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

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

[Out]

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

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Rubi [A]  time = 0.0256217, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6097, 212, 206, 203} \[ -\frac{a+b \tanh ^{-1}\left (c x^2\right )}{x}+b \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )+b \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^2])/x^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0170497, size = 75, normalized size = 1.63 \[ -\frac{a}{x}-\frac{b \tanh ^{-1}\left (c x^2\right )}{x}-\frac{1}{2} b \sqrt{c} \log \left (1-\sqrt{c} x\right )+\frac{1}{2} b \sqrt{c} \log \left (\sqrt{c} x+1\right )+b \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 42, normalized size = 0.9 \begin{align*} -{\frac{a}{x}}-{\frac{b{\it Artanh} \left ( c{x}^{2} \right ) }{x}}+b\arctan \left ( x\sqrt{c} \right ) \sqrt{c}+b{\it Artanh} \left ( x\sqrt{c} \right ) \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*arctanh(x*c^(1/2))*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 = 2.12528, size = 385, normalized size = 8.37 \begin{align*} \left [\frac{2 \, b \sqrt{c} x \arctan \left (\sqrt{c} x\right ) + b \sqrt{c} x \log \left (\frac{c x^{2} + 2 \, \sqrt{c} x + 1}{c x^{2} - 1}\right ) - b \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) - 2 \, a}{2 \, x}, -\frac{2 \, b \sqrt{-c} x \arctan \left (\sqrt{-c} x\right ) - b \sqrt{-c} x \log \left (\frac{c x^{2} + 2 \, \sqrt{-c} x - 1}{c x^{2} + 1}\right ) + b \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a}{2 \, 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(sqrt(c)*x) + b*sqrt(c)*x*log((c*x^2 + 2*sqrt(c)*x + 1)/(c*x^2 - 1)) - b*log(-(c*x^2
+ 1)/(c*x^2 - 1)) - 2*a)/x, -1/2*(2*b*sqrt(-c)*x*arctan(sqrt(-c)*x) - b*sqrt(-c)*x*log((c*x^2 + 2*sqrt(-c)*x
- 1)/(c*x^2 + 1)) + b*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*a)/x]

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Sympy [A]  time = 15.1448, size = 699, normalized size = 15.2 \begin{align*} \text{result too large to display} \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 - oo*b)/x, Eq(c, -1/x**2)), (-(a + oo*b)/x, Eq(c, x**(-2))), (-a/x, Eq(c, 0)), (-4*a*x**4/(4*x*
*5 - 4*x/c**2) + 4*a/(4*c**2*x**5 - 4*x) - b*c**2*x**5*(1/c)**(3/2)*log(x + I*sqrt(1/c))/(4*x**5 - 4*x/c**2) -
I*b*c**2*x**5*(1/c)**(3/2)*log(x + I*sqrt(1/c))/(4*x**5 - 4*x/c**2) + 2*b*c*x**5*sqrt(1/c)*log(x - I*sqrt(1/c
))/(4*x**5 - 4*x/c**2) - 2*I*b*c*x**5*sqrt(1/c)*log(x - I*sqrt(1/c))/(4*x**5 - 4*x/c**2) + 3*b*c*x**5*sqrt(1/c
)*log(x + I*sqrt(1/c))/(4*x**5 - 4*x/c**2) + 3*I*b*c*x**5*sqrt(1/c)*log(x + I*sqrt(1/c))/(4*x**5 - 4*x/c**2) -
4*b*c*x**5*sqrt(1/c)*log(x - sqrt(1/c))/(4*x**5 - 4*x/c**2) - 4*b*c*x**5*sqrt(1/c)*atanh(c*x**2)/(4*x**5 - 4*
x/c**2) - 4*b*x**4*atanh(c*x**2)/(4*x**5 - 4*x/c**2) - 2*b*x*sqrt(1/c)*log(x - I*sqrt(1/c))/(4*c*x**5 - 4*x/c)
+ 2*I*b*x*sqrt(1/c)*log(x - I*sqrt(1/c))/(4*c*x**5 - 4*x/c) - 3*b*x*sqrt(1/c)*log(x + I*sqrt(1/c))/(4*c*x**5
- 4*x/c) - 3*I*b*x*sqrt(1/c)*log(x + I*sqrt(1/c))/(4*c*x**5 - 4*x/c) + 4*b*x*sqrt(1/c)*log(x - sqrt(1/c))/(4*c
*x**5 - 4*x/c) + 4*b*x*sqrt(1/c)*atanh(c*x**2)/(4*c*x**5 - 4*x/c) + b*x*(1/c)**(3/2)*log(x + I*sqrt(1/c))/(4*x
**5 - 4*x/c**2) + I*b*x*(1/c)**(3/2)*log(x + I*sqrt(1/c))/(4*x**5 - 4*x/c**2) + 4*b*atanh(c*x**2)/(4*c**2*x**5
- 4*x), True))

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Giac [B]  time = 1.22457, size = 107, normalized size = 2.33 \begin{align*} \frac{1}{2} \, b c{\left (\frac{2 \, \arctan \left (x \sqrt{{\left | c \right |}}\right )}{\sqrt{{\left | c \right |}}} + \frac{\log \left ({\left | x + \frac{1}{\sqrt{{\left | c \right |}}} \right |}\right )}{\sqrt{{\left | c \right |}}} - \frac{\log \left ({\left | x - \frac{1}{\sqrt{{\left | c \right |}}} \right |}\right )}{\sqrt{{\left | c \right |}}}\right )} - \frac{b \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\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]

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