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

Optimal. Leaf size=121 \[ \frac{(1-a) (a+b x+1)^{5/2}}{b^2 \sqrt{-a-b x+1}}+\frac{(3-2 a) \sqrt{-a-b x+1} (a+b x+1)^{3/2}}{2 b^2}+\frac{3 (3-2 a) \sqrt{-a-b x+1} \sqrt{a+b x+1}}{2 b^2}-\frac{3 (3-2 a) \sin ^{-1}(a+b x)}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0955477, antiderivative size = 121, 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} \[ \frac{(1-a) (a+b x+1)^{5/2}}{b^2 \sqrt{-a-b x+1}}+\frac{(3-2 a) \sqrt{-a-b x+1} (a+b x+1)^{3/2}}{2 b^2}+\frac{3 (3-2 a) \sqrt{-a-b x+1} \sqrt{a+b x+1}}{2 b^2}-\frac{3 (3-2 a) \sin ^{-1}(a+b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a + b*x])*x,x]

[Out]

(3*(3 - 2*a)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*b^2) + ((3 - 2*a)*Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/
(2*b^2) + ((1 - a)*(1 + a + b*x)^(5/2))/(b^2*Sqrt[1 - a - b*x]) - (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{(3-2 a) \sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac{(1-a) (1+a+b x)^{5/2}}{b^2 \sqrt{1-a-b x}}-\frac{(3 (3-2 a)) \int \frac{\sqrt{1+a+b x}}{\sqrt{1-a-b x}} \, dx}{2 b}\\ &=\frac{3 (3-2 a) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^2}+\frac{(3-2 a) \sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac{(1-a) (1+a+b x)^{5/2}}{b^2 \sqrt{1-a-b x}}-\frac{(3 (3-2 a)) \int \frac{1}{\sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{2 b}\\ &=\frac{3 (3-2 a) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^2}+\frac{(3-2 a) \sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac{(1-a) (1+a+b x)^{5/2}}{b^2 \sqrt{1-a-b x}}-\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{3 (3-2 a) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^2}+\frac{(3-2 a) \sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac{(1-a) (1+a+b x)^{5/2}}{b^2 \sqrt{1-a-b x}}+\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{3 (3-2 a) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^2}+\frac{(3-2 a) \sqrt{1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac{(1-a) (1+a+b x)^{5/2}}{b^2 \sqrt{1-a-b x}}-\frac{3 (3-2 a) \sin ^{-1}(a+b x)}{2 b^2}\\ \end{align*}

Mathematica [A]  time = 0.15619, size = 142, normalized size = 1.17 \[ \frac{\frac{\sqrt{b} \sqrt{a+b x+1} \left (a^2-15 a-b^2 x^2-5 b x+14\right )}{\sqrt{-a-b x+1}}+12 a \sqrt{-b} \sinh ^{-1}\left (\frac{\sqrt{-b} \sqrt{-a-b x+1}}{\sqrt{2} \sqrt{b}}\right )+18 \sqrt{-b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{-a-b x+1}}{\sqrt{2} \sqrt{-b}}\right )}{2 b^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a + b*x])*x,x]

[Out]

((Sqrt[b]*Sqrt[1 + a + b*x]*(14 - 15*a + a^2 - 5*b*x - b^2*x^2))/Sqrt[1 - a - b*x] + 12*a*Sqrt[-b]*ArcSinh[(Sq
rt[-b]*Sqrt[1 - a - b*x])/(Sqrt[2]*Sqrt[b])] + 18*Sqrt[-b]*ArcSinh[(Sqrt[b]*Sqrt[1 - a - b*x])/(Sqrt[2]*Sqrt[-
b])])/(2*b^(5/2))

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Maple [B]  time = 0.042, size = 381, normalized size = 3.2 \begin{align*} -{\frac{b{x}^{3}}{2}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+{\frac{{a}^{2}x}{2\,b}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}-{\frac{9}{2\,b}\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{a}{b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+7\,{\frac{1}{{b}^{2}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}-10\,{\frac{ax}{b\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}+3\,{\frac{a}{b\sqrt{{b}^{2}}}\arctan \left ({\frac{\sqrt{{b}^{2}}}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}} \left ( x+{\frac{a}{b}} \right ) } \right ) }-7\,{\frac{{a}^{2}}{{b}^{2}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}+{\frac{9\,x}{2\,b}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}-{\frac{a{x}^{2}}{2}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+{\frac{{a}^{3}}{2\,{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}-{\frac{a}{2\,{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}-3\,{\frac{{x}^{2}}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*x^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+1/2/b*a^2*x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-9/2/b/(b^2)^(1/2)*arctan(
(b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+7/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-10*a/b/(-b^2*x^2-2*a*
b*x-a^2+1)^(1/2)*x+3*a/b/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-7*a^2/b^2/(-b^
2*x^2-2*a*b*x-a^2+1)^(1/2)+9/2/b*x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/2*a*x^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+1/2
/b^2*a^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/2/b^2*a/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3*x^2/(-b^2*x^2-2*a*b*x-a^2+1
)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.18579, size = 300, normalized size = 2.48 \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((b*x+a+1)^3/(1-(b*x+a)^2)^(3/2)*x,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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (a + b x + 1\right )^{3}}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.2323, size = 147, normalized size = 1.21 \begin{align*} \frac{1}{2} \, \sqrt{-{\left (b x + a\right )}^{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{-{\left (b x + a\right )}^{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((b*x+a+1)^3/(1-(b*x+a)^2)^(3/2)*x,x, algorithm="giac")

[Out]

1/2*sqrt(-(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*x + a)^2 + 1)*abs(b) + b)/(b^2*x + a*b) - 1)*abs(b))