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

Optimal. Leaf size=60 \[ \frac{1-a x}{a c \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2}}{a c}+\frac{\sin ^{-1}(a x)}{a c} \]

[Out]

(1 - a*x)/(a*c*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a*c) + ArcSin[a*x]/(a*c)

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Rubi [A]  time = 0.140548, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6157, 6149, 797, 641, 216, 637} \[ \frac{1-a x}{a c \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2}}{a c}+\frac{\sin ^{-1}(a x)}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - a*x)/(a*c*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a*c) + ArcSin[a*x]/(a*c)

Rule 6157

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

Rule 6149

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

Rule 797

Int[(x_)^2*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/c, Int[(f + g*x)*(a + c*x^2)^(p
+ 1), x], x] - Dist[a/c, Int[(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, f, g, p}, x] && EqQ[a*g^2 + f^2*
c, 0]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 637

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rubi steps

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

Mathematica [A]  time = 0.0288781, size = 54, normalized size = 0.9 \[ \frac{-a^2 x^2+\sqrt{1-a^2 x^2} \sin ^{-1}(a x)-a x+2}{a c \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2 - a*x - a^2*x^2 + Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(a*c*Sqrt[1 - a^2*x^2])

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Maple [B]  time = 0.047, size = 192, normalized size = 3.2 \begin{align*}{\frac{1}{2\,{a}^{3}c \left ( x+{a}^{-1} \right ) ^{2}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{5}{4\,ac}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}+{\frac{5}{4\,c}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{1}{4\,ac}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}-{\frac{1}{4\,c}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.00184, size = 153, normalized size = 2.55 \begin{align*} \frac{2 \, a x - 2 \,{\left (a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt{-a^{2} x^{2} + 1}{\left (a x + 2\right )} + 2}{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^2*x^2+1)^(1/2)/(c-c/a^2/x^2),x, algorithm="fricas")

[Out]

(2*a*x - 2*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*x + 2) + 2)/(a^2*c*x + a*c
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{2} \int \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + a^{2} x^{2} - a x - 1}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*Integral(x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + a**2*x**2 - a*x - 1), x)/c

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Giac [A]  time = 1.26101, size = 96, normalized size = 1.6 \begin{align*} \frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{c{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a c} - \frac{2}{c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(a*x)*sgn(a)/(c*abs(a)) + sqrt(-a^2*x^2 + 1)/(a*c) - 2/(c*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*
abs(a))