3.1151 \(\int \frac{e^{3 \tanh ^{-1}(a x)} x}{c-a^2 c x^2} \, dx\)

Optimal. Leaf size=70 \[ \frac{(a x+1)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (a x+1)}{a^2 c \sqrt{1-a^2 x^2}}+\frac{\sin ^{-1}(a x)}{a^2 c} \]

[Out]

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

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Rubi [A]  time = 0.0860757, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6148, 789, 653, 216} \[ \frac{(a x+1)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (a x+1)}{a^2 c \sqrt{1-a^2 x^2}}+\frac{\sin ^{-1}(a x)}{a^2 c} \]

Antiderivative was successfully verified.

[In]

Int[(E^(3*ArcTanh[a*x])*x)/(c - a^2*c*x^2),x]

[Out]

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

Rule 6148

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] || Gt
Q[c, 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]

Rule 789

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

Rule 653

Int[((d_) + (e_.)*(x_))^2*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)*(a + c*x^2)^(p + 1))/(c*(
p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && LtQ[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 \frac{e^{3 \tanh ^{-1}(a x)} x}{c-a^2 c x^2} \, dx &=\frac{\int \frac{x (1+a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c}\\ &=\frac{(1+a x)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac{\int \frac{(1+a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a c}\\ &=\frac{(1+a x)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (1+a x)}{a^2 c \sqrt{1-a^2 x^2}}+\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a c}\\ &=\frac{(1+a x)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (1+a x)}{a^2 c \sqrt{1-a^2 x^2}}+\frac{\sin ^{-1}(a x)}{a^2 c}\\ \end{align*}

Mathematica [A]  time = 0.0617958, size = 70, normalized size = 1. \[ \frac{-7 a^2 x^2+3 (a x-1) \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-2 a x+5}{3 a^2 c (a x-1) \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(3*ArcTanh[a*x])*x)/(c - a^2*c*x^2),x]

[Out]

(5 - 2*a*x - 7*a^2*x^2 + 3*(-1 + a*x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*a^2*c*(-1 + a*x)*Sqrt[1 - a^2*x^2])

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Maple [B]  time = 0.038, size = 155, normalized size = 2.2 \begin{align*} -5\,{\frac{x}{ac\sqrt{-{a}^{2}{x}^{2}+1}}}+{\frac{1}{ac}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-3\,{\frac{1}{{a}^{2}c\sqrt{-{a}^{2}{x}^{2}+1}}}-{\frac{4}{3\,{a}^{3}c} \left ( x-{a}^{-1} \right ) ^{-1}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{8\,x}{3\,ac}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.75188, size = 572, normalized size = 8.17 \begin{align*} -\frac{a^{2} c{\left (\frac{2 \, c}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{3} c x + \sqrt{-a^{2} x^{2} + 1} a^{3} c^{2}} + \frac{2 \, c}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{3} c x - \sqrt{-a^{2} x^{2} + 1} a^{3} c^{2}} - \frac{2 \, \sqrt{a^{2} c^{2}}}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{4} c x + \sqrt{-a^{2} x^{2} + 1} a^{4} c^{2}} + \frac{2 \, \sqrt{a^{2} c^{2}}}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{4} c x - \sqrt{-a^{2} x^{2} + 1} a^{4} c^{2}} - \frac{8 \, x}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a} + \frac{8}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{2}} + \frac{12 \, \sqrt{a^{2} c^{2}} x}{\sqrt{-a^{2} x^{2} + 1} a^{3} c^{2}} + \frac{\sqrt{a^{2} c^{2}}}{\sqrt{-a^{2} x^{2} + 1} a^{4} c^{2}} + \frac{3 \, \left (a^{2} c^{2}\right )^{\frac{3}{2}} x}{\sqrt{-a^{2} x^{2} + 1} a^{5} c^{4}} - \frac{3 \, \sqrt{a^{2} c^{2}} \arcsin \left (\frac{x}{c \sqrt{\frac{1}{a^{2} c^{2}}}}\right )}{a^{5} c^{3} \sqrt{\frac{1}{a^{2} c^{2}}}}\right )}}{3 \, \sqrt{a^{2} c^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a^2*c*(2*c/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^3*c*x + sqrt(-a^2*x^2 + 1)*a^3*c^2) + 2*c/(sqrt(a^2*c^2)*s
qrt(-a^2*x^2 + 1)*a^3*c*x - sqrt(-a^2*x^2 + 1)*a^3*c^2) - 2*sqrt(a^2*c^2)/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^
4*c*x + sqrt(-a^2*x^2 + 1)*a^4*c^2) + 2*sqrt(a^2*c^2)/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^4*c*x - sqrt(-a^2*x^
2 + 1)*a^4*c^2) - 8*x/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a) + 8/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^2) + 12*sqr
t(a^2*c^2)*x/(sqrt(-a^2*x^2 + 1)*a^3*c^2) + sqrt(a^2*c^2)/(sqrt(-a^2*x^2 + 1)*a^4*c^2) + 3*(a^2*c^2)^(3/2)*x/(
sqrt(-a^2*x^2 + 1)*a^5*c^4) - 3*sqrt(a^2*c^2)*arcsin(x/(c*sqrt(1/(a^2*c^2))))/(a^5*c^3*sqrt(1/(a^2*c^2))))/sqr
t(a^2*c^2)

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Fricas [A]  time = 2.62036, size = 217, normalized size = 3.1 \begin{align*} -\frac{5 \, a^{2} x^{2} - 10 \, a x + 6 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt{-a^{2} x^{2} + 1}{\left (7 \, a x - 5\right )} + 5}{3 \,{\left (a^{4} c x^{2} - 2 \, a^{3} c x + a^{2} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(5*a^2*x^2 - 10*a*x + 6*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - sqrt(-a^2*x^2 + 1)
*(7*a*x - 5) + 5)/(a^4*c*x^2 - 2*a^3*c*x + a^2*c)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a x^{2}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{3}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{4}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + I
ntegral(3*a*x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x
) + Integral(3*a**2*x**3/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2
+ 1)), x) + Integral(a**3*x**4/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2
*x**2 + 1)), x))/c

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Giac [A]  time = 1.19037, size = 151, normalized size = 2.16 \begin{align*} \frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a c{\left | a \right |}} + \frac{2 \,{\left (\frac{12 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} - 5\right )}}{3 \, a c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{3}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(a*x)*sgn(a)/(a*c*abs(a)) + 2/3*(12*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 3*(sqrt(-a^2*x^2 + 1)*abs(
a) + a)^2/(a^4*x^2) - 5)/(a*c*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^3*abs(a))