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

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Rubi [A]  time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.66, size = 48, normalized size = 1.37 \[ -\frac {b x \log \left (-c^{2} + x^{2}\right ) - 2 \, b x \log \relax (x) + b c \log \left (-\frac {c + x}{c - x}\right ) + 2 \, a c}{2 \, c x} \]

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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giac [B]  time = 0.20, size = 87, normalized size = 2.49 \[ \frac {b \log \left (-\frac {c + x}{c - x} + 1\right ) - b \log \left (-\frac {c + x}{c - x}\right ) - \frac {b \log \left (-\frac {c + x}{c - x}\right )}{\frac {c + x}{c - x} - 1} - \frac {2 \, a}{\frac {c + x}{c - x} - 1}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 37, normalized size = 1.06 \[ -\frac {a}{x}-\frac {b \arctanh \left (\frac {c}{x}\right )}{x}-\frac {b \ln \left (1-\frac {c^{2}}{x^{2}}\right )}{2 c} \]

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 = 0.31, size = 37, normalized size = 1.06 \[ -\frac {b {\left (\frac {2 \, c \operatorname {artanh}\left (\frac {c}{x}\right )}{x} + \log \left (-\frac {c^{2}}{x^{2}} + 1\right )\right )}}{2 \, c} - \frac {a}{x} \]

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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mupad [B]  time = 0.74, size = 43, normalized size = 1.23 \[ \frac {b\,x\,\ln \relax (x)-\frac {b\,x\,\ln \left (x^2-c^2\right )}{2}}{c\,x}-\frac {a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x))/x^2,x)

[Out]

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

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sympy [A]  time = 0.79, size = 39, normalized size = 1.11 \[ \begin {cases} - \frac {a}{x} - \frac {b \operatorname {atanh}{\left (\frac {c}{x} \right )}}{x} + \frac {b \log {\relax (x )}}{c} - \frac {b \log {\left (- c + x \right )}}{c} - \frac {b \operatorname {atanh}{\left (\frac {c}{x} \right )}}{c} & \text {for}\: c \neq 0 \\- \frac {a}{x} & \text {otherwise} \end {cases} \]

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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