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

Optimal. Leaf size=37 \[ \frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac{b \log \left (1-c^2 x^4\right )}{4 c} \]

[Out]

(x^2*(a + b*ArcTanh[c*x^2]))/2 + (b*Log[1 - c^2*x^4])/(4*c)

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Rubi [A]  time = 0.0149314, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6097, 260} \[ \frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac{b \log \left (1-c^2 x^4\right )}{4 c} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcTanh[c*x^2]),x]

[Out]

(x^2*(a + b*ArcTanh[c*x^2]))/2 + (b*Log[1 - c^2*x^4])/(4*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 \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-(b c) \int \frac{x^3}{1-c^2 x^4} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac{b \log \left (1-c^2 x^4\right )}{4 c}\\ \end{align*}

Mathematica [A]  time = 0.0069701, size = 42, normalized size = 1.14 \[ \frac{a x^2}{2}+\frac{b \log \left (1-c^2 x^4\right )}{4 c}+\frac{1}{2} b x^2 \tanh ^{-1}\left (c x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcTanh[c*x^2]),x]

[Out]

(a*x^2)/2 + (b*x^2*ArcTanh[c*x^2])/2 + (b*Log[1 - c^2*x^4])/(4*c)

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Maple [A]  time = 0.003, size = 37, normalized size = 1. \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b{x}^{2}{\it Artanh} \left ( c{x}^{2} \right ) }{2}}+{\frac{b\ln \left ( -{c}^{2}{x}^{4}+1 \right ) }{4\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2+1/2*b*x^2*arctanh(c*x^2)+1/4*b*ln(-c^2*x^4+1)/c

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Maxima [A]  time = 0.955621, size = 50, normalized size = 1.35 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{{\left (2 \, c x^{2} \operatorname{artanh}\left (c x^{2}\right ) + \log \left (-c^{2} x^{4} + 1\right )\right )} b}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2 + 1/4*(2*c*x^2*arctanh(c*x^2) + log(-c^2*x^4 + 1))*b/c

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Fricas [A]  time = 2.06531, size = 108, normalized size = 2.92 \begin{align*} \frac{b c x^{2} \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a c x^{2} + b \log \left (c^{2} x^{4} - 1\right )}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b*c*x^2*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*a*c*x^2 + b*log(c^2*x^4 - 1))/c

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Sympy [A]  time = 10.0682, size = 71, normalized size = 1.92 \begin{align*} \begin{cases} \frac{a x^{2}}{2} + \frac{b x^{2} \operatorname{atanh}{\left (c x^{2} \right )}}{2} + \frac{b \log{\left (x - i \sqrt{\frac{1}{c}} \right )}}{2 c} + \frac{b \log{\left (x + i \sqrt{\frac{1}{c}} \right )}}{2 c} - \frac{b \operatorname{atanh}{\left (c x^{2} \right )}}{2 c} & \text{for}\: c \neq 0 \\\frac{a x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*x**2/2 + b*x**2*atanh(c*x**2)/2 + b*log(x - I*sqrt(1/c))/(2*c) + b*log(x + I*sqrt(1/c))/(2*c) - b
*atanh(c*x**2)/(2*c), Ne(c, 0)), (a*x**2/2, True))

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Giac [A]  time = 1.18776, size = 66, normalized size = 1.78 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{4} \,{\left (x^{2} \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + \frac{\log \left ({\left | c^{2} x^{4} - 1 \right |}\right )}{c}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2 + 1/4*(x^2*log(-(c*x^2 + 1)/(c*x^2 - 1)) + log(abs(c^2*x^4 - 1))/c)*b