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

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

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 60, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.03, size = 54, normalized size = 0.90 \[ \frac {-a^2 x^2-\sqrt {1-a^2 x^2} \sin ^{-1}(a x)+a x+2}{a^3 c \sqrt {1-a^2 x^2}} \]

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])

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fricas [A]  time = 0.57, size = 69, normalized size = 1.15 \[ \frac {2 \, a x + 2 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x - 2\right )} - 2}{a^{4} c x - a^{3} c} \]

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)

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giac [A]  time = 0.50, size = 78, normalized size = 1.30 \[ -\frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{a^{2} c {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3} c} + \frac {2}{a^{2} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]

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))

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maple [A]  time = 0.04, size = 98, normalized size = 1.63 \[ \frac {\sqrt {-a^{2} x^{2}+1}}{a^{3} c}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c \,a^{2} \sqrt {a^{2}}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{c \,a^{4} \left (x -\frac {1}{a}\right )} \]

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)

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maxima [B]  time = 0.46, size = 157, normalized size = 2.62 \[ -\frac {1}{2} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} c}{a^{5} c^{2} x + a^{4} c^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} c}{a^{5} c^{2} x - a^{4} c^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{5} c x + a^{4} c} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{5} c x - a^{4} c} + \frac {2 \, \arcsin \left (a x\right )}{a^{4} c} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4} c}\right )} \]

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

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mupad [B]  time = 0.07, size = 93, normalized size = 1.55 \[ \frac {\sqrt {1-a^2\,x^2}}{a^3\,c}-\frac {\sqrt {1-a^2\,x^2}}{\left (a\,c\,\sqrt {-a^2}-a^2\,c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,c\,\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1 - a^2*x^2)^(1/2)/(a^3*c) - (1 - a^2*x^2)^(1/2)/((a*c*(-a^2)^(1/2) - a^2*c*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - a
sinh(x*(-a^2)^(1/2))/(a^2*c*(-a^2)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{2}}{- a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{3}}{- a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]

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

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