x)**2/(6*a**6) - 23*acoth(a*x)/(45*a**
6), Ne(a, 0)), (-pi**2*x**6/24, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{arcoth}\left (a x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccoth(a*x)^2,x, algorithm="giac")

[Out]

integrate(x^5*arccoth(a*x)^2, x)