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

Optimal. Leaf size=64 \[ -\frac {\text {Int}\left (\frac {1}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2},x\right )}{a^2}+\frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )}{a^3}-\frac {1}{a^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.27, 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 = {} \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(1/(a^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])) + SinhIntegral[ArcTanh[a*x]]/a^3 - Defer[Int][1/(Sqrt[1 - a^2*x^2]*A
rcTanh[a*x]^2), x]/a^2

Rubi steps

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

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Mathematica [A]  time = 3.26, size = 0, normalized size = 0.00 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.49, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctanh \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2/(-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 \[ \int \frac {x^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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