3.1068 \(\int \frac{e^{2 \tanh ^{-1}(a x)} x^4}{(c-a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=86 \[ \frac{11}{16 a^5 c^3 (1-a x)}-\frac{1}{16 a^5 c^3 (a x+1)}-\frac{3}{8 a^5 c^3 (1-a x)^2}+\frac{1}{12 a^5 c^3 (1-a x)^3}-\frac{\tanh ^{-1}(a x)}{4 a^5 c^3} \]

[Out]

1/(12*a^5*c^3*(1 - a*x)^3) - 3/(8*a^5*c^3*(1 - a*x)^2) + 11/(16*a^5*c^3*(1 - a*x)) - 1/(16*a^5*c^3*(1 + a*x))
- ArcTanh[a*x]/(4*a^5*c^3)

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Rubi [A]  time = 0.137553, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6150, 88, 207} \[ \frac{11}{16 a^5 c^3 (1-a x)}-\frac{1}{16 a^5 c^3 (a x+1)}-\frac{3}{8 a^5 c^3 (1-a x)^2}+\frac{1}{12 a^5 c^3 (1-a x)^3}-\frac{\tanh ^{-1}(a x)}{4 a^5 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*x^4)/(c - a^2*c*x^2)^3,x]

[Out]

1/(12*a^5*c^3*(1 - a*x)^3) - 3/(8*a^5*c^3*(1 - a*x)^2) + 11/(16*a^5*c^3*(1 - a*x)) - 1/(16*a^5*c^3*(1 + a*x))
- ArcTanh[a*x]/(4*a^5*c^3)

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])

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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^4}{(1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=\frac{\int \left (\frac{1}{4 a^4 (-1+a x)^4}+\frac{3}{4 a^4 (-1+a x)^3}+\frac{11}{16 a^4 (-1+a x)^2}+\frac{1}{16 a^4 (1+a x)^2}+\frac{1}{4 a^4 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3}\\ &=\frac{1}{12 a^5 c^3 (1-a x)^3}-\frac{3}{8 a^5 c^3 (1-a x)^2}+\frac{11}{16 a^5 c^3 (1-a x)}-\frac{1}{16 a^5 c^3 (1+a x)}+\frac{\int \frac{1}{-1+a^2 x^2} \, dx}{4 a^4 c^3}\\ &=\frac{1}{12 a^5 c^3 (1-a x)^3}-\frac{3}{8 a^5 c^3 (1-a x)^2}+\frac{11}{16 a^5 c^3 (1-a x)}-\frac{1}{16 a^5 c^3 (1+a x)}-\frac{\tanh ^{-1}(a x)}{4 a^5 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0411151, size = 64, normalized size = 0.74 \[ \frac{-9 a^3 x^3+6 a^2 x^2+5 a x-3 (a x-1)^3 (a x+1) \tanh ^{-1}(a x)-4}{12 a^5 c^3 (a x-1)^3 (a x+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*ArcTanh[a*x])*x^4)/(c - a^2*c*x^2)^3,x]

[Out]

(-4 + 5*a*x + 6*a^2*x^2 - 9*a^3*x^3 - 3*(-1 + a*x)^3*(1 + a*x)*ArcTanh[a*x])/(12*a^5*c^3*(-1 + a*x)^3*(1 + a*x
))

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Maple [A]  time = 0.036, size = 90, normalized size = 1.1 \begin{align*} -{\frac{1}{16\,{a}^{5}{c}^{3} \left ( ax+1 \right ) }}-{\frac{\ln \left ( ax+1 \right ) }{8\,{a}^{5}{c}^{3}}}-{\frac{1}{12\,{a}^{5}{c}^{3} \left ( ax-1 \right ) ^{3}}}-{\frac{3}{8\,{a}^{5}{c}^{3} \left ( ax-1 \right ) ^{2}}}-{\frac{11}{16\,{a}^{5}{c}^{3} \left ( ax-1 \right ) }}+{\frac{\ln \left ( ax-1 \right ) }{8\,{a}^{5}{c}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*x^4/(-a^2*c*x^2+c)^3,x)

[Out]

-1/16/a^5/c^3/(a*x+1)-1/8/c^3/a^5*ln(a*x+1)-1/12/c^3/a^5/(a*x-1)^3-3/8/c^3/a^5/(a*x-1)^2-11/16/c^3/a^5/(a*x-1)
+1/8/c^3/a^5*ln(a*x-1)

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Maxima [A]  time = 0.969717, size = 127, normalized size = 1.48 \begin{align*} -\frac{9 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 5 \, a x + 4}{12 \,{\left (a^{9} c^{3} x^{4} - 2 \, a^{8} c^{3} x^{3} + 2 \, a^{6} c^{3} x - a^{5} c^{3}\right )}} - \frac{\log \left (a x + 1\right )}{8 \, a^{5} c^{3}} + \frac{\log \left (a x - 1\right )}{8 \, a^{5} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^4/(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/12*(9*a^3*x^3 - 6*a^2*x^2 - 5*a*x + 4)/(a^9*c^3*x^4 - 2*a^8*c^3*x^3 + 2*a^6*c^3*x - a^5*c^3) - 1/8*log(a*x
+ 1)/(a^5*c^3) + 1/8*log(a*x - 1)/(a^5*c^3)

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Fricas [A]  time = 2.38175, size = 273, normalized size = 3.17 \begin{align*} -\frac{18 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 10 \, a x + 3 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x + 1\right ) - 3 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x - 1\right ) + 8}{24 \,{\left (a^{9} c^{3} x^{4} - 2 \, a^{8} c^{3} x^{3} + 2 \, a^{6} c^{3} x - a^{5} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^4/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/24*(18*a^3*x^3 - 12*a^2*x^2 - 10*a*x + 3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x + 1) - 3*(a^4*x^4 - 2*a^
3*x^3 + 2*a*x - 1)*log(a*x - 1) + 8)/(a^9*c^3*x^4 - 2*a^8*c^3*x^3 + 2*a^6*c^3*x - a^5*c^3)

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Sympy [A]  time = 0.685253, size = 88, normalized size = 1.02 \begin{align*} - \frac{9 a^{3} x^{3} - 6 a^{2} x^{2} - 5 a x + 4}{12 a^{9} c^{3} x^{4} - 24 a^{8} c^{3} x^{3} + 24 a^{6} c^{3} x - 12 a^{5} c^{3}} + \frac{\frac{\log{\left (x - \frac{1}{a} \right )}}{8} - \frac{\log{\left (x + \frac{1}{a} \right )}}{8}}{a^{5} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(9*a**3*x**3 - 6*a**2*x**2 - 5*a*x + 4)/(12*a**9*c**3*x**4 - 24*a**8*c**3*x**3 + 24*a**6*c**3*x - 12*a**5*c**
3) + (log(x - 1/a)/8 - log(x + 1/a)/8)/(a**5*c**3)

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Giac [A]  time = 1.13778, size = 101, normalized size = 1.17 \begin{align*} -\frac{\log \left ({\left | a x + 1 \right |}\right )}{8 \, a^{5} c^{3}} + \frac{\log \left ({\left | a x - 1 \right |}\right )}{8 \, a^{5} c^{3}} - \frac{9 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 5 \, a x + 4}{12 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{3} a^{5} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^4/(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

-1/8*log(abs(a*x + 1))/(a^5*c^3) + 1/8*log(abs(a*x - 1))/(a^5*c^3) - 1/12*(9*a^3*x^3 - 6*a^2*x^2 - 5*a*x + 4)/
((a*x + 1)*(a*x - 1)^3*a^5*c^3)