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

Optimal. Leaf size=43 \[ \frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}-\frac {\sin ^{-1}(a x)}{a c} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6127, 663, 216} \[ \frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}-\frac {\sin ^{-1}(a x)}{a c} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(c - a*c*x),x]

[Out]

(2*Sqrt[1 - a^2*x^2])/(a*c*(1 - a*x)) - ArcSin[a*x]/(a*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 663

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

Rule 6127

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^n, Int[(c + d*x)^(p - n)*(1 -
a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 46, normalized size = 1.07 \[ \frac {2 \left (\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+\sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{a c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.79, size = 62, normalized size = 1.44 \[ \frac {2 \, {\left (a x + {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} - 1\right )}}{a^{2} c x - a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.21, size = 53, normalized size = 1.23 \[ -\frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c {\left | a \right |}} + \frac {4}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsin(a*x)*sgn(a)/(c*abs(a)) + 4/(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 = 76, normalized size = 1.77 \[ -\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c \sqrt {a^{2}}}-\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{c \,a^{2} \left (x -\frac {1}{a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.44, size = 40, normalized size = 0.93 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2} c x - a c} - \frac {\arcsin \left (a x\right )}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.82, size = 71, normalized size = 1.65 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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