3.54 \(\int e^{-3 \tanh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=56 \[ -\frac {2 (1-a x)^2}{a \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a}-\frac {3 \sin ^{-1}(a x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6123, 853, 669, 641, 216} \[ -\frac {2 (1-a x)^2}{a \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a}-\frac {3 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^(-3*ArcTanh[a*x]),x]

[Out]

(-2*(1 - a*x)^2)/(a*Sqrt[1 - a^2*x^2]) - (3*Sqrt[1 - a^2*x^2])/a - (3*ArcSin[a*x])/a

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 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 669

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(p + 1)), x] - Dist[(e^2*(m + p))/(c*(p + 1)), Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 853

Int[((d_) + (e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a^
m, Int[((f + g*x)^n*(a + c*x^2)^(m + p))/(d - e*x)^m, x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[m, 0] && IntegerQ[n]

Rule 6123

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.)), x_Symbol] :> Int[(1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)*Sqrt[1 - a^2*
x^2]), x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 39, normalized size = 0.70 \[ \frac {\sqrt {1-a^2 x^2} \left (-\frac {4}{a x+1}-1\right )}{a}-\frac {3 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(-3*ArcTanh[a*x]),x]

[Out]

(Sqrt[1 - a^2*x^2]*(-1 - 4/(1 + a*x)))/a - (3*ArcSin[a*x])/a

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fricas [A]  time = 0.40, size = 64, normalized size = 1.14 \[ -\frac {5 \, a x - 6 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x + 5\right )} + 5}{a^{2} x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

-(5*a*x - 6*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*x + 5) + 5)/(a^2*x + a)

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giac [A]  time = 0.20, size = 64, normalized size = 1.14 \[ -\frac {3 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a} + \frac {8}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

-3*arcsin(a*x)*sgn(a)/abs(a) - sqrt(-a^2*x^2 + 1)/a + 8/(((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a))

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maple [B]  time = 0.04, size = 164, normalized size = 2.93 \[ -\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{4} \left (x +\frac {1}{a}\right )^{3}}-\frac {2 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{3} \left (x +\frac {1}{a}\right )^{2}}-\frac {2 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a}-3 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x -\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)

[Out]

-1/a^4/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-2/a^3/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-2/a*(-a
^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)*x-3/(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 [A]  time = 0.43, size = 63, normalized size = 1.12 \[ \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{3} x^{2} + 2 \, a^{2} x + a} - \frac {3 \, \arcsin \left (a x\right )}{a} - \frac {6 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2} x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxima")

[Out]

(-a^2*x^2 + 1)^(3/2)/(a^3*x^2 + 2*a^2*x + a) - 3*arcsin(a*x)/a - 6*sqrt(-a^2*x^2 + 1)/(a^2*x + a)

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mupad [B]  time = 0.05, size = 81, normalized size = 1.45 \[ \frac {4\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - a^2*x^2)^(3/2)/(a*x + 1)^3,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)

[Out]

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

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