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

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

[Out]

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

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Rubi [A]  time = 0.0202372, 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} \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{2 x^2}-\frac{b \log \left (1-\frac{c^2}{x^4}\right )}{4 c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x^2])/x^3,x]

[Out]

-(a + b*ArcTanh[c/x^2])/(2*x^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 \frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{x^3} \, dx &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{2 x^2}-(b c) \int \frac{1}{\left (1-\frac{c^2}{x^4}\right ) x^5} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{2 x^2}-\frac{b \log \left (1-\frac{c^2}{x^4}\right )}{4 c}\\ \end{align*}

Mathematica [A]  time = 0.0089931, size = 42, normalized size = 1.14 \[ -\frac{a}{2 x^2}-\frac{b \log \left (1-\frac{c^2}{x^4}\right )}{4 c}-\frac{b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 37, normalized size = 1. \begin{align*} -{\frac{a}{2\,{x}^{2}}}-{\frac{b}{2\,{x}^{2}}{\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) }-{\frac{b}{4\,c}\ln \left ( 1-{\frac{{c}^{2}}{{x}^{4}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.969016, size = 50, normalized size = 1.35 \begin{align*} -\frac{b{\left (\frac{2 \, c \operatorname{artanh}\left (\frac{c}{x^{2}}\right )}{x^{2}} + \log \left (-\frac{c^{2}}{x^{4}} + 1\right )\right )}}{4 \, c} - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.77503, size = 126, normalized size = 3.41 \begin{align*} -\frac{b x^{2} \log \left (x^{4} - c^{2}\right ) - 4 \, b x^{2} \log \left (x\right ) + b c \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) + 2 \, a c}{4 \, c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 18.5034, size = 76, normalized size = 2.05 \begin{align*} \begin{cases} - \frac{a}{2 x^{2}} - \frac{b \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{2 x^{2}} + \frac{b \log{\left (x \right )}}{c} - \frac{b \log{\left (- i \sqrt{c} + x \right )}}{2 c} - \frac{b \log{\left (i \sqrt{c} + x \right )}}{2 c} + \frac{b \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{2 c} & \text{for}\: c \neq 0 \\- \frac{a}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.27518, size = 70, normalized size = 1.89 \begin{align*} -\frac{b \log \left (x^{4} - c^{2}\right )}{4 \, c} + \frac{b \log \left (x\right )}{c} - \frac{b \log \left (\frac{x^{2} + c}{x^{2} - c}\right )}{4 \, x^{2}} - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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