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

Optimal. Leaf size=60 \[ \frac{a x+1}{a^3 c \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2}}{a^3 c}-\frac{\sin ^{-1}(a x)}{a^3 c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.1074, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6148, 797, 641, 216, 637} \[ \frac{a x+1}{a^3 c \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2}}{a^3 c}-\frac{\sin ^{-1}(a x)}{a^3 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + a*x)/(a^3*c*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a^3*c) - ArcSin[a*x]/(a^3*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 797

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

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 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 637

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rubi steps

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

Mathematica [A]  time = 0.0312228, size = 54, normalized size = 0.9 \[ \frac{-a^2 x^2-\sqrt{1-a^2 x^2} \sin ^{-1}(a x)+a x+2}{a^3 c \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.036, size = 98, normalized size = 1.6 \begin{align*}{\frac{1}{{a}^{3}c}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{1}{{a}^{2}c}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{c{a}^{4}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.60072, size = 320, normalized size = 5.33 \begin{align*} -\frac{a^{2} c{\left (\frac{\sqrt{-a^{2} x^{2} + 1} c}{\sqrt{a^{2} c^{2}} a^{4} c x + a^{4} c^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} c}{\sqrt{a^{2} c^{2}} a^{4} c x - a^{4} c^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{5} c x + \sqrt{a^{2} c^{2}} a^{3}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{5} c x - \sqrt{a^{2} c^{2}} a^{3}} - \frac{2 \, \sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1}}{a^{5} c^{2}} + \frac{2 \, \sqrt{a^{2} c^{2}} \arcsin \left (\frac{x}{c \sqrt{\frac{1}{a^{2} c^{2}}}}\right )}{a^{6} c^{3} \sqrt{\frac{1}{a^{2} c^{2}}}}\right )}}{2 \, \sqrt{a^{2} c^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.57227, size = 155, normalized size = 2.58 \begin{align*} \frac{2 \, a x + 2 \,{\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 - 2\right )} - 2}{a^{4} c x - a^{3} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.24204, size = 105, normalized size = 1.75 \begin{align*} -\frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a^{2} c{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3} c} + \frac{2}{a^{2} c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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