3.102 \(\int x^2 (a+b \tanh ^{-1}(c x^3)) \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^2 x^6\right )}{6 c} \]

[Out]

(x^3*(a + b*ArcTanh[c*x^3]))/3 + (b*Log[1 - c^2*x^6])/(6*c)

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Rubi [A]  time = 0.020476, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6097, 260} \[ \frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^2 x^6\right )}{6 c} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcTanh[c*x^3]),x]

[Out]

(x^3*(a + b*ArcTanh[c*x^3]))/3 + (b*Log[1 - c^2*x^6])/(6*c)

Rule 6097

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
nh[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 - c^2*x^(2*n)), x], x
] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

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]

Rubi steps

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

Mathematica [A]  time = 0.008322, size = 42, normalized size = 1.14 \[ \frac{a x^3}{3}+\frac{b \log \left (1-c^2 x^6\right )}{6 c}+\frac{1}{3} b x^3 \tanh ^{-1}\left (c x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*ArcTanh[c*x^3]),x]

[Out]

(a*x^3)/3 + (b*x^3*ArcTanh[c*x^3])/3 + (b*Log[1 - c^2*x^6])/(6*c)

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Maple [A]  time = 0.003, size = 37, normalized size = 1. \begin{align*}{\frac{{x}^{3}a}{3}}+{\frac{b{x}^{3}{\it Artanh} \left ( c{x}^{3} \right ) }{3}}+{\frac{b\ln \left ( -{c}^{2}{x}^{6}+1 \right ) }{6\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arctanh(c*x^3)),x)

[Out]

1/3*x^3*a+1/3*b*x^3*arctanh(c*x^3)+1/6*b*ln(-c^2*x^6+1)/c

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Maxima [A]  time = 1.01133, size = 50, normalized size = 1.35 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{{\left (2 \, c x^{3} \operatorname{artanh}\left (c x^{3}\right ) + \log \left (-c^{2} x^{6} + 1\right )\right )} b}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3 + 1/6*(2*c*x^3*arctanh(c*x^3) + log(-c^2*x^6 + 1))*b/c

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Fricas [A]  time = 1.9613, size = 108, normalized size = 2.92 \begin{align*} \frac{b c x^{3} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + 2 \, a c x^{3} + b \log \left (c^{2} x^{6} - 1\right )}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*c*x^3*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 2*a*c*x^3 + b*log(c^2*x^6 - 1))/c

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: KeyError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*atanh(c*x**3)),x)

[Out]

Exception raised: KeyError

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Giac [A]  time = 1.1694, size = 66, normalized size = 1.78 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{6} \,{\left (x^{3} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + \frac{\log \left ({\left | c^{2} x^{6} - 1 \right |}\right )}{c}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3 + 1/6*(x^3*log(-(c*x^3 + 1)/(c*x^3 - 1)) + log(abs(c^2*x^6 - 1))/c)*b