3.162 \(\int x^3 (1-a^2 x^2) \tanh ^{-1}(a x) \, dx\)

Optimal. Leaf size=63 \[ -\frac{1}{6} a^2 x^6 \tanh ^{-1}(a x)+\frac{x}{12 a^3}-\frac{\tanh ^{-1}(a x)}{12 a^4}-\frac{a x^5}{30}+\frac{x^3}{36 a}+\frac{1}{4} x^4 \tanh ^{-1}(a x) \]

[Out]

x/(12*a^3) + x^3/(36*a) - (a*x^5)/30 - ArcTanh[a*x]/(12*a^4) + (x^4*ArcTanh[a*x])/4 - (a^2*x^6*ArcTanh[a*x])/6

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Rubi [A]  time = 0.0807167, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6014, 5916, 302, 206} \[ -\frac{1}{6} a^2 x^6 \tanh ^{-1}(a x)+\frac{x}{12 a^3}-\frac{\tanh ^{-1}(a x)}{12 a^4}-\frac{a x^5}{30}+\frac{x^3}{36 a}+\frac{1}{4} x^4 \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^3*(1 - a^2*x^2)*ArcTanh[a*x],x]

[Out]

x/(12*a^3) + x^3/(36*a) - (a*x^5)/30 - ArcTanh[a*x]/(12*a^4) + (x^4*ArcTanh[a*x])/4 - (a^2*x^6*ArcTanh[a*x])/6

Rule 6014

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Dist
[d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[(c^2*d)/f^2, Int[(f*x)^(m + 2)*(d +
e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q
, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))

Rule 5916

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcT
anh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 206

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

Rubi steps

\begin{align*} \int x^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx &=-\left (a^2 \int x^5 \tanh ^{-1}(a x) \, dx\right )+\int x^3 \tanh ^{-1}(a x) \, dx\\ &=\frac{1}{4} x^4 \tanh ^{-1}(a x)-\frac{1}{6} a^2 x^6 \tanh ^{-1}(a x)-\frac{1}{4} a \int \frac{x^4}{1-a^2 x^2} \, dx+\frac{1}{6} a^3 \int \frac{x^6}{1-a^2 x^2} \, dx\\ &=\frac{1}{4} x^4 \tanh ^{-1}(a x)-\frac{1}{6} a^2 x^6 \tanh ^{-1}(a x)-\frac{1}{4} a \int \left (-\frac{1}{a^4}-\frac{x^2}{a^2}+\frac{1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx+\frac{1}{6} a^3 \int \left (-\frac{1}{a^6}-\frac{x^2}{a^4}-\frac{x^4}{a^2}+\frac{1}{a^6 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac{x}{12 a^3}+\frac{x^3}{36 a}-\frac{a x^5}{30}+\frac{1}{4} x^4 \tanh ^{-1}(a x)-\frac{1}{6} a^2 x^6 \tanh ^{-1}(a x)+\frac{\int \frac{1}{1-a^2 x^2} \, dx}{6 a^3}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{4 a^3}\\ &=\frac{x}{12 a^3}+\frac{x^3}{36 a}-\frac{a x^5}{30}-\frac{\tanh ^{-1}(a x)}{12 a^4}+\frac{1}{4} x^4 \tanh ^{-1}(a x)-\frac{1}{6} a^2 x^6 \tanh ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0174197, size = 79, normalized size = 1.25 \[ -\frac{1}{6} a^2 x^6 \tanh ^{-1}(a x)+\frac{x}{12 a^3}+\frac{\log (1-a x)}{24 a^4}-\frac{\log (a x+1)}{24 a^4}-\frac{a x^5}{30}+\frac{x^3}{36 a}+\frac{1}{4} x^4 \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(1 - a^2*x^2)*ArcTanh[a*x],x]

[Out]

x/(12*a^3) + x^3/(36*a) - (a*x^5)/30 + (x^4*ArcTanh[a*x])/4 - (a^2*x^6*ArcTanh[a*x])/6 + Log[1 - a*x]/(24*a^4)
- Log[1 + a*x]/(24*a^4)

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Maple [A]  time = 0.027, size = 65, normalized size = 1. \begin{align*} -{\frac{{a}^{2}{x}^{6}{\it Artanh} \left ( ax \right ) }{6}}+{\frac{{x}^{4}{\it Artanh} \left ( ax \right ) }{4}}-{\frac{a{x}^{5}}{30}}+{\frac{{x}^{3}}{36\,a}}+{\frac{x}{12\,{a}^{3}}}+{\frac{\ln \left ( ax-1 \right ) }{24\,{a}^{4}}}-{\frac{\ln \left ( ax+1 \right ) }{24\,{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-a^2*x^2+1)*arctanh(a*x),x)

[Out]

-1/6*a^2*x^6*arctanh(a*x)+1/4*x^4*arctanh(a*x)-1/30*a*x^5+1/36*x^3/a+1/12*x/a^3+1/24/a^4*ln(a*x-1)-1/24/a^4*ln
(a*x+1)

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Maxima [A]  time = 0.949963, size = 97, normalized size = 1.54 \begin{align*} -\frac{1}{360} \, a{\left (\frac{2 \,{\left (6 \, a^{4} x^{5} - 5 \, a^{2} x^{3} - 15 \, x\right )}}{a^{4}} + \frac{15 \, \log \left (a x + 1\right )}{a^{5}} - \frac{15 \, \log \left (a x - 1\right )}{a^{5}}\right )} - \frac{1}{12} \,{\left (2 \, a^{2} x^{6} - 3 \, x^{4}\right )} \operatorname{artanh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-a^2*x^2+1)*arctanh(a*x),x, algorithm="maxima")

[Out]

-1/360*a*(2*(6*a^4*x^5 - 5*a^2*x^3 - 15*x)/a^4 + 15*log(a*x + 1)/a^5 - 15*log(a*x - 1)/a^5) - 1/12*(2*a^2*x^6
- 3*x^4)*arctanh(a*x)

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Fricas [A]  time = 2.25821, size = 143, normalized size = 2.27 \begin{align*} -\frac{12 \, a^{5} x^{5} - 10 \, a^{3} x^{3} - 30 \, a x + 15 \,{\left (2 \, a^{6} x^{6} - 3 \, a^{4} x^{4} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{360 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-a^2*x^2+1)*arctanh(a*x),x, algorithm="fricas")

[Out]

-1/360*(12*a^5*x^5 - 10*a^3*x^3 - 30*a*x + 15*(2*a^6*x^6 - 3*a^4*x^4 + 1)*log(-(a*x + 1)/(a*x - 1)))/a^4

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Sympy [A]  time = 2.2913, size = 54, normalized size = 0.86 \begin{align*} \begin{cases} - \frac{a^{2} x^{6} \operatorname{atanh}{\left (a x \right )}}{6} - \frac{a x^{5}}{30} + \frac{x^{4} \operatorname{atanh}{\left (a x \right )}}{4} + \frac{x^{3}}{36 a} + \frac{x}{12 a^{3}} - \frac{\operatorname{atanh}{\left (a x \right )}}{12 a^{4}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(-a**2*x**2+1)*atanh(a*x),x)

[Out]

Piecewise((-a**2*x**6*atanh(a*x)/6 - a*x**5/30 + x**4*atanh(a*x)/4 + x**3/(36*a) + x/(12*a**3) - atanh(a*x)/(1
2*a**4), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.17286, size = 113, normalized size = 1.79 \begin{align*} -\frac{1}{24} \,{\left (2 \, a^{2} x^{6} - 3 \, x^{4}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - \frac{\log \left ({\left | a x + 1 \right |}\right )}{24 \, a^{4}} + \frac{\log \left ({\left | a x - 1 \right |}\right )}{24 \, a^{4}} - \frac{6 \, a^{11} x^{5} - 5 \, a^{9} x^{3} - 15 \, a^{7} x}{180 \, a^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-a^2*x^2+1)*arctanh(a*x),x, algorithm="giac")

[Out]

-1/24*(2*a^2*x^6 - 3*x^4)*log(-(a*x + 1)/(a*x - 1)) - 1/24*log(abs(a*x + 1))/a^4 + 1/24*log(abs(a*x - 1))/a^4
- 1/180*(6*a^11*x^5 - 5*a^9*x^3 - 15*a^7*x)/a^10