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

Optimal. Leaf size=87 \[ \frac{x^3 \sqrt{1-a^2 x^2}}{4 a}-\frac{x^2 \sqrt{1-a^2 x^2}}{3 a^2}-\frac{(16-9 a x) \sqrt{1-a^2 x^2}}{24 a^4}-\frac{3 \sin ^{-1}(a x)}{8 a^4} \]

[Out]

-(x^2*Sqrt[1 - a^2*x^2])/(3*a^2) + (x^3*Sqrt[1 - a^2*x^2])/(4*a) - ((16 - 9*a*x)*Sqrt[1 - a^2*x^2])/(24*a^4) -
(3*ArcSin[a*x])/(8*a^4)

________________________________________________________________________________________

Rubi [A]  time = 0.076842, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6124, 833, 780, 216} \[ \frac{x^3 \sqrt{1-a^2 x^2}}{4 a}-\frac{x^2 \sqrt{1-a^2 x^2}}{3 a^2}-\frac{(16-9 a x) \sqrt{1-a^2 x^2}}{24 a^4}-\frac{3 \sin ^{-1}(a x)}{8 a^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/E^ArcTanh[a*x],x]

[Out]

-(x^2*Sqrt[1 - a^2*x^2])/(3*a^2) + (x^3*Sqrt[1 - a^2*x^2])/(4*a) - ((16 - 9*a*x)*Sqrt[1 - a^2*x^2])/(24*a^4) -
(3*ArcSin[a*x])/(8*a^4)

Rule 6124

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

Rule 833

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

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[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]

Rubi steps

\begin{align*} \int e^{-\tanh ^{-1}(a x)} x^3 \, dx &=\int \frac{x^3 (1-a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{x^3 \sqrt{1-a^2 x^2}}{4 a}-\frac{\int \frac{x^2 \left (3 a-4 a^2 x\right )}{\sqrt{1-a^2 x^2}} \, dx}{4 a^2}\\ &=-\frac{x^2 \sqrt{1-a^2 x^2}}{3 a^2}+\frac{x^3 \sqrt{1-a^2 x^2}}{4 a}+\frac{\int \frac{x \left (8 a^2-9 a^3 x\right )}{\sqrt{1-a^2 x^2}} \, dx}{12 a^4}\\ &=-\frac{x^2 \sqrt{1-a^2 x^2}}{3 a^2}+\frac{x^3 \sqrt{1-a^2 x^2}}{4 a}-\frac{(16-9 a x) \sqrt{1-a^2 x^2}}{24 a^4}-\frac{3 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{8 a^3}\\ &=-\frac{x^2 \sqrt{1-a^2 x^2}}{3 a^2}+\frac{x^3 \sqrt{1-a^2 x^2}}{4 a}-\frac{(16-9 a x) \sqrt{1-a^2 x^2}}{24 a^4}-\frac{3 \sin ^{-1}(a x)}{8 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0413456, size = 51, normalized size = 0.59 \[ \frac{\sqrt{1-a^2 x^2} \left (6 a^3 x^3-8 a^2 x^2+9 a x-16\right )-9 \sin ^{-1}(a x)}{24 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/E^ArcTanh[a*x],x]

[Out]

(Sqrt[1 - a^2*x^2]*(-16 + 9*a*x - 8*a^2*x^2 + 6*a^3*x^3) - 9*ArcSin[a*x])/(24*a^4)

________________________________________________________________________________________

Maple [B]  time = 0.043, size = 154, normalized size = 1.8 \begin{align*} -{\frac{x}{4\,{a}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{5\,x}{8\,{a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{5}{8\,{a}^{3}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{1}{3\,{a}^{4}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{{a}^{4}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}-{\frac{1}{{a}^{3}}\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(x^3/(a*x+1)*(-a^2*x^2+1)^(1/2),x)

[Out]

-1/4/a^3*x*(-a^2*x^2+1)^(3/2)+5/8/a^3*x*(-a^2*x^2+1)^(1/2)+5/8/a^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+
1)^(1/2))+1/3/a^4*(-a^2*x^2+1)^(3/2)-1/a^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)-1/a^3/(a^2)^(1/2)*arctan((a^2)^(
1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.43909, size = 108, normalized size = 1.24 \begin{align*} -\frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{4 \, a^{3}} + \frac{5 \, \sqrt{-a^{2} x^{2} + 1} x}{8 \, a^{3}} + \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{3 \, a^{4}} - \frac{3 \, \arcsin \left (a x\right )}{8 \, a^{4}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(-a^2*x^2 + 1)^(3/2)*x/a^3 + 5/8*sqrt(-a^2*x^2 + 1)*x/a^3 + 1/3*(-a^2*x^2 + 1)^(3/2)/a^4 - 3/8*arcsin(a*x
)/a^4 - sqrt(-a^2*x^2 + 1)/a^4

________________________________________________________________________________________

Fricas [A]  time = 1.95591, size = 151, normalized size = 1.74 \begin{align*} \frac{{\left (6 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 9 \, a x - 16\right )} \sqrt{-a^{2} x^{2} + 1} + 18 \, \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{24 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((6*a^3*x^3 - 8*a^2*x^2 + 9*a*x - 16)*sqrt(-a^2*x^2 + 1) + 18*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a^4

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.19229, size = 80, normalized size = 0.92 \begin{align*} \frac{1}{24} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \, x{\left (\frac{3 \, x}{a} - \frac{4}{a^{2}}\right )} + \frac{9}{a^{3}}\right )} x - \frac{16}{a^{4}}\right )} - \frac{3 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \, a^{3}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt(-a^2*x^2 + 1)*((2*x*(3*x/a - 4/a^2) + 9/a^3)*x - 16/a^4) - 3/8*arcsin(a*x)*sgn(a)/(a^3*abs(a))