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

Optimal. Leaf size=53 \[ \frac{1}{3 a c^4 (1-a x)^3}-\frac{1}{a c^4 (1-a x)^4}+\frac{4}{5 a c^4 (1-a x)^5} \]

[Out]

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

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Rubi [A]  time = 0.0669647, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6129, 43} \[ \frac{1}{3 a c^4 (1-a x)^3}-\frac{1}{a c^4 (1-a x)^4}+\frac{4}{5 a c^4 (1-a x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(5*a*c^4*(1 - a*x)^5) - 1/(a*c^4*(1 - a*x)^4) + 1/(3*a*c^4*(1 - a*x)^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 6129

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

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0187029, size = 31, normalized size = 0.58 \[ -\frac{5 a^2 x^2+5 a x+2}{15 a c^4 (a x-1)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2 + 5*a*x + 5*a^2*x^2)/(15*a*c^4*(-1 + a*x)^5)

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Maple [A]  time = 0.045, size = 42, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{4}} \left ( -{\frac{1}{3\,a \left ( ax-1 \right ) ^{3}}}-{\frac{1}{a \left ( ax-1 \right ) ^{4}}}-{\frac{4}{5\,a \left ( ax-1 \right ) ^{5}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(-1/3/a/(a*x-1)^3-1/a/(a*x-1)^4-4/5/a/(a*x-1)^5)

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Maxima [A]  time = 1.02298, size = 104, normalized size = 1.96 \begin{align*} -\frac{5 \, a^{2} x^{2} + 5 \, a x + 2}{15 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, 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*c*x+c)^4,x, algorithm="maxima")

[Out]

-1/15*(5*a^2*x^2 + 5*a*x + 2)/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a
*c^4)

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Fricas [A]  time = 1.46517, size = 158, normalized size = 2.98 \begin{align*} -\frac{5 \, a^{2} x^{2} + 5 \, a x + 2}{15 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, 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*c*x+c)^4,x, algorithm="fricas")

[Out]

-1/15*(5*a^2*x^2 + 5*a*x + 2)/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a
*c^4)

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Sympy [A]  time = 1.59646, size = 80, normalized size = 1.51 \begin{align*} - \frac{5 a^{2} x^{2} + 5 a x + 2}{15 a^{6} c^{4} x^{5} - 75 a^{5} c^{4} x^{4} + 150 a^{4} c^{4} x^{3} - 150 a^{3} c^{4} x^{2} + 75 a^{2} c^{4} x - 15 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*c*x+c)**4,x)

[Out]

-(5*a**2*x**2 + 5*a*x + 2)/(15*a**6*c**4*x**5 - 75*a**5*c**4*x**4 + 150*a**4*c**4*x**3 - 150*a**3*c**4*x**2 +
75*a**2*c**4*x - 15*a*c**4)

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Giac [A]  time = 1.15388, size = 57, normalized size = 1.08 \begin{align*} -\frac{\frac{5}{{\left (a x - 1\right )}^{3} a} + \frac{15}{{\left (a x - 1\right )}^{4} a} + \frac{12}{{\left (a x - 1\right )}^{5} a}}{15 \, 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*c*x+c)^4,x, algorithm="giac")

[Out]

-1/15*(5/((a*x - 1)^3*a) + 15/((a*x - 1)^4*a) + 12/((a*x - 1)^5*a))/c^4