3.329 \(\int \frac{x^5}{(1-a^2 x^2)^3 \tanh ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=101 \[ \frac{\text{Unintegrable}\left (\frac{1}{\tanh ^{-1}(a x)},x\right )}{a^5}-\frac{3 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^6}+\frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{2 a^6}+\frac{2 x}{a^5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{x}{a^5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac{x}{a^5 \tanh ^{-1}(a x)} \]

[Out]

-(x/(a^5*ArcTanh[a*x])) - x/(a^5*(1 - a^2*x^2)^2*ArcTanh[a*x]) + (2*x)/(a^5*(1 - a^2*x^2)*ArcTanh[a*x]) - (3*C
oshIntegral[2*ArcTanh[a*x]])/(2*a^6) + CoshIntegral[4*ArcTanh[a*x]]/(2*a^6) + Unintegrable[ArcTanh[a*x]^(-1),
x]/a^5

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Rubi [A]  time = 0.904577, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^5}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]

[Out]

-(x/(a^5*ArcTanh[a*x])) - x/(a^5*(1 - a^2*x^2)^2*ArcTanh[a*x]) + (2*x)/(a^5*(1 - a^2*x^2)*ArcTanh[a*x]) - (3*C
oshIntegral[2*ArcTanh[a*x]])/(2*a^6) + CoshIntegral[4*ArcTanh[a*x]]/(2*a^6) + Defer[Int][ArcTanh[a*x]^(-1), x]
/a^5

Rubi steps

\begin{align*} \int \frac{x^5}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx &=\frac{\int \frac{x^3}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx}{a^2}-\frac{\int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx}{a^2}\\ &=\frac{\int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx}{a^4}-2 \frac{\int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx}{a^4}+\frac{\int \frac{x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx}{a^4}\\ &=-\frac{x}{a^5 \tanh ^{-1}(a x)}-\frac{x}{a^5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{\int \frac{1}{\tanh ^{-1}(a x)} \, dx}{a^5}+\frac{\int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx}{a^5}-2 \left (-\frac{x}{a^5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx}{a^5}+\frac{\int \frac{x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx}{a^3}\right )+\frac{3 \int \frac{x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx}{a^3}\\ &=-\frac{x}{a^5 \tanh ^{-1}(a x)}-\frac{x}{a^5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh ^4(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^6}-2 \left (-\frac{x}{a^5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^6}+\frac{\operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^6}\right )+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^6}+\frac{\int \frac{1}{\tanh ^{-1}(a x)} \, dx}{a^5}\\ &=-\frac{x}{a^5 \tanh ^{-1}(a x)}-\frac{x}{a^5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-2 \left (-\frac{x}{a^5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^6}+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^6}\right )+\frac{\operatorname{Subst}\left (\int \left (\frac{3}{8 x}+\frac{\cosh (2 x)}{2 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^6}+\frac{3 \operatorname{Subst}\left (\int \left (-\frac{1}{8 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^6}+\frac{\int \frac{1}{\tanh ^{-1}(a x)} \, dx}{a^5}\\ &=-\frac{x}{a^5 \tanh ^{-1}(a x)}-\frac{x}{a^5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^6}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^6}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^6}-2 \left (-\frac{x}{a^5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^6}\right )+\frac{\int \frac{1}{\tanh ^{-1}(a x)} \, dx}{a^5}\\ &=-\frac{x}{a^5 \tanh ^{-1}(a x)}-\frac{x}{a^5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^6}-2 \left (-\frac{x}{a^5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^6}\right )+\frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{2 a^6}+\frac{\int \frac{1}{\tanh ^{-1}(a x)} \, dx}{a^5}\\ \end{align*}

Mathematica [A]  time = 11.9583, size = 0, normalized size = 0. \[ \int \frac{x^5}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]

[Out]

Integrate[x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]^2), x]

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Maple [A]  time = 0.199, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{ \left ( -{a}^{2}{x}^{2}+1 \right ) ^{3} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(-a^2*x^2+1)^3/arctanh(a*x)^2,x)

[Out]

int(x^5/(-a^2*x^2+1)^3/arctanh(a*x)^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, x^{5}}{{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right ) -{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-a x + 1\right )} - \int -\frac{2 \,{\left (a^{2} x^{6} - 5 \, x^{4}\right )}}{{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) -{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-a^2*x^2+1)^3/arctanh(a*x)^2,x, algorithm="maxima")

[Out]

-2*x^5/((a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1) - (a^5*x^4 - 2*a^3*x^2 + a)*log(-a*x + 1)) - integrate(-2*(a^2*
x^6 - 5*x^4)/((a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)*log(a*x + 1) - (a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)*log(-
a*x + 1)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{x^{5}}{{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname{artanh}\left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-a^2*x^2+1)^3/arctanh(a*x)^2,x, algorithm="fricas")

[Out]

integral(-x^5/((a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*arctanh(a*x)^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{5}}{a^{6} x^{6} \operatorname{atanh}^{2}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname{atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname{atanh}^{2}{\left (a x \right )} - \operatorname{atanh}^{2}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(-a**2*x**2+1)**3/atanh(a*x)**2,x)

[Out]

-Integral(x**5/(a**6*x**6*atanh(a*x)**2 - 3*a**4*x**4*atanh(a*x)**2 + 3*a**2*x**2*atanh(a*x)**2 - atanh(a*x)**
2), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{5}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname{artanh}\left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-a^2*x^2+1)^3/arctanh(a*x)^2,x, algorithm="giac")

[Out]

integrate(-x^5/((a^2*x^2 - 1)^3*arctanh(a*x)^2), x)