3.2.63 \(\int x^2 (1-a^2 x^2) \tanh ^{-1}(a x) \, dx\) [163]

Optimal. Leaf size=62 \[ \frac {x^2}{15 a}-\frac {a x^4}{20}+\frac {1}{3} x^3 \tanh ^{-1}(a x)-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{15 a^3} \]

[Out]

1/15*x^2/a-1/20*a*x^4+1/3*x^3*arctanh(a*x)-1/5*a^2*x^5*arctanh(a*x)+1/15*ln(-a^2*x^2+1)/a^3

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 = {6161, 6037, 272, 45} \begin {gather*} -\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{15 a^3}-\frac {a x^4}{20}+\frac {1}{3} x^3 \tanh ^{-1}(a x)+\frac {x^2}{15 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(1 - a^2*x^2)*ArcTanh[a*x],x]

[Out]

x^2/(15*a) - (a*x^4)/20 + (x^3*ArcTanh[a*x])/3 - (a^2*x^5*ArcTanh[a*x])/5 + Log[1 - a^2*x^2]/(15*a^3)

Rule 45

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 272

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 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 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^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx &=-\left (a^2 \int x^4 \tanh ^{-1}(a x) \, dx\right )+\int x^2 \tanh ^{-1}(a x) \, dx\\ &=\frac {1}{3} x^3 \tanh ^{-1}(a x)-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)-\frac {1}{3} a \int \frac {x^3}{1-a^2 x^2} \, dx+\frac {1}{5} a^3 \int \frac {x^5}{1-a^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \tanh ^{-1}(a x)-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)-\frac {1}{6} a \text {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right )+\frac {1}{10} a^3 \text {Subst}\left (\int \frac {x^2}{1-a^2 x} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \tanh ^{-1}(a x)-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)-\frac {1}{6} a \text {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 {1}{10} a^3 \text {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 {x^2}{15 a}-\frac {a x^4}{20}+\frac {1}{3} x^3 \tanh ^{-1}(a x)-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{15 a^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 62, normalized size = 1.00 \begin {gather*} \frac {x^2}{15 a}-\frac {a x^4}{20}+\frac {1}{3} x^3 \tanh ^{-1}(a x)-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{15 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/(15*a) - (a*x^4)/20 + (x^3*ArcTanh[a*x])/3 - (a^2*x^5*ArcTanh[a*x])/5 + Log[1 - a^2*x^2]/(15*a^3)

________________________________________________________________________________________

Maple [A]
time = 0.28, size = 62, normalized size = 1.00

method	result	size
derivativedivides	\(\frac {-\frac {\arctanh \left (a x \right ) a^{5} x^{5}}{5}+\frac {a^{3} x^{3} \arctanh \left (a x \right )}{3}-\frac {a^{4} x^{4}}{20}+\frac {a^{2} x^{2}}{15}+\frac {\ln \left (a x -1\right )}{15}+\frac {\ln \left (a x +1\right )}{15}}{a^{3}}\)	\(62\)
default	\(\frac {-\frac {\arctanh \left (a x \right ) a^{5} x^{5}}{5}+\frac {a^{3} x^{3} \arctanh \left (a x \right )}{3}-\frac {a^{4} x^{4}}{20}+\frac {a^{2} x^{2}}{15}+\frac {\ln \left (a x -1\right )}{15}+\frac {\ln \left (a x +1\right )}{15}}{a^{3}}\)	\(62\)
risch	\(\left (-\frac {1}{10} a^{2} x^{5}+\frac {1}{6} x^{3}\right ) \ln \left (a x +1\right )+\frac {a^{2} x^{5} \ln \left (-a x +1\right )}{10}-\frac {x^{4} a}{20}-\frac {x^{3} \ln \left (-a x +1\right )}{6}+\frac {x^{2}}{15 a}+\frac {\ln \left (a^{2} x^{2}-1\right )}{15 a^{3}}-\frac {1}{45 a^{3}}\)	\(84\)
meijerg	\(\frac {-\frac {a^{2} x^{2} \left (3 a^{2} x^{2}+6\right )}{15}+\frac {2 a^{6} x^{6} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{5}}{4 a^{3}}+\frac {\frac {2 a^{2} x^{2}}{3}-\frac {2 a^{4} x^{4} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{3}}{4 a^{3}}\)	\(158\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3*(-1/5*arctanh(a*x)*a^5*x^5+1/3*a^3*x^3*arctanh(a*x)-1/20*a^4*x^4+1/15*a^2*x^2+1/15*ln(a*x-1)+1/15*ln(a*x
+1))

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 65, normalized size = 1.05 \begin {gather*} -\frac {1}{60} \, a {\left (\frac {3 \, a^{2} x^{4} - 4 \, x^{2}}{a^{2}} - \frac {4 \, \log \left (a x + 1\right )}{a^{4}} - \frac {4 \, \log \left (a x - 1\right )}{a^{4}}\right )} - \frac {1}{15} \, {\left (3 \, a^{2} x^{5} - 5 \, x^{3}\right )} \operatorname {artanh}\left (a x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*a*((3*a^2*x^4 - 4*x^2)/a^2 - 4*log(a*x + 1)/a^4 - 4*log(a*x - 1)/a^4) - 1/15*(3*a^2*x^5 - 5*x^3)*arctanh
(a*x)

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 68, normalized size = 1.10 \begin {gather*} -\frac {3 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 2 \, {\left (3 \, a^{5} x^{5} - 5 \, a^{3} x^{3}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 4 \, \log \left (a^{2} x^{2} - 1\right )}{60 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(3*a^4*x^4 - 4*a^2*x^2 + 2*(3*a^5*x^5 - 5*a^3*x^3)*log(-(a*x + 1)/(a*x - 1)) - 4*log(a^2*x^2 - 1))/a^3

________________________________________________________________________________________

Sympy [A]
time = 0.27, size = 63, normalized size = 1.02 \begin {gather*} \begin {cases} - \frac {a^{2} x^{5} \operatorname {atanh}{\left (a x \right )}}{5} - \frac {a x^{4}}{20} + \frac {x^{3} \operatorname {atanh}{\left (a x \right )}}{3} + \frac {x^{2}}{15 a} + \frac {2 \log {\left (x - \frac {1}{a} \right )}}{15 a^{3}} + \frac {2 \operatorname {atanh}{\left (a x \right )}}{15 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*x**5*atanh(a*x)/5 - a*x**4/20 + x**3*atanh(a*x)/3 + x**2/(15*a) + 2*log(x - 1/a)/(15*a**3) +
2*atanh(a*x)/(15*a**3), Ne(a, 0)), (0, True))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (52) = 104\).
time = 0.39, size = 268, normalized size = 4.32 \begin {gather*} \frac {2}{15} \, a {\left (\frac {\log \left (\frac {{\left | -a x - 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{4}} - \frac {\log \left ({\left | -\frac {a x + 1}{a x - 1} + 1 \right |}\right )}{a^{4}} - \frac {\frac {{\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {4 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {a x + 1}{a x - 1}}{a^{4} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{4}} - \frac {{\left (\frac {15 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {5 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {5 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right )}{a^{4} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{5}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*a*(log(abs(-a*x - 1)/abs(a*x - 1))/a^4 - log(abs(-(a*x + 1)/(a*x - 1) + 1))/a^4 - ((a*x + 1)^3/(a*x - 1)^
3 + 4*(a*x + 1)^2/(a*x - 1)^2 + (a*x + 1)/(a*x - 1))/(a^4*((a*x + 1)/(a*x - 1) - 1)^4) - (15*(a*x + 1)^3/(a*x
- 1)^3 + 5*(a*x + 1)^2/(a*x - 1)^2 + 5*(a*x + 1)/(a*x - 1) - 1)*log(-(a*((a*x + 1)/(a*x - 1) + 1)/((a*x + 1)*a
/(a*x - 1) - a) + 1)/(a*((a*x + 1)/(a*x - 1) + 1)/((a*x + 1)*a/(a*x - 1) - a) - 1))/(a^4*((a*x + 1)/(a*x - 1)
- 1)^5))

________________________________________________________________________________________

Mupad [B]
time = 0.91, size = 53, normalized size = 0.85 \begin {gather*} \frac {\frac {\ln \left (a^2\,x^2-1\right )}{15}+\frac {a^2\,x^2}{15}}{a^3}-\frac {a\,x^4}{20}+\frac {x^3\,\mathrm {atanh}\left (a\,x\right )}{3}-\frac {a^2\,x^5\,\mathrm {atanh}\left (a\,x\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^2*atanh(a*x)*(a^2*x^2 - 1),x)

[Out]

(log(a^2*x^2 - 1)/15 + (a^2*x^2)/15)/a^3 - (a*x^4)/20 + (x^3*atanh(a*x))/3 - (a^2*x^5*atanh(a*x))/5

________________________________________________________________________________________