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

Optimal. Leaf size=46 \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{x}+\frac{b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c}}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c}} \]

[Out]

(b*ArcTan[x/Sqrt[c]])/Sqrt[c] - (a + b*ArcTanh[c/x^2])/x + (b*ArcTanh[x/Sqrt[c]])/Sqrt[c]

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Rubi [A]  time = 0.0309239, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {6097, 263, 212, 206, 203} \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{x}+\frac{b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c}}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x^2])/x^2,x]

[Out]

(b*ArcTan[x/Sqrt[c]])/Sqrt[c] - (a + b*ArcTanh[c/x^2])/x + (b*ArcTanh[x/Sqrt[c]])/Sqrt[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 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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0180695, size = 72, normalized size = 1.57 \[ -\frac{a}{x}-\frac{b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{x}-\frac{b \log \left (\sqrt{c}-x\right )}{2 \sqrt{c}}+\frac{b \log \left (\sqrt{c}+x\right )}{2 \sqrt{c}}+\frac{b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 44, normalized size = 1. \begin{align*} -{\frac{a}{x}}-{\frac{b}{x}{\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) }+{b\arctan \left ({x{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}+{b{\it Artanh} \left ({\frac{1}{x}\sqrt{c}} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.7739, size = 385, normalized size = 8.37 \begin{align*} \left [\frac{2 \, b \sqrt{c} x \arctan \left (\frac{x}{\sqrt{c}}\right ) + b \sqrt{c} x \log \left (\frac{x^{2} + 2 \, \sqrt{c} x + c}{x^{2} - c}\right ) - b c \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) - 2 \, a c}{2 \, c x}, -\frac{2 \, b \sqrt{-c} x \arctan \left (\frac{\sqrt{-c} x}{c}\right ) + b \sqrt{-c} x \log \left (\frac{x^{2} - 2 \, \sqrt{-c} x - c}{x^{2} + c}\right ) + b c \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) + 2 \, a c}{2 \, c x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*b*sqrt(c)*x*arctan(x/sqrt(c)) + b*sqrt(c)*x*log((x^2 + 2*sqrt(c)*x + c)/(x^2 - c)) - b*c*log((x^2 + c)
/(x^2 - c)) - 2*a*c)/(c*x), -1/2*(2*b*sqrt(-c)*x*arctan(sqrt(-c)*x/c) + b*sqrt(-c)*x*log((x^2 - 2*sqrt(-c)*x -
c)/(x^2 + c)) + b*c*log((x^2 + c)/(x^2 - c)) + 2*a*c)/(c*x)]

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Sympy [A]  time = 13.8796, size = 593, normalized size = 12.89 \begin{align*} \begin{cases} - \frac{a}{x} & \text{for}\: c = 0 \\- \frac{a - \infty b}{x} & \text{for}\: c = - x^{2} \\- \frac{a + \infty b}{x} & \text{for}\: c = x^{2} \\\frac{2 a c^{\frac{39}{2}}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} - \frac{2 a c^{\frac{35}{2}} x^{4}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} + \frac{2 b c^{\frac{39}{2}} \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} - \frac{2 b c^{\frac{35}{2}} x^{4} \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} + \frac{2 b c^{19} x \log{\left (- \sqrt{c} + x \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} - \frac{b c^{19} x \log{\left (- i \sqrt{c} + x \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} + \frac{i b c^{19} x \log{\left (- i \sqrt{c} + x \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} - \frac{b c^{19} x \log{\left (i \sqrt{c} + x \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} - \frac{i b c^{19} x \log{\left (i \sqrt{c} + x \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} + \frac{2 b c^{19} x \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} - \frac{2 b c^{17} x^{5} \log{\left (- \sqrt{c} + x \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} + \frac{b c^{17} x^{5} \log{\left (- i \sqrt{c} + x \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} - \frac{i b c^{17} x^{5} \log{\left (- i \sqrt{c} + x \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} + \frac{b c^{17} x^{5} \log{\left (i \sqrt{c} + x \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} + \frac{i b c^{17} x^{5} \log{\left (i \sqrt{c} + x \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} - \frac{2 b c^{17} x^{5} \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{- 2 c^{\frac{39}{2}} x + 2 c^{\frac{35}{2}} x^{5}} & \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**2,x)

[Out]

Piecewise((-a/x, Eq(c, 0)), (-(a - oo*b)/x, Eq(c, -x**2)), (-(a + oo*b)/x, Eq(c, x**2)), (2*a*c**(39/2)/(-2*c*
*(39/2)*x + 2*c**(35/2)*x**5) - 2*a*c**(35/2)*x**4/(-2*c**(39/2)*x + 2*c**(35/2)*x**5) + 2*b*c**(39/2)*atanh(c
/x**2)/(-2*c**(39/2)*x + 2*c**(35/2)*x**5) - 2*b*c**(35/2)*x**4*atanh(c/x**2)/(-2*c**(39/2)*x + 2*c**(35/2)*x*
*5) + 2*b*c**19*x*log(-sqrt(c) + x)/(-2*c**(39/2)*x + 2*c**(35/2)*x**5) - b*c**19*x*log(-I*sqrt(c) + x)/(-2*c*
*(39/2)*x + 2*c**(35/2)*x**5) + I*b*c**19*x*log(-I*sqrt(c) + x)/(-2*c**(39/2)*x + 2*c**(35/2)*x**5) - b*c**19*
x*log(I*sqrt(c) + x)/(-2*c**(39/2)*x + 2*c**(35/2)*x**5) - I*b*c**19*x*log(I*sqrt(c) + x)/(-2*c**(39/2)*x + 2*
c**(35/2)*x**5) + 2*b*c**19*x*atanh(c/x**2)/(-2*c**(39/2)*x + 2*c**(35/2)*x**5) - 2*b*c**17*x**5*log(-sqrt(c)
+ x)/(-2*c**(39/2)*x + 2*c**(35/2)*x**5) + b*c**17*x**5*log(-I*sqrt(c) + x)/(-2*c**(39/2)*x + 2*c**(35/2)*x**5
) - I*b*c**17*x**5*log(-I*sqrt(c) + x)/(-2*c**(39/2)*x + 2*c**(35/2)*x**5) + b*c**17*x**5*log(I*sqrt(c) + x)/(
-2*c**(39/2)*x + 2*c**(35/2)*x**5) + I*b*c**17*x**5*log(I*sqrt(c) + x)/(-2*c**(39/2)*x + 2*c**(35/2)*x**5) - 2
*b*c**17*x**5*atanh(c/x**2)/(-2*c**(39/2)*x + 2*c**(35/2)*x**5), True))

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Giac [A]  time = 1.23177, size = 84, normalized size = 1.83 \begin{align*} -b c{\left (\frac{\arctan \left (\frac{x}{\sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{\arctan \left (\frac{x}{\sqrt{c}}\right )}{c^{\frac{3}{2}}}\right )} - \frac{b \log \left (\frac{x^{2} + c}{x^{2} - c}\right )}{2 \, x} - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*c*(arctan(x/sqrt(-c))/(sqrt(-c)*c) - arctan(x/sqrt(c))/c^(3/2)) - 1/2*b*log((x^2 + c)/(x^2 - c))/x - a/x