3.1196 \(\int \frac {e^{-\tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac {e^{-\tanh ^{-1}(a x)}}{a c} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6137} \[ -\frac {e^{-\tanh ^{-1}(a x)}}{a c} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcTanh[a*x]*(c - a^2*c*x^2)),x]

[Out]

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

Rule 6137

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcTanh[a*x])/(a*c*n), x] /; F
reeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.69 \[ -\frac {\sqrt {1-a x}}{a c \sqrt {a x+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^ArcTanh[a*x]*(c - a^2*c*x^2)),x]

[Out]

-(Sqrt[1 - a*x]/(a*c*Sqrt[1 + a*x]))

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fricas [A]  time = 0.59, size = 31, normalized size = 1.94 \[ -\frac {a x + \sqrt {-a^{2} x^{2} + 1} + 1}{a^{2} c x + a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 37, normalized size = 2.31 \[ \frac {2}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 28, normalized size = 1.75 \[ -\frac {\sqrt {-a^{2} x^{2}+1}}{a c \left (a x +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )} {\left (a x + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 46, normalized size = 2.88 \[ \frac {\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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