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

Optimal. Leaf size=48 \[ \frac {1}{2 a^4 (1-a x)}+\frac {5 \log (1-a x)}{4 a^4}-\frac {\log (a x+1)}{4 a^4}+\frac {x}{a^3} \]

[Out]

x/a^3+1/2/a^4/(-a*x+1)+5/4*ln(-a*x+1)/a^4-1/4*ln(a*x+1)/a^4

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Rubi [A]  time = 0.11, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6150, 88} \[ \frac {x}{a^3}+\frac {1}{2 a^4 (1-a x)}+\frac {5 \log (1-a x)}{4 a^4}-\frac {\log (a x+1)}{4 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/a^3 + 1/(2*a^4*(1 - a*x)) + (5*Log[1 - a*x])/(4*a^4) - Log[1 + a*x]/(4*a^4)

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 6150

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p
] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\int \frac {x^3}{(1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac {1}{a^3}+\frac {1}{2 a^3 (-1+a x)^2}+\frac {5}{4 a^3 (-1+a x)}-\frac {1}{4 a^3 (1+a x)}\right ) \, dx\\ &=\frac {x}{a^3}+\frac {1}{2 a^4 (1-a x)}+\frac {5 \log (1-a x)}{4 a^4}-\frac {\log (1+a x)}{4 a^4}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 39, normalized size = 0.81 \[ \frac {4 a x+\frac {2}{1-a x}+5 \log (1-a x)-\log (a x+1)}{4 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*x + 2/(1 - a*x) + 5*Log[1 - a*x] - Log[1 + a*x])/(4*a^4)

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fricas [A]  time = 0.57, size = 55, normalized size = 1.15 \[ \frac {4 \, a^{2} x^{2} - 4 \, a x - {\left (a x - 1\right )} \log \left (a x + 1\right ) + 5 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \, {\left (a^{5} x - a^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*a^2*x^2 - 4*a*x - (a*x - 1)*log(a*x + 1) + 5*(a*x - 1)*log(a*x - 1) - 2)/(a^5*x - a^4)

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giac [A]  time = 0.19, size = 42, normalized size = 0.88 \[ \frac {x}{a^{3}} - \frac {\log \left ({\left | a x + 1 \right |}\right )}{4 \, a^{4}} + \frac {5 \, \log \left ({\left | a x - 1 \right |}\right )}{4 \, a^{4}} - \frac {1}{2 \, {\left (a x - 1\right )} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a^3 - 1/4*log(abs(a*x + 1))/a^4 + 5/4*log(abs(a*x - 1))/a^4 - 1/2/((a*x - 1)*a^4)

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maple [A]  time = 0.03, size = 41, normalized size = 0.85 \[ \frac {x}{a^{3}}-\frac {1}{2 a^{4} \left (a x -1\right )}+\frac {5 \ln \left (a x -1\right )}{4 a^{4}}-\frac {\ln \left (a x +1\right )}{4 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a^3-1/2/a^4/(a*x-1)+5/4/a^4*ln(a*x-1)-1/4*ln(a*x+1)/a^4

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maxima [A]  time = 0.31, size = 43, normalized size = 0.90 \[ -\frac {1}{2 \, {\left (a^{5} x - a^{4}\right )}} + \frac {x}{a^{3}} - \frac {\log \left (a x + 1\right )}{4 \, a^{4}} + \frac {5 \, \log \left (a x - 1\right )}{4 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a^5*x - a^4) + x/a^3 - 1/4*log(a*x + 1)/a^4 + 5/4*log(a*x - 1)/a^4

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mupad [B]  time = 0.07, size = 46, normalized size = 0.96 \[ \frac {5\,\ln \left (a\,x-1\right )}{4\,a^4}-\frac {\ln \left (a\,x+1\right )}{4\,a^4}-\frac {1}{2\,a\,\left (a^4\,x-a^3\right )}+\frac {x}{a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.25, size = 39, normalized size = 0.81 \[ - \frac {1}{2 a^{5} x - 2 a^{4}} + \frac {x}{a^{3}} + \frac {\frac {5 \log {\left (x - \frac {1}{a} \right )}}{4} - \frac {\log {\left (x + \frac {1}{a} \right )}}{4}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*a**5*x - 2*a**4) + x/a**3 + (5*log(x - 1/a)/4 - log(x + 1/a)/4)/a**4

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