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

Optimal. Leaf size=37 \[ \frac{1}{3 a c^4 (1-a x)^3}-\frac{1}{2 a c^4 (1-a x)^4} \]

[Out]

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

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Rubi [A]  time = 0.0607729, antiderivative size = 37, 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} \[ \frac{1}{3 a c^4 (1-a x)^3}-\frac{1}{2 a c^4 (1-a x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0162178, size = 23, normalized size = 0.62 \[ -\frac{2 a x+1}{6 a c^4 (a x-1)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 + 2*a*x)/(6*a*c^4*(-1 + a*x)^4)

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Maple [A]  time = 0.046, size = 30, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{4}} \left ( -{\frac{1}{3\,a \left ( ax-1 \right ) ^{3}}}-{\frac{1}{2\,a \left ( ax-1 \right ) ^{4}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02396, size = 77, normalized size = 2.08 \begin{align*} -\frac{2 \, a x + 1}{6 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, 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)*(a*x+1)/(-a*c*x+c)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47311, size = 116, normalized size = 3.14 \begin{align*} -\frac{2 \, a x + 1}{6 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, 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)*(a*x+1)/(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.548321, size = 60, normalized size = 1.62 \begin{align*} - \frac{2 a x + 1}{6 a^{5} c^{4} x^{4} - 24 a^{4} c^{4} x^{3} + 36 a^{3} c^{4} x^{2} - 24 a^{2} c^{4} x + 6 a c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*x + 1)/(6*a**5*c**4*x**4 - 24*a**4*c**4*x**3 + 36*a**3*c**4*x**2 - 24*a**2*c**4*x + 6*a*c**4)

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Giac [A]  time = 1.16371, size = 28, normalized size = 0.76 \begin{align*} -\frac{2 \, a x + 1}{6 \,{\left (a x - 1\right )}^{4} a c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*a*x + 1)/((a*x - 1)^4*a*c^4)