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

Optimal. Leaf size=87 \[ \frac{1}{16 a c^3 (1-a x)}+\frac{1}{16 a c^3 (1-a x)^2}+\frac{1}{12 a c^3 (1-a x)^3}+\frac{1}{8 a c^3 (1-a x)^4}+\frac{\tanh ^{-1}(a x)}{16 a c^3} \]

[Out]

1/(8*a*c^3*(1 - a*x)^4) + 1/(12*a*c^3*(1 - a*x)^3) + 1/(16*a*c^3*(1 - a*x)^2) + 1/(16*a*c^3*(1 - a*x)) + ArcTa
nh[a*x]/(16*a*c^3)

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Rubi [A]  time = 0.0933569, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6140, 44, 207} \[ \frac{1}{16 a c^3 (1-a x)}+\frac{1}{16 a c^3 (1-a x)^2}+\frac{1}{12 a c^3 (1-a x)^3}+\frac{1}{8 a c^3 (1-a x)^4}+\frac{\tanh ^{-1}(a x)}{16 a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(8*a*c^3*(1 - a*x)^4) + 1/(12*a*c^3*(1 - a*x)^3) + 1/(16*a*c^3*(1 - a*x)^2) + 1/(16*a*c^3*(1 - a*x)) + ArcTa
nh[a*x]/(16*a*c^3)

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6140

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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

Mathematica [A]  time = 0.0375491, size = 52, normalized size = 0.6 \[ \frac{-3 a^3 x^3+12 a^2 x^2-19 a x+3 (a x-1)^4 \tanh ^{-1}(a x)+16}{48 a c^3 (a x-1)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16 - 19*a*x + 12*a^2*x^2 - 3*a^3*x^3 + 3*(-1 + a*x)^4*ArcTanh[a*x])/(48*a*c^3*(-1 + a*x)^4)

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Maple [A]  time = 0.049, size = 90, normalized size = 1. \begin{align*}{\frac{\ln \left ( ax+1 \right ) }{32\,a{c}^{3}}}+{\frac{1}{8\,a{c}^{3} \left ( ax-1 \right ) ^{4}}}-{\frac{1}{12\,a{c}^{3} \left ( ax-1 \right ) ^{3}}}+{\frac{1}{16\,a{c}^{3} \left ( ax-1 \right ) ^{2}}}-{\frac{1}{16\,a{c}^{3} \left ( ax-1 \right ) }}-{\frac{\ln \left ( ax-1 \right ) }{32\,a{c}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*ln(a*x+1)/a/c^3+1/8/c^3/a/(a*x-1)^4-1/12/a/c^3/(a*x-1)^3+1/16/a/c^3/(a*x-1)^2-1/16/a/c^3/(a*x-1)-1/32/a/c
^3*ln(a*x-1)

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Maxima [A]  time = 1.04561, size = 138, normalized size = 1.59 \begin{align*} -\frac{3 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 19 \, a x - 16}{48 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} + \frac{\log \left (a x + 1\right )}{32 \, a c^{3}} - \frac{\log \left (a x - 1\right )}{32 \, a c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(3*a^3*x^3 - 12*a^2*x^2 + 19*a*x - 16)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^
3) + 1/32*log(a*x + 1)/(a*c^3) - 1/32*log(a*x - 1)/(a*c^3)

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Fricas [A]  time = 1.58107, size = 324, normalized size = 3.72 \begin{align*} -\frac{6 \, a^{3} x^{3} - 24 \, a^{2} x^{2} + 38 \, a x - 3 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x - 1\right ) - 32}{96 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(6*a^3*x^3 - 24*a^2*x^2 + 38*a*x - 3*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(a*x + 1) + 3*(a^4
*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(a*x - 1) - 32)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*
a^2*c^3*x + a*c^3)

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Sympy [A]  time = 0.681876, size = 99, normalized size = 1.14 \begin{align*} - \frac{3 a^{3} x^{3} - 12 a^{2} x^{2} + 19 a x - 16}{48 a^{5} c^{3} x^{4} - 192 a^{4} c^{3} x^{3} + 288 a^{3} c^{3} x^{2} - 192 a^{2} c^{3} x + 48 a c^{3}} - \frac{\frac{\log{\left (x - \frac{1}{a} \right )}}{32} - \frac{\log{\left (x + \frac{1}{a} \right )}}{32}}{a c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*a**3*x**3 - 12*a**2*x**2 + 19*a*x - 16)/(48*a**5*c**3*x**4 - 192*a**4*c**3*x**3 + 288*a**3*c**3*x**2 - 192
*a**2*c**3*x + 48*a*c**3) - (log(x - 1/a)/32 - log(x + 1/a)/32)/(a*c**3)

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Giac [A]  time = 1.14152, size = 123, normalized size = 1.41 \begin{align*} \frac{\log \left ({\left | -\frac{2}{a x - 1} - 1 \right |}\right )}{32 \, a c^{3}} - \frac{\frac{3 \, a^{3} c^{9}}{a x - 1} - \frac{3 \, a^{3} c^{9}}{{\left (a x - 1\right )}^{2}} + \frac{4 \, a^{3} c^{9}}{{\left (a x - 1\right )}^{3}} - \frac{6 \, a^{3} c^{9}}{{\left (a x - 1\right )}^{4}}}{48 \, a^{4} c^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*log(abs(-2/(a*x - 1) - 1))/(a*c^3) - 1/48*(3*a^3*c^9/(a*x - 1) - 3*a^3*c^9/(a*x - 1)^2 + 4*a^3*c^9/(a*x -
1)^3 - 6*a^3*c^9/(a*x - 1)^4)/(a^4*c^12)