3.2.72 \(\int x^4 (1-a^2 x^2) \tanh ^{-1}(a x)^2 \, dx\) [172]

Optimal. Leaf size=162 \[ \frac {4 x}{105 a^4}-\frac {2 x^3}{315 a^2}-\frac {x^5}{105}-\frac {4 \tanh ^{-1}(a x)}{105 a^5}+\frac {2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {4 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a^5}-\frac {2 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{35 a^5} \]

[Out]

4/105*x/a^4-2/315*x^3/a^2-1/105*x^5-4/105*arctanh(a*x)/a^5+2/35*x^2*arctanh(a*x)/a^3+1/35*x^4*arctanh(a*x)/a-1
/21*a*x^6*arctanh(a*x)+2/35*arctanh(a*x)^2/a^5+1/5*x^5*arctanh(a*x)^2-1/7*a^2*x^7*arctanh(a*x)^2-4/35*arctanh(
a*x)*ln(2/(-a*x+1))/a^5-2/35*polylog(2,1-2/(-a*x+1))/a^5

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Rubi [A]
time = 0.43, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6161, 6037, 6127, 308, 212, 327, 6131, 6055, 2449, 2352} \begin {gather*} -\frac {2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a^5}+\frac {2 \tanh ^{-1}(a x)^2}{35 a^5}-\frac {4 \tanh ^{-1}(a x)}{105 a^5}-\frac {4 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{35 a^5}+\frac {4 x}{105 a^4}+\frac {2 x^2 \tanh ^{-1}(a x)}{35 a^3}-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {2 x^3}{315 a^2}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {x^5}{105} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x)/(105*a^4) - (2*x^3)/(315*a^2) - x^5/105 - (4*ArcTanh[a*x])/(105*a^5) + (2*x^2*ArcTanh[a*x])/(35*a^3) + (
x^4*ArcTanh[a*x])/(35*a) - (a*x^6*ArcTanh[a*x])/21 + (2*ArcTanh[a*x]^2)/(35*a^5) + (x^5*ArcTanh[a*x]^2)/5 - (a
^2*x^7*ArcTanh[a*x]^2)/7 - (4*ArcTanh[a*x]*Log[2/(1 - a*x)])/(35*a^5) - (2*PolyLog[2, 1 - 2/(1 - a*x)])/(35*a^
5)

Rule 212

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

Rule 308

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 327

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 6037

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

Rule 6055

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

Rule 6127

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

Rule 6131

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

Rule 6161

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]))

Rubi steps

\begin {align*} \int x^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2 \, dx &=-\left (a^2 \int x^6 \tanh ^{-1}(a x)^2 \, dx\right )+\int x^4 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {1}{5} (2 a) \int \frac {x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {1}{7} \left (2 a^3\right ) \int \frac {x^7 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {2 \int x^3 \tanh ^{-1}(a x) \, dx}{5 a}-\frac {2 \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a}-\frac {1}{7} (2 a) \int x^5 \tanh ^{-1}(a x) \, dx+\frac {1}{7} (2 a) \int \frac {x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {x^4 \tanh ^{-1}(a x)}{10 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {1}{10} \int \frac {x^4}{1-a^2 x^2} \, dx+\frac {2 \int x \tanh ^{-1}(a x) \, dx}{5 a^3}-\frac {2 \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^3}-\frac {2 \int x^3 \tanh ^{-1}(a x) \, dx}{7 a}+\frac {2 \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{7 a}+\frac {1}{21} a^2 \int \frac {x^6}{1-a^2 x^2} \, dx\\ &=\frac {x^2 \tanh ^{-1}(a x)}{5 a^3}+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {\tanh ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {1}{14} \int \frac {x^4}{1-a^2 x^2} \, dx-\frac {1}{10} \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 {2 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{5 a^4}-\frac {2 \int x \tanh ^{-1}(a x) \, dx}{7 a^3}+\frac {2 \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{7 a^3}-\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{5 a^2}+\frac {1}{21} a^2 \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 {53 x}{210 a^4}+\frac {11 x^3}{630 a^2}-\frac {x^5}{105}+\frac {2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {2 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}+\frac {1}{14} \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 {\int \frac {1}{1-a^2 x^2} \, dx}{21 a^4}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{10 a^4}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{5 a^4}+\frac {2 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{7 a^4}+\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^4}+\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{7 a^2}\\ &=\frac {4 x}{105 a^4}-\frac {2 x^3}{315 a^2}-\frac {x^5}{105}-\frac {53 \tanh ^{-1}(a x)}{210 a^5}+\frac {2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {4 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a^5}-\frac {2 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{5 a^5}+\frac {\int \frac {1}{1-a^2 x^2} \, dx}{14 a^4}+\frac {\int \frac {1}{1-a^2 x^2} \, dx}{7 a^4}-\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{7 a^4}\\ &=\frac {4 x}{105 a^4}-\frac {2 x^3}{315 a^2}-\frac {x^5}{105}-\frac {4 \tanh ^{-1}(a x)}{105 a^5}+\frac {2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {4 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a^5}-\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{5 a^5}+\frac {2 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{7 a^5}\\ &=\frac {4 x}{105 a^4}-\frac {2 x^3}{315 a^2}-\frac {x^5}{105}-\frac {4 \tanh ^{-1}(a x)}{105 a^5}+\frac {2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {4 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a^5}-\frac {2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a^5}\\ \end {align*}

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Mathematica [A]
time = 0.68, size = 113, normalized size = 0.70 \begin {gather*} -\frac {-12 a x+2 a^3 x^3+3 a^5 x^5+9 \left (2-7 a^5 x^5+5 a^7 x^7\right ) \tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x) \left (4-6 a^2 x^2-3 a^4 x^4+5 a^6 x^6+12 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )\right )-18 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )}{315 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/315*(-12*a*x + 2*a^3*x^3 + 3*a^5*x^5 + 9*(2 - 7*a^5*x^5 + 5*a^7*x^7)*ArcTanh[a*x]^2 + 3*ArcTanh[a*x]*(4 - 6
*a^2*x^2 - 3*a^4*x^4 + 5*a^6*x^6 + 12*Log[1 + E^(-2*ArcTanh[a*x])]) - 18*PolyLog[2, -E^(-2*ArcTanh[a*x])])/a^5

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Maple [A]
time = 0.93, size = 199, normalized size = 1.23

method	result	size
derivativedivides	\(\frac {-\frac {\arctanh \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {\arctanh \left (a x \right )^{2} a^{5} x^{5}}{5}-\frac {\arctanh \left (a x \right ) a^{6} x^{6}}{21}+\frac {a^{4} x^{4} \arctanh \left (a x \right )}{35}+\frac {2 a^{2} x^{2} \arctanh \left (a x \right )}{35}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{35}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{35}+\frac {\ln \left (a x -1\right )^{2}}{70}-\frac {2 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {\ln \left (a x +1\right )^{2}}{70}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {a^{5} x^{5}}{105}-\frac {2 a^{3} x^{3}}{315}+\frac {4 a x}{105}+\frac {2 \ln \left (a x -1\right )}{105}-\frac {2 \ln \left (a x +1\right )}{105}}{a^{5}}\)	\(199\)
default	\(\frac {-\frac {\arctanh \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {\arctanh \left (a x \right )^{2} a^{5} x^{5}}{5}-\frac {\arctanh \left (a x \right ) a^{6} x^{6}}{21}+\frac {a^{4} x^{4} \arctanh \left (a x \right )}{35}+\frac {2 a^{2} x^{2} \arctanh \left (a x \right )}{35}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{35}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{35}+\frac {\ln \left (a x -1\right )^{2}}{70}-\frac {2 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {\ln \left (a x +1\right )^{2}}{70}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {a^{5} x^{5}}{105}-\frac {2 a^{3} x^{3}}{315}+\frac {4 a x}{105}+\frac {2 \ln \left (a x -1\right )}{105}-\frac {2 \ln \left (a x +1\right )}{105}}{a^{5}}\)	\(199\)
risch	\(-\frac {\ln \left (-a x +1\right )^{2}}{70 a^{5}}-\frac {4177 \ln \left (-a x +1\right )}{29400 a^{5}}+\frac {\ln \left (-a x +1\right )^{2} x^{5}}{20}-\frac {\ln \left (-a x +1\right ) x^{5}}{50}+\frac {a^{2} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{7}}{14}+\frac {4 x}{105 a^{4}}-\frac {2 x^{3}}{315 a^{2}}-\frac {\left (\left (-\frac {1}{25}+\frac {\ln \left (a x +1\right )}{5}\right ) \left (a x +1\right )^{5}+\left (\frac {1}{4}-\ln \left (a x +1\right )\right ) \left (a x +1\right )^{4}+\left (-\frac {2}{3}+2 \ln \left (a x +1\right )\right ) \left (a x +1\right )^{3}+\left (1-2 \ln \left (a x +1\right )\right ) \left (a x +1\right )^{2}+\left (-1+\ln \left (a x +1\right )\right ) \left (a x +1\right )\right ) \ln \left (-a x +1\right )}{2 a^{5}}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{10 a^{5}}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{5 a^{5}}-\frac {\left (a x +1\right )^{2} \ln \left (a x +1\right )}{10 a^{5}}-\frac {3 \left (a x +1\right )^{4} \ln \left (a x +1\right )}{40 a^{5}}+\frac {\left (a x +1\right )^{5} \ln \left (a x +1\right )}{50 a^{5}}+\frac {2 \left (a x +1\right )^{3} \ln \left (a x +1\right )}{15 a^{5}}+\frac {\ln \left (-a x +1\right ) \ln \left (a x +1\right )}{14 a^{5}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{7 a^{5}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{7 a^{5}}-\frac {409 \ln \left (a x +1\right )}{4200 a^{5}}+\frac {\ln \left (a x +1\right )^{2}}{70 a^{5}}+\frac {\ln \left (a x +1\right )^{2} x^{5}}{20}-\frac {\ln \left (a x +1\right ) x^{5}}{50}-\frac {a^{2} \ln \left (-a x +1\right )^{2} x^{7}}{28}+\frac {a \ln \left (-a x +1\right ) x^{6}}{42}+\frac {3 \ln \left (-a x +1\right ) x^{4}}{280 a}-\frac {\ln \left (-a x +1\right ) x^{3}}{30 a^{2}}+\frac {3 \ln \left (-a x +1\right ) x^{2}}{140 a^{3}}-\frac {\ln \left (-a x +1\right ) x}{10 a^{4}}-\frac {3 \ln \left (a x +1\right ) x^{2}}{140 a^{3}}-\frac {\ln \left (a x +1\right ) x}{10 a^{4}}-\frac {a^{2} \ln \left (a x +1\right )^{2} x^{7}}{28}-\frac {a \ln \left (a x +1\right ) x^{6}}{42}-\frac {3 \ln \left (a x +1\right ) x^{4}}{280 a}-\frac {\ln \left (a x +1\right ) x^{3}}{30 a^{2}}-\frac {14083}{257250 a^{5}}-\frac {247 \ln \left (a x -1\right )}{3675 a^{5}}-\frac {x^{5}}{105}-\frac {2 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{35 a^{5}}\)	\(610\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^5*(-1/7*arctanh(a*x)^2*a^7*x^7+1/5*arctanh(a*x)^2*a^5*x^5-1/21*arctanh(a*x)*a^6*x^6+1/35*a^4*x^4*arctanh(a
*x)+2/35*a^2*x^2*arctanh(a*x)+2/35*arctanh(a*x)*ln(a*x-1)+2/35*arctanh(a*x)*ln(a*x+1)+1/70*ln(a*x-1)^2-2/35*di
log(1/2*a*x+1/2)-1/35*ln(a*x-1)*ln(1/2*a*x+1/2)-1/70*ln(a*x+1)^2+1/35*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+
1/2)-1/105*a^5*x^5-2/315*a^3*x^3+4/105*a*x+2/105*ln(a*x-1)-2/105*ln(a*x+1))

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Maxima [A]
time = 0.26, size = 190, normalized size = 1.17 \begin {gather*} -\frac {1}{630} \, a^{2} {\left (\frac {6 \, a^{5} x^{5} + 4 \, a^{3} x^{3} - 24 \, a x + 9 \, \log \left (a x + 1\right )^{2} - 18 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 9 \, \log \left (a x - 1\right )^{2} - 12 \, \log \left (a x - 1\right )}{a^{7}} + \frac {36 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{7}} + \frac {12 \, \log \left (a x + 1\right )}{a^{7}}\right )} - \frac {1}{105} \, a {\left (\frac {5 \, a^{4} x^{6} - 3 \, a^{2} x^{4} - 6 \, x^{2}}{a^{4}} - \frac {6 \, \log \left (a x + 1\right )}{a^{6}} - \frac {6 \, \log \left (a x - 1\right )}{a^{6}}\right )} \operatorname {artanh}\left (a x\right ) - \frac {1}{35} \, {\left (5 \, a^{2} x^{7} - 7 \, x^{5}\right )} \operatorname {artanh}\left (a x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/630*a^2*((6*a^5*x^5 + 4*a^3*x^3 - 24*a*x + 9*log(a*x + 1)^2 - 18*log(a*x + 1)*log(a*x - 1) - 9*log(a*x - 1)
^2 - 12*log(a*x - 1))/a^7 + 36*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a^7 + 12*log(a*x + 1)
/a^7) - 1/105*a*((5*a^4*x^6 - 3*a^2*x^4 - 6*x^2)/a^4 - 6*log(a*x + 1)/a^6 - 6*log(a*x - 1)/a^6)*arctanh(a*x) -
1/35*(5*a^2*x^7 - 7*x^5)*arctanh(a*x)^2

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^2*x^6 - x^4)*arctanh(a*x)^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- x^{4} \operatorname {atanh}^{2}{\left (a x \right )}\right )\, dx - \int a^{2} x^{6} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-a**2*x**2+1)*atanh(a*x)**2,x)

[Out]

-Integral(-x**4*atanh(a*x)**2, x) - Integral(a**2*x**6*atanh(a*x)**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int x^4\,{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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