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

Optimal. Leaf size=102 \[ \frac {\text {Int}\left (\frac {1}{\tanh ^{-1}(a x)},x\right )}{a^5}-\frac {3 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^6}+\frac {\text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{2 a^6}-\frac {x}{a^5 \tanh ^{-1}(a x)}+\frac {2 x}{a^5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {x}{a^5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.90, 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 = {} \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx \]

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*}

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Mathematica [A]  time = 12.13, size = 0, normalized size = 0.00 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx \]

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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fricas [A]  time = 1.36, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x^{5}}{{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x^{5}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]

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)

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maple [A]  time = 0.51, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\left (-a^{2} x^{2}+1\right )^{3} \arctanh \left (a x \right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, x^{5}}{{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right ) - {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-a x + 1\right )} - \int -\frac {2 \, {\left (a^{2} x^{6} - 5 \, x^{4}\right )}}{{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) - {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x^5}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x^{5}}{a^{6} x^{6} \operatorname {atanh}^{2}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{2}{\left (a x \right )} - \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]

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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