### 3.602 $$\int \frac{e^{-2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx$$

Optimal. Leaf size=14 $\frac{1}{a c (a x+1)}$

[Out]

1/(a*c*(1 + a*x))

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Rubi [A]  time = 0.0611401, antiderivative size = 14, 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} $\frac{1}{a c (a x+1)}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(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 = 0.0144369, size = 18, normalized size = 1.29 $-\frac{e^{-2 \coth ^{-1}(a x)}}{2 a c}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)),x]

[Out]

-1/(2*a*c*E^(2*ArcCoth[a*x]))

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Maple [A]  time = 0.039, size = 15, normalized size = 1.1 \begin{align*}{\frac{1}{ac \left ( ax+1 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03499, size = 16, normalized size = 1.14 \begin{align*} \frac{1}{a^{2} c x + a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((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 = 1.49258, size = 26, normalized size = 1.86 \begin{align*} \frac{1}{a^{2} c x + a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((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 = 0.291627, size = 10, normalized size = 0.71 \begin{align*} \frac{1}{a^{2} c x + a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((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 = 1.11098, size = 19, normalized size = 1.36 \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((a*x-1)/(a*x+1)/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

1/((a*x + 1)*a*c)