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

Optimal. Leaf size=35 \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x}\right )}{x}-\frac{b \log \left (1-\frac{c^2}{x^2}\right )}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.0208549, antiderivative size = 35, 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} \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x}\right )}{x}-\frac{b \log \left (1-\frac{c^2}{x^2}\right )}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x])/x^2,x]

[Out]

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

Mathematica [A]  time = 0.0082962, size = 38, normalized size = 1.09 \[ -\frac{a}{x}-\frac{b \log \left (1-\frac{c^2}{x^2}\right )}{2 c}-\frac{b \tanh ^{-1}\left (\frac{c}{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c/x])/x^2,x]

[Out]

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

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Maple [A]  time = 0.004, size = 37, normalized size = 1.1 \begin{align*} -{\frac{a}{x}}-{\frac{b}{x}{\it Artanh} \left ({\frac{c}{x}} \right ) }-{\frac{b}{2\,c}\ln \left ( 1-{\frac{{c}^{2}}{{x}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.956861, size = 50, normalized size = 1.43 \begin{align*} -\frac{b{\left (\frac{2 \, c \operatorname{artanh}\left (\frac{c}{x}\right )}{x} + \log \left (-\frac{c^{2}}{x^{2}} + 1\right )\right )}}{2 \, c} - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.69369, size = 115, normalized size = 3.29 \begin{align*} -\frac{b x \log \left (-c^{2} + x^{2}\right ) - 2 \, b x \log \left (x\right ) + b c \log \left (-\frac{c + x}{c - x}\right ) + 2 \, a c}{2 \, c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.8518, size = 39, normalized size = 1.11 \begin{align*} \begin{cases} - \frac{a}{x} - \frac{b \operatorname{atanh}{\left (\frac{c}{x} \right )}}{x} + \frac{b \log{\left (x \right )}}{c} - \frac{b \log{\left (- c + x \right )}}{c} - \frac{b \operatorname{atanh}{\left (\frac{c}{x} \right )}}{c} & \text{for}\: c \neq 0 \\- \frac{a}{x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13314, size = 66, normalized size = 1.89 \begin{align*} -\frac{1}{2} \, b{\left (\frac{\log \left (-\frac{\frac{c}{x} + 1}{\frac{c}{x} - 1}\right )}{x} + \frac{\log \left ({\left | \frac{c^{2}}{x^{2}} - 1 \right |}\right )}{c}\right )} - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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