### 3.569 $$\int \frac{e^{2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx$$

Optimal. Leaf size=16 $-\frac{1}{a c (1-a x)}$

[Out]

-(1/(a*c*(1 - a*x)))

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Rubi [A]  time = 0.0653081, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.136, Rules used = {6167, 6140, 32} $-\frac{1}{a c (1-a x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(1/(a*c*(1 - a*x)))

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx\\ &=-\frac{\int \frac{1}{(1-a x)^2} \, dx}{c}\\ &=-\frac{1}{a c (1-a x)}\\ \end{align*}

Mathematica [C]  time = 0.0145529, size = 18, normalized size = 1.12 $\frac{e^{2 \coth ^{-1}(a x)}}{2 a c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

E^(2*ArcCoth[a*x])/(2*a*c)

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Maple [A]  time = 0.039, size = 15, normalized size = 0.9 \begin{align*}{\frac{1}{ac \left ( ax-1 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c/a/(a*x-1)

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Maxima [A]  time = 1.02187, size = 18, normalized size = 1.12 \begin{align*} \frac{1}{a^{2} c x - a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a^2*c*x - a*c)

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Fricas [A]  time = 1.49786, size = 26, normalized size = 1.62 \begin{align*} \frac{1}{a^{2} c x - a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a^2*c*x - a*c)

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Sympy [A]  time = 0.292063, size = 10, normalized size = 0.62 \begin{align*} \frac{1}{a^{2} c x - a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a**2*c*x - a*c)

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Giac [A]  time = 1.12216, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{{\left (a x - 1\right )} a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/((a*x - 1)*a*c)