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

Optimal. Leaf size=34 \[ \frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )+\frac{1}{4} b c \log \left (c^2-x^4\right ) \]

[Out]

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

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Rubi [A]  time = 0.0180373, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6097, 263, 260} \[ \frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )+\frac{1}{4} b c \log \left (c^2-x^4\right ) \]

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*c*Log[c^2 - x^4])/4

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 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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

Mathematica [A]  time = 0.0068589, size = 39, normalized size = 1.15 \[ \frac{a x^2}{2}+\frac{1}{4} b c \log \left (x^4-c^2\right )+\frac{1}{2} b x^2 \tanh ^{-1}\left (\frac{c}{x^2}\right ) \]

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*c*Log[-c^2 + x^4])/4

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Maple [A]  time = 0.016, size = 52, normalized size = 1.5 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b{x}^{2}}{2}{\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) }+{\frac{bc}{4}\ln \left ( 1+{\frac{c}{{x}^{2}}} \right ) }-bc\ln \left ({x}^{-1} \right ) +{\frac{bc}{4}\ln \left ({\frac{c}{{x}^{2}}}-1 \right ) } \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*c*ln(1+c/x^2)-b*c*ln(1/x)+1/4*b*c*ln(c/x^2-1)

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Maxima [A]  time = 0.981602, size = 46, normalized size = 1.35 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \operatorname{artanh}\left (\frac{c}{x^{2}}\right ) + c \log \left (x^{4} - c^{2}\right )\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="maxima")

[Out]

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

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Fricas [A]  time = 1.64731, size = 99, normalized size = 2.91 \begin{align*} \frac{1}{4} \, b x^{2} \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) + \frac{1}{2} \, a x^{2} + \frac{1}{4} \, b c \log \left (x^{4} - c^{2}\right ) \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*x^2*log((x^2 + c)/(x^2 - c)) + 1/2*a*x^2 + 1/4*b*c*log(x^4 - c^2)

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Sympy [C]  time = 10.1228, size = 61, normalized size = 1.79 \begin{align*} \frac{a x^{2}}{2} + \frac{b c \log{\left (- i \sqrt{c} + x \right )}}{2} + \frac{b c \log{\left (i \sqrt{c} + x \right )}}{2} - \frac{b c \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{2} + \frac{b x^{2} \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + b*c*log(-I*sqrt(c) + x)/2 + b*c*log(I*sqrt(c) + x)/2 - b*c*atanh(c/x**2)/2 + b*x**2*atanh(c/x**2)/2

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Giac [A]  time = 1.38631, size = 63, normalized size = 1.85 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{4} \,{\left (x^{2} \log \left (-\frac{\frac{c}{x^{2}} + 1}{\frac{c}{x^{2}} - 1}\right ) + c \log \left ({\left | x^{4} - c^{2} \right |}\right )\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)) + c*log(abs(x^4 - c^2)))*b