### 3.589 $$\int \frac{e^{4 \coth ^{-1}(a x)}}{(c-a^2 c x^2)^4} \, dx$$

Optimal. Leaf size=122 $\frac{5}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (a x+1)}+\frac{1}{16 a c^4 (1-a x)^2}+\frac{1}{16 a c^4 (1-a x)^3}+\frac{1}{16 a c^4 (1-a x)^4}+\frac{1}{20 a c^4 (1-a x)^5}+\frac{3 \tanh ^{-1}(a x)}{32 a c^4}$

[Out]

1/(20*a*c^4*(1 - a*x)^5) + 1/(16*a*c^4*(1 - a*x)^4) + 1/(16*a*c^4*(1 - a*x)^3) + 1/(16*a*c^4*(1 - a*x)^2) + 5/
(64*a*c^4*(1 - a*x)) - 1/(64*a*c^4*(1 + a*x)) + (3*ArcTanh[a*x])/(32*a*c^4)

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Rubi [A]  time = 0.117967, antiderivative size = 122, 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} $\frac{5}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (a x+1)}+\frac{1}{16 a c^4 (1-a x)^2}+\frac{1}{16 a c^4 (1-a x)^3}+\frac{1}{16 a c^4 (1-a x)^4}+\frac{1}{20 a c^4 (1-a x)^5}+\frac{3 \tanh ^{-1}(a x)}{32 a c^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

1/(20*a*c^4*(1 - a*x)^5) + 1/(16*a*c^4*(1 - a*x)^4) + 1/(16*a*c^4*(1 - a*x)^3) + 1/(16*a*c^4*(1 - a*x)^2) + 5/
(64*a*c^4*(1 - a*x)) - 1/(64*a*c^4*(1 + a*x)) + (3*ArcTanh[a*x])/(32*a*c^4)

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 )^4} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx\\ &=\frac{\int \frac{1}{(1-a x)^6 (1+a x)^2} \, dx}{c^4}\\ &=\frac{\int \left (\frac{1}{4 (-1+a x)^6}-\frac{1}{4 (-1+a x)^5}+\frac{3}{16 (-1+a x)^4}-\frac{1}{8 (-1+a x)^3}+\frac{5}{64 (-1+a x)^2}+\frac{1}{64 (1+a x)^2}-\frac{3}{32 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^4}\\ &=\frac{1}{20 a c^4 (1-a x)^5}+\frac{1}{16 a c^4 (1-a x)^4}+\frac{1}{16 a c^4 (1-a x)^3}+\frac{1}{16 a c^4 (1-a x)^2}+\frac{5}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (1+a x)}-\frac{3 \int \frac{1}{-1+a^2 x^2} \, dx}{32 c^4}\\ &=\frac{1}{20 a c^4 (1-a x)^5}+\frac{1}{16 a c^4 (1-a x)^4}+\frac{1}{16 a c^4 (1-a x)^3}+\frac{1}{16 a c^4 (1-a x)^2}+\frac{5}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (1+a x)}+\frac{3 \tanh ^{-1}(a x)}{32 a c^4}\\ \end{align*}

Mathematica [A]  time = 0.0611854, size = 80, normalized size = 0.66 $\frac{-15 a^5 x^5+60 a^4 x^4-80 a^3 x^3+20 a^2 x^2+47 a x+15 (a x-1)^5 (a x+1) \tanh ^{-1}(a x)-48}{160 a c^4 (a x-1)^5 (a x+1)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-48 + 47*a*x + 20*a^2*x^2 - 80*a^3*x^3 + 60*a^4*x^4 - 15*a^5*x^5 + 15*(-1 + a*x)^5*(1 + a*x)*ArcTanh[a*x])/(1
60*a*c^4*(-1 + a*x)^5*(1 + a*x))

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Maple [A]  time = 0.055, size = 120, normalized size = 1. \begin{align*} -{\frac{1}{64\,a{c}^{4} \left ( ax+1 \right ) }}+{\frac{3\,\ln \left ( ax+1 \right ) }{64\,a{c}^{4}}}-{\frac{1}{20\,a{c}^{4} \left ( ax-1 \right ) ^{5}}}+{\frac{1}{16\,a{c}^{4} \left ( ax-1 \right ) ^{4}}}-{\frac{1}{16\,a{c}^{4} \left ( ax-1 \right ) ^{3}}}+{\frac{1}{16\,a{c}^{4} \left ( ax-1 \right ) ^{2}}}-{\frac{5}{64\,a{c}^{4} \left ( ax-1 \right ) }}-{\frac{3\,\ln \left ( ax-1 \right ) }{64\,a{c}^{4}}} \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)^4,x)

[Out]

-1/64/a/c^4/(a*x+1)+3/64*ln(a*x+1)/a/c^4-1/20/c^4/a/(a*x-1)^5+1/16/c^4/a/(a*x-1)^4-1/16/c^4/a/(a*x-1)^3+1/16/c
^4/a/(a*x-1)^2-5/64/c^4/a/(a*x-1)-3/64/c^4/a*ln(a*x-1)

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Maxima [A]  time = 1.08779, size = 176, normalized size = 1.44 \begin{align*} -\frac{15 \, a^{5} x^{5} - 60 \, a^{4} x^{4} + 80 \, a^{3} x^{3} - 20 \, a^{2} x^{2} - 47 \, a x + 48}{160 \,{\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} + \frac{3 \, \log \left (a x + 1\right )}{64 \, a c^{4}} - \frac{3 \, \log \left (a x - 1\right )}{64 \, a c^{4}} \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)^4,x, algorithm="maxima")

[Out]

-1/160*(15*a^5*x^5 - 60*a^4*x^4 + 80*a^3*x^3 - 20*a^2*x^2 - 47*a*x + 48)/(a^7*c^4*x^6 - 4*a^6*c^4*x^5 + 5*a^5*
c^4*x^4 - 5*a^3*c^4*x^2 + 4*a^2*c^4*x - a*c^4) + 3/64*log(a*x + 1)/(a*c^4) - 3/64*log(a*x - 1)/(a*c^4)

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Fricas [A]  time = 1.48673, size = 421, normalized size = 3.45 \begin{align*} -\frac{30 \, a^{5} x^{5} - 120 \, a^{4} x^{4} + 160 \, a^{3} x^{3} - 40 \, a^{2} x^{2} - 94 \, a x - 15 \,{\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (a x + 1\right ) + 15 \,{\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (a x - 1\right ) + 96}{320 \,{\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\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)^4,x, algorithm="fricas")

[Out]

-1/320*(30*a^5*x^5 - 120*a^4*x^4 + 160*a^3*x^3 - 40*a^2*x^2 - 94*a*x - 15*(a^6*x^6 - 4*a^5*x^5 + 5*a^4*x^4 - 5
*a^2*x^2 + 4*a*x - 1)*log(a*x + 1) + 15*(a^6*x^6 - 4*a^5*x^5 + 5*a^4*x^4 - 5*a^2*x^2 + 4*a*x - 1)*log(a*x - 1)
+ 96)/(a^7*c^4*x^6 - 4*a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^3*c^4*x^2 + 4*a^2*c^4*x - a*c^4)

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Sympy [A]  time = 0.957356, size = 129, normalized size = 1.06 \begin{align*} - \frac{15 a^{5} x^{5} - 60 a^{4} x^{4} + 80 a^{3} x^{3} - 20 a^{2} x^{2} - 47 a x + 48}{160 a^{7} c^{4} x^{6} - 640 a^{6} c^{4} x^{5} + 800 a^{5} c^{4} x^{4} - 800 a^{3} c^{4} x^{2} + 640 a^{2} c^{4} x - 160 a c^{4}} + \frac{- \frac{3 \log{\left (x - \frac{1}{a} \right )}}{64} + \frac{3 \log{\left (x + \frac{1}{a} \right )}}{64}}{a c^{4}} \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)**4,x)

[Out]

-(15*a**5*x**5 - 60*a**4*x**4 + 80*a**3*x**3 - 20*a**2*x**2 - 47*a*x + 48)/(160*a**7*c**4*x**6 - 640*a**6*c**4
*x**5 + 800*a**5*c**4*x**4 - 800*a**3*c**4*x**2 + 640*a**2*c**4*x - 160*a*c**4) + (-3*log(x - 1/a)/64 + 3*log(
x + 1/a)/64)/(a*c**4)

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Giac [A]  time = 1.1502, size = 171, normalized size = 1.4 \begin{align*} \frac{3 \, \log \left ({\left | -\frac{2}{a x - 1} - 1 \right |}\right )}{64 \, a c^{4}} + \frac{1}{128 \, a c^{4}{\left (\frac{2}{a x - 1} + 1\right )}} - \frac{\frac{25 \, a^{9} c^{16}}{a x - 1} - \frac{20 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{2}} + \frac{20 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{3}} - \frac{20 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{4}} + \frac{16 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{5}}}{320 \, a^{10} c^{20}} \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)^4,x, algorithm="giac")

[Out]

3/64*log(abs(-2/(a*x - 1) - 1))/(a*c^4) + 1/128/(a*c^4*(2/(a*x - 1) + 1)) - 1/320*(25*a^9*c^16/(a*x - 1) - 20*
a^9*c^16/(a*x - 1)^2 + 20*a^9*c^16/(a*x - 1)^3 - 20*a^9*c^16/(a*x - 1)^4 + 16*a^9*c^16/(a*x - 1)^5)/(a^10*c^20
)