3.5.20 \(\int \frac {1}{x (1-a^2 x^2)^{3/2} \tanh ^{-1}(a x)^2} \, dx\) [420]

Optimal. Leaf size=90 \[ -\frac {a x}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}-\frac {\sqrt {1-a^2 x^2}}{a x \tanh ^{-1}(a x)}+\text {Chi}\left (\tanh ^{-1}(a x)\right )-\frac {\text {Int}\left (\frac {1}{x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)},x\right )}{a} \]

[Out]

Chi(arctanh(a*x))-a*x/arctanh(a*x)/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)/a/x/arctanh(a*x)-Unintegrable(1/x^2/a
rctanh(a*x)/(-a^2*x^2+1)^(1/2),x)/a

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Rubi [A]
time = 0.40, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2),x]

[Out]

-((a*x)/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])) - Sqrt[1 - a^2*x^2]/(a*x*ArcTanh[a*x]) + CoshIntegral[ArcTanh[a*x]]
- Defer[Int][1/(x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]), x]/a

Rubi steps

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

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Mathematica [A]
time = 4.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/(x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2),x]

[Out]

Integrate[1/(x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2), x]

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Maple [A]
time = 2.67, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctanh \left (a x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^2,x)

[Out]

int(1/x/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^2,x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^2,x, algorithm="maxima")

[Out]

integrate(1/((-a^2*x^2 + 1)^(3/2)*x*arctanh(a*x)^2), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^2,x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*x^2 + 1)/((a^4*x^5 - 2*a^2*x^3 + x)*arctanh(a*x)^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*atanh(a*x)^2*(1 - a^2*x^2)^(3/2)),x)

[Out]

int(1/(x*atanh(a*x)^2*(1 - a^2*x^2)^(3/2)), x)

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