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

Optimal. Leaf size=119 \[ \frac{(a+1) (-a-b x+1)^{5/2}}{b^2 \sqrt{a+b x+1}}+\frac{(2 a+3) \sqrt{a+b x+1} (-a-b x+1)^{3/2}}{2 b^2}+\frac{3 (2 a+3) \sqrt{a+b x+1} \sqrt{-a-b x+1}}{2 b^2}+\frac{3 (2 a+3) \sin ^{-1}(a+b x)}{2 b^2} \]

[Out]

((1 + a)*(1 - a - b*x)^(5/2))/(b^2*Sqrt[1 + a + b*x]) + (3*(3 + 2*a)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*b
^2) + ((3 + 2*a)*(1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x])/(2*b^2) + (3*(3 + 2*a)*ArcSin[a + b*x])/(2*b^2)

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Rubi [A]  time = 0.0921294, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6163, 78, 50, 53, 619, 216} \[ \frac{(a+1) (-a-b x+1)^{5/2}}{b^2 \sqrt{a+b x+1}}+\frac{(2 a+3) \sqrt{a+b x+1} (-a-b x+1)^{3/2}}{2 b^2}+\frac{3 (2 a+3) \sqrt{a+b x+1} \sqrt{-a-b x+1}}{2 b^2}+\frac{3 (2 a+3) \sin ^{-1}(a+b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x/E^(3*ArcTanh[a + b*x]),x]

[Out]

((1 + a)*(1 - a - b*x)^(5/2))/(b^2*Sqrt[1 + a + b*x]) + (3*(3 + 2*a)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*b
^2) + ((3 + 2*a)*(1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x])/(2*b^2) + (3*(3 + 2*a)*ArcSin[a + b*x])/(2*b^2)

Rule 6163

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[((d + e*x)^m*(1
+ a*c + b*c*x)^(n/2))/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 53

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

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^{-3 \tanh ^{-1}(a+b x)} x \, dx &=\int \frac{x (1-a-b x)^{3/2}}{(1+a+b x)^{3/2}} \, dx\\ &=\frac{(1+a) (1-a-b x)^{5/2}}{b^2 \sqrt{1+a+b x}}+\frac{(3+2 a) \int \frac{(1-a-b x)^{3/2}}{\sqrt{1+a+b x}} \, dx}{b}\\ &=\frac{(1+a) (1-a-b x)^{5/2}}{b^2 \sqrt{1+a+b x}}+\frac{(3+2 a) (1-a-b x)^{3/2} \sqrt{1+a+b x}}{2 b^2}+\frac{(3 (3+2 a)) \int \frac{\sqrt{1-a-b x}}{\sqrt{1+a+b x}} \, dx}{2 b}\\ &=\frac{(1+a) (1-a-b x)^{5/2}}{b^2 \sqrt{1+a+b x}}+\frac{3 (3+2 a) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^2}+\frac{(3+2 a) (1-a-b x)^{3/2} \sqrt{1+a+b x}}{2 b^2}+\frac{(3 (3+2 a)) \int \frac{1}{\sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{2 b}\\ &=\frac{(1+a) (1-a-b x)^{5/2}}{b^2 \sqrt{1+a+b x}}+\frac{3 (3+2 a) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^2}+\frac{(3+2 a) (1-a-b x)^{3/2} \sqrt{1+a+b x}}{2 b^2}+\frac{(3 (3+2 a)) \int \frac{1}{\sqrt{(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{2 b}\\ &=\frac{(1+a) (1-a-b x)^{5/2}}{b^2 \sqrt{1+a+b x}}+\frac{3 (3+2 a) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^2}+\frac{(3+2 a) (1-a-b x)^{3/2} \sqrt{1+a+b x}}{2 b^2}-\frac{(3 (3+2 a)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{4 b^3}\\ &=\frac{(1+a) (1-a-b x)^{5/2}}{b^2 \sqrt{1+a+b x}}+\frac{3 (3+2 a) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^2}+\frac{(3+2 a) (1-a-b x)^{3/2} \sqrt{1+a+b x}}{2 b^2}+\frac{3 (3+2 a) \sin ^{-1}(a+b x)}{2 b^2}\\ \end{align*}

Mathematica [A]  time = 0.135023, size = 157, normalized size = 1.32 \[ \frac{\sqrt{-b} \left (-a^2 (b x+14)-a^3+a \left (b^2 x^2-20 b x+1\right )+b^3 x^3-6 b^2 x^2-9 b x+14\right )+6 (2 a+3) \sqrt{b} \sqrt{-a^2-2 a b x-b^2 x^2+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{-a-b x+1}}{\sqrt{2} \sqrt{-b}}\right )}{2 (-b)^{5/2} \sqrt{-(a+b x-1) (a+b x+1)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/E^(3*ArcTanh[a + b*x]),x]

[Out]

(Sqrt[-b]*(14 - a^3 - 9*b*x - 6*b^2*x^2 + b^3*x^3 - a^2*(14 + b*x) + a*(1 - 20*b*x + b^2*x^2)) + 6*(3 + 2*a)*S
qrt[b]*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSinh[(Sqrt[b]*Sqrt[1 - a - b*x])/(Sqrt[2]*Sqrt[-b])])/(2*(-b)^(5/2
)*Sqrt[-((-1 + a + b*x)*(1 + a + b*x))])

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Maple [B]  time = 0.043, size = 543, normalized size = 4.6 \begin{align*} 3\,{\frac{1}{{b}^{4}} \left ( - \left ( x+{\frac{1+a}{b}} \right ) ^{2}{b}^{2}+2\,b \left ( x+{\frac{1+a}{b}} \right ) \right ) ^{5/2} \left ( x+{b}^{-1}+{\frac{a}{b}} \right ) ^{-2}}+3\,{\frac{1}{{b}^{2}} \left ( - \left ( x+{\frac{1+a}{b}} \right ) ^{2}{b}^{2}+2\,b \left ( x+{\frac{1+a}{b}} \right ) \right ) ^{3/2}}+{\frac{9\,x}{2\,b}\sqrt{- \left ( x+{\frac{1+a}{b}} \right ) ^{2}{b}^{2}+2\,b \left ( x+{\frac{1+a}{b}} \right ) }}+{\frac{9\,a}{2\,{b}^{2}}\sqrt{- \left ( x+{\frac{1+a}{b}} \right ) ^{2}{b}^{2}+2\,b \left ( x+{\frac{1+a}{b}} \right ) }}+{\frac{9}{2\,b}\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{1+a}{b}}-{b}^{-1} \right ){\frac{1}{\sqrt{- \left ( x+{\frac{1+a}{b}} \right ) ^{2}{b}^{2}+2\,b \left ( x+{\frac{1+a}{b}} \right ) }}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+{\frac{1}{{b}^{5}} \left ( - \left ( x+{\frac{1+a}{b}} \right ) ^{2}{b}^{2}+2\,b \left ( x+{\frac{1+a}{b}} \right ) \right ) ^{{\frac{5}{2}}} \left ( x+{b}^{-1}+{\frac{a}{b}} \right ) ^{-3}}+{\frac{a}{{b}^{5}} \left ( - \left ( x+{\frac{1+a}{b}} \right ) ^{2}{b}^{2}+2\,b \left ( x+{\frac{1+a}{b}} \right ) \right ) ^{{\frac{5}{2}}} \left ( x+{b}^{-1}+{\frac{a}{b}} \right ) ^{-3}}+2\,{\frac{a}{{b}^{4}} \left ( - \left ( x+{\frac{1+a}{b}} \right ) ^{2}{b}^{2}+2\,b \left ( x+{\frac{1+a}{b}} \right ) \right ) ^{5/2} \left ( x+{b}^{-1}+{\frac{a}{b}} \right ) ^{-2}}+2\,{\frac{a}{{b}^{2}} \left ( - \left ( x+{\frac{1+a}{b}} \right ) ^{2}{b}^{2}+2\,b \left ( x+{\frac{1+a}{b}} \right ) \right ) ^{3/2}}+3\,{\frac{ax}{b}\sqrt{- \left ( x+{\frac{1+a}{b}} \right ) ^{2}{b}^{2}+2\,b \left ( x+{\frac{1+a}{b}} \right ) }}+3\,{\frac{{a}^{2}}{{b}^{2}}\sqrt{- \left ( x+{\frac{1+a}{b}} \right ) ^{2}{b}^{2}+2\,b \left ( x+{\frac{1+a}{b}} \right ) }}+3\,{\frac{a}{b\sqrt{{b}^{2}}}\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{1+a}{b}}-{b}^{-1} \right ){\frac{1}{\sqrt{- \left ( x+{\frac{1+a}{b}} \right ) ^{2}{b}^{2}+2\,b \left ( x+{\frac{1+a}{b}} \right ) }}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/b^4/(x+1/b+a/b)^2*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(5/2)+3/b^2*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(3/2
)+9/2/b*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2)*x+9/2/b^2*a*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2)+9/2/
b/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+(1+a)/b-1/b)/(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2))+1/b^5/(x+1/b+a/b)
^3*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(5/2)+1/b^5*a/(x+1/b+a/b)^3*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(5/2)
+2/b^4*a/(x+1/b+a/b)^2*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(5/2)+2/b^2*a*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))
^(3/2)+3/b*a*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2)*x+3/b^2*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2)*a^2
+3/b*a/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+(1+a)/b-1/b)/(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2))

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Maxima [B]  time = 1.45439, size = 404, normalized size = 3.39 \begin{align*} -\frac{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}} a}{b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2} + 2 \, b^{3} x + 2 \, a b^{2} + b^{2}} - \frac{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}}}{b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2} + 2 \, b^{3} x + 2 \, a b^{2} + b^{2}} + \frac{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}}}{2 \,{\left (b^{3} x + a b^{2} + b^{2}\right )}} + \frac{6 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{3} x + a b^{2} + b^{2}} + \frac{3 \, a \arcsin \left (b x + a\right )}{b^{2}} + \frac{6 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{3} x + a b^{2} + b^{2}} + \frac{9 \, \arcsin \left (b x + a\right )}{2 \, b^{2}} + \frac{3 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*a/(b^4*x^2 + 2*a*b^3*x + a^2*b^2 + 2*b^3*x + 2*a*b^2 + b^2) - (-b^2*x^2
- 2*a*b*x - a^2 + 1)^(3/2)/(b^4*x^2 + 2*a*b^3*x + a^2*b^2 + 2*b^3*x + 2*a*b^2 + b^2) + 1/2*(-b^2*x^2 - 2*a*b*x
- a^2 + 1)^(3/2)/(b^3*x + a*b^2 + b^2) + 6*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a/(b^3*x + a*b^2 + b^2) + 3*a*a
rcsin(b*x + a)/b^2 + 6*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/(b^3*x + a*b^2 + b^2) + 9/2*arcsin(b*x + a)/b^2 + 3/
2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/b^2

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Fricas [A]  time = 1.8327, size = 300, normalized size = 2.52 \begin{align*} -\frac{3 \,{\left ({\left (2 \, a + 3\right )} b x + 2 \, a^{2} + 5 \, a + 3\right )} \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) +{\left (b^{2} x^{2} - a^{2} - 5 \, b x - 15 \, a - 14\right )} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \,{\left (b^{3} x +{\left (a + 1\right )} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*((2*a + 3)*b*x + 2*a^2 + 5*a + 3)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b
*x + a^2 - 1)) + (b^2*x^2 - a^2 - 5*b*x - 15*a - 14)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/(b^3*x + (a + 1)*b^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.22523, size = 171, normalized size = 1.44 \begin{align*} -\frac{1}{2} \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (\frac{x}{b} - \frac{a b^{2} + 6 \, b^{2}}{b^{4}}\right )} - \frac{3 \,{\left (2 \, a + 3\right )} \arcsin \left (-b x - a\right ) \mathrm{sgn}\left (b\right )}{2 \, b{\left | b \right |}} - \frac{8 \,{\left (a + 1\right )}}{b{\left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b}{b^{2} x + a b} + 1\right )}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(x/b - (a*b^2 + 6*b^2)/b^4) - 3/2*(2*a + 3)*arcsin(-b*x - a)*sgn(b)/(b
*abs(b)) - 8*(a + 1)/(b*((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b) + 1)*abs(b))