3.1232 \(\int \frac{e^{-2 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx\)

Optimal. Leaf size=15 \[ -\frac{1}{a c (a x+1)} \]

[Out]

-(1/(a*c*(1 + a*x)))

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Rubi [A]  time = 0.0325568, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6140, 32} \[ -\frac{1}{a c (a x+1)} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)),x]

[Out]

-(1/(a*c*(1 + a*x)))

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 \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 = 0.0164715, size = 18, normalized size = 1.2 \[ -\frac{e^{-2 \tanh ^{-1}(a x)}}{2 a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.024, size = 16, 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)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c),x)

[Out]

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

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Maxima [A]  time = 0.958754, size = 19, normalized size = 1.27 \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)^2*(-a^2*x^2+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 = 2.06551, size = 27, normalized size = 1.8 \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)^2*(-a^2*x^2+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 = 0.340862, size = 12, normalized size = 0.8 \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)**2*(-a**2*x**2+1)/(-a**2*c*x**2+c),x)

[Out]

-1/(a**2*c*x + a*c)

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Giac [A]  time = 1.14865, size = 20, normalized size = 1.33 \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)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

-1/((a*x + 1)*a*c)