3.192 \(\int x^4 (1-a^2 x^2)^2 \tanh ^{-1}(a x) \, dx\)

Optimal. Leaf size=96 \[ \frac{a^3 x^8}{72}+\frac{4 x^2}{315 a^3}+\frac{4 \log \left (1-a^2 x^2\right )}{315 a^5}+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)-\frac{11 a x^6}{378}+\frac{2 x^4}{315 a}+\frac{1}{5} x^5 \tanh ^{-1}(a x) \]

[Out]

(4*x^2)/(315*a^3) + (2*x^4)/(315*a) - (11*a*x^6)/378 + (a^3*x^8)/72 + (x^5*ArcTanh[a*x])/5 - (2*a^2*x^7*ArcTan
h[a*x])/7 + (a^4*x^9*ArcTanh[a*x])/9 + (4*Log[1 - a^2*x^2])/(315*a^5)

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Rubi [A]  time = 0.188829, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6012, 5916, 266, 43} \[ \frac{a^3 x^8}{72}+\frac{4 x^2}{315 a^3}+\frac{4 \log \left (1-a^2 x^2\right )}{315 a^5}+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)-\frac{11 a x^6}{378}+\frac{2 x^4}{315 a}+\frac{1}{5} x^5 \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^4*(1 - a^2*x^2)^2*ArcTanh[a*x],x]

[Out]

(4*x^2)/(315*a^3) + (2*x^4)/(315*a) - (11*a*x^6)/378 + (a^3*x^8)/72 + (x^5*ArcTanh[a*x])/5 - (2*a^2*x^7*ArcTan
h[a*x])/7 + (a^4*x^9*ArcTanh[a*x])/9 + (4*Log[1 - a^2*x^2])/(315*a^5)

Rule 6012

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

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 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])

Rubi steps

\begin{align*} \int x^4 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x) \, dx &=\int \left (x^4 \tanh ^{-1}(a x)-2 a^2 x^6 \tanh ^{-1}(a x)+a^4 x^8 \tanh ^{-1}(a x)\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x^6 \tanh ^{-1}(a x) \, dx\right )+a^4 \int x^8 \tanh ^{-1}(a x) \, dx+\int x^4 \tanh ^{-1}(a x) \, dx\\ &=\frac{1}{5} x^5 \tanh ^{-1}(a x)-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)-\frac{1}{5} a \int \frac{x^5}{1-a^2 x^2} \, dx+\frac{1}{7} \left (2 a^3\right ) \int \frac{x^7}{1-a^2 x^2} \, dx-\frac{1}{9} a^5 \int \frac{x^9}{1-a^2 x^2} \, dx\\ &=\frac{1}{5} x^5 \tanh ^{-1}(a x)-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)-\frac{1}{10} a \operatorname{Subst}\left (\int \frac{x^2}{1-a^2 x} \, dx,x,x^2\right )+\frac{1}{7} a^3 \operatorname{Subst}\left (\int \frac{x^3}{1-a^2 x} \, dx,x,x^2\right )-\frac{1}{18} a^5 \operatorname{Subst}\left (\int \frac{x^4}{1-a^2 x} \, dx,x,x^2\right )\\ &=\frac{1}{5} x^5 \tanh ^{-1}(a x)-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)-\frac{1}{10} a \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{1}{7} a^3 \operatorname{Subst}\left (\int \left (-\frac{1}{a^6}-\frac{x}{a^4}-\frac{x^2}{a^2}-\frac{1}{a^6 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{18} a^5 \operatorname{Subst}\left (\int \left (-\frac{1}{a^8}-\frac{x}{a^6}-\frac{x^2}{a^4}-\frac{x^3}{a^2}-\frac{1}{a^8 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{4 x^2}{315 a^3}+\frac{2 x^4}{315 a}-\frac{11 a x^6}{378}+\frac{a^3 x^8}{72}+\frac{1}{5} x^5 \tanh ^{-1}(a x)-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)+\frac{4 \log \left (1-a^2 x^2\right )}{315 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0266366, size = 96, normalized size = 1. \[ \frac{a^3 x^8}{72}+\frac{4 x^2}{315 a^3}+\frac{4 \log \left (1-a^2 x^2\right )}{315 a^5}+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)-\frac{11 a x^6}{378}+\frac{2 x^4}{315 a}+\frac{1}{5} x^5 \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(1 - a^2*x^2)^2*ArcTanh[a*x],x]

[Out]

(4*x^2)/(315*a^3) + (2*x^4)/(315*a) - (11*a*x^6)/378 + (a^3*x^8)/72 + (x^5*ArcTanh[a*x])/5 - (2*a^2*x^7*ArcTan
h[a*x])/7 + (a^4*x^9*ArcTanh[a*x])/9 + (4*Log[1 - a^2*x^2])/(315*a^5)

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Maple [A]  time = 0.028, size = 87, normalized size = 0.9 \begin{align*}{\frac{{a}^{4}{x}^{9}{\it Artanh} \left ( ax \right ) }{9}}-{\frac{2\,{a}^{2}{x}^{7}{\it Artanh} \left ( ax \right ) }{7}}+{\frac{{x}^{5}{\it Artanh} \left ( ax \right ) }{5}}+{\frac{{a}^{3}{x}^{8}}{72}}-{\frac{11\,{x}^{6}a}{378}}+{\frac{2\,{x}^{4}}{315\,a}}+{\frac{4\,{x}^{2}}{315\,{a}^{3}}}+{\frac{4\,\ln \left ( ax-1 \right ) }{315\,{a}^{5}}}+{\frac{4\,\ln \left ( ax+1 \right ) }{315\,{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-a^2*x^2+1)^2*arctanh(a*x),x)

[Out]

1/9*a^4*x^9*arctanh(a*x)-2/7*a^2*x^7*arctanh(a*x)+1/5*x^5*arctanh(a*x)+1/72*a^3*x^8-11/378*x^6*a+2/315*x^4/a+4
/315*x^2/a^3+4/315/a^5*ln(a*x-1)+4/315/a^5*ln(a*x+1)

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Maxima [A]  time = 0.960876, size = 120, normalized size = 1.25 \begin{align*} \frac{1}{7560} \, a{\left (\frac{105 \, a^{6} x^{8} - 220 \, a^{4} x^{6} + 48 \, a^{2} x^{4} + 96 \, x^{2}}{a^{4}} + \frac{96 \, \log \left (a x + 1\right )}{a^{6}} + \frac{96 \, \log \left (a x - 1\right )}{a^{6}}\right )} + \frac{1}{315} \,{\left (35 \, a^{4} x^{9} - 90 \, a^{2} x^{7} + 63 \, x^{5}\right )} \operatorname{artanh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-a^2*x^2+1)^2*arctanh(a*x),x, algorithm="maxima")

[Out]

1/7560*a*((105*a^6*x^8 - 220*a^4*x^6 + 48*a^2*x^4 + 96*x^2)/a^4 + 96*log(a*x + 1)/a^6 + 96*log(a*x - 1)/a^6) +
1/315*(35*a^4*x^9 - 90*a^2*x^7 + 63*x^5)*arctanh(a*x)

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Fricas [A]  time = 1.98492, size = 213, normalized size = 2.22 \begin{align*} \frac{105 \, a^{8} x^{8} - 220 \, a^{6} x^{6} + 48 \, a^{4} x^{4} + 96 \, a^{2} x^{2} + 12 \,{\left (35 \, a^{9} x^{9} - 90 \, a^{7} x^{7} + 63 \, a^{5} x^{5}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) + 96 \, \log \left (a^{2} x^{2} - 1\right )}{7560 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-a^2*x^2+1)^2*arctanh(a*x),x, algorithm="fricas")

[Out]

1/7560*(105*a^8*x^8 - 220*a^6*x^6 + 48*a^4*x^4 + 96*a^2*x^2 + 12*(35*a^9*x^9 - 90*a^7*x^7 + 63*a^5*x^5)*log(-(
a*x + 1)/(a*x - 1)) + 96*log(a^2*x^2 - 1))/a^5

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Sympy [A]  time = 4.934, size = 100, normalized size = 1.04 \begin{align*} \begin{cases} \frac{a^{4} x^{9} \operatorname{atanh}{\left (a x \right )}}{9} + \frac{a^{3} x^{8}}{72} - \frac{2 a^{2} x^{7} \operatorname{atanh}{\left (a x \right )}}{7} - \frac{11 a x^{6}}{378} + \frac{x^{5} \operatorname{atanh}{\left (a x \right )}}{5} + \frac{2 x^{4}}{315 a} + \frac{4 x^{2}}{315 a^{3}} + \frac{8 \log{\left (x - \frac{1}{a} \right )}}{315 a^{5}} + \frac{8 \operatorname{atanh}{\left (a x \right )}}{315 a^{5}} & \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**4*(-a**2*x**2+1)**2*atanh(a*x),x)

[Out]

Piecewise((a**4*x**9*atanh(a*x)/9 + a**3*x**8/72 - 2*a**2*x**7*atanh(a*x)/7 - 11*a*x**6/378 + x**5*atanh(a*x)/
5 + 2*x**4/(315*a) + 4*x**2/(315*a**3) + 8*log(x - 1/a)/(315*a**5) + 8*atanh(a*x)/(315*a**5), Ne(a, 0)), (0, T
rue))

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Giac [A]  time = 1.15844, size = 127, normalized size = 1.32 \begin{align*} \frac{1}{630} \,{\left (35 \, a^{4} x^{9} - 90 \, a^{2} x^{7} + 63 \, x^{5}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) + \frac{4 \, \log \left ({\left | a^{2} x^{2} - 1 \right |}\right )}{315 \, a^{5}} + \frac{105 \, a^{11} x^{8} - 220 \, a^{9} x^{6} + 48 \, a^{7} x^{4} + 96 \, a^{5} x^{2}}{7560 \, a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-a^2*x^2+1)^2*arctanh(a*x),x, algorithm="giac")

[Out]

1/630*(35*a^4*x^9 - 90*a^2*x^7 + 63*x^5)*log(-(a*x + 1)/(a*x - 1)) + 4/315*log(abs(a^2*x^2 - 1))/a^5 + 1/7560*
(105*a^11*x^8 - 220*a^9*x^6 + 48*a^7*x^4 + 96*a^5*x^2)/a^8