3.468 \(\int \frac {\tanh ^{-1}(a x)}{(c-a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=48 \[ \frac {x \tanh ^{-1}(a x)}{c \sqrt {c-a^2 c x^2}}-\frac {1}{a c \sqrt {c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {5958} \[ \frac {x \tanh ^{-1}(a x)}{c \sqrt {c-a^2 c x^2}}-\frac {1}{a c \sqrt {c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]/(c - a^2*c*x^2)^(3/2),x]

[Out]

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

Rule 5958

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[b/(c*d*Sqrt[d + e*x^2]
), x] + Simp[(x*(a + b*ArcTanh[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
]

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=-\frac {1}{a c \sqrt {c-a^2 c x^2}}+\frac {x \tanh ^{-1}(a x)}{c \sqrt {c-a^2 c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 43, normalized size = 0.90 \[ \frac {\sqrt {c-a^2 c x^2} \left (1-a x \tanh ^{-1}(a x)\right )}{a c^2 \left (a^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 54, normalized size = 1.12 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - 2\right )}}{2 \, {\left (a^{3} c^{2} x^{2} - a c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.69, size = 70, normalized size = 1.46 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} x \log \left (-\frac {a x + 1}{a x - 1}\right )}{2 \, {\left (a^{2} c x^{2} - c\right )} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.47, size = 74, normalized size = 1.54 \[ -\frac {\left (\arctanh \left (a x \right )-1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{2 a \left (a x -1\right ) c^{2}}-\frac {\left (\arctanh \left (a x \right )+1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{2 a \left (a x +1\right ) c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.43, size = 90, normalized size = 1.88 \[ -\frac {a^{2} {\left (\frac {\sqrt {-a^{2} c x^{2} + c}}{a^{4} c x + a^{3} c} - \frac {\sqrt {-a^{2} c x^{2} + c}}{a^{4} c x - a^{3} c}\right )}}{2 \, c} + \frac {x \operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} c x^{2} + c} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2*(sqrt(-a^2*c*x^2 + c)/(a^4*c*x + a^3*c) - sqrt(-a^2*c*x^2 + c)/(a^4*c*x - a^3*c))/c + x*arctanh(a*x)/
(sqrt(-a^2*c*x^2 + c)*c)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)/(c - a^2*c*x^2)^(3/2),x)

[Out]

int(atanh(a*x)/(c - a^2*c*x^2)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atanh(a*x)/(-c*(a*x - 1)*(a*x + 1))**(3/2), x)

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