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

Optimal. Leaf size=39 \[ \frac {a x+1}{a^2 c \sqrt {1-a^2 x^2}}-\frac {\sin ^{-1}(a x)}{a^2 c} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6148, 778, 216} \[ \frac {a x+1}{a^2 c \sqrt {1-a^2 x^2}}-\frac {\sin ^{-1}(a x)}{a^2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + a*x)/(a^2*c*Sqrt[1 - a^2*x^2]) - ArcSin[a*x]/(a^2*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 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

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

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Mathematica [A]  time = 0.03, size = 45, normalized size = 1.15 \[ \frac {\frac {a x}{\sqrt {1-a^2 x^2}}+\frac {1}{\sqrt {1-a^2 x^2}}-\sin ^{-1}(a x)}{a^2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.75, size = 64, normalized size = 1.64 \[ \frac {a x + 2 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} - 1}{a^{3} c x - a^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 59, normalized size = 1.51 \[ -\frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{a c {\left | a \right |}} + \frac {2}{a c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.04, size = 79, normalized size = 2.03 \[ -\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c a \sqrt {a^{2}}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{c \,a^{3} \left (x -\frac {1}{a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.44, size = 137, normalized size = 3.51 \[ -\frac {1}{2} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} c}{a^{4} c^{2} x + a^{3} c^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} c}{a^{4} c^{2} x - a^{3} c^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{4} c x + a^{3} c} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{4} c x - a^{3} c} + \frac {2 \, \arcsin \left (a x\right )}{a^{3} c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.90, size = 64, normalized size = 1.64 \[ \frac {1}{a^2\,c\,\sqrt {1-a^2\,x^2}}+\frac {x}{a\,c\,\sqrt {1-a^2\,x^2}}+\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{a^3\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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