3.3.87 \(\int \frac {x^3}{(1-a^2 x^2)^2 \tanh ^{-1}(a x)^2} \, dx\) [287]

Optimal. Leaf size=61 \[ \frac {x}{a^3 \tanh ^{-1}(a x)}-\frac {x}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^4}-\frac {\text {Int}\left (\frac {1}{\tanh ^{-1}(a x)},x\right )}{a^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.23, 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 {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x/(a^3*ArcTanh[a*x]) - x/(a^3*(1 - a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[2*ArcTanh[a*x]]/a^4 - Defer[Int][ArcT
anh[a*x]^(-1), x]/a^3

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 2.37, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 32.05, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (-a^{2} x^{2}+1\right )^{2} \arctanh \left (a x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________