3.1258 \(\int \frac {e^{-3 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx\)

Optimal. Leaf size=18 \[ -\frac {e^{-3 \tanh ^{-1}(a x)}}{3 a c} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 18, 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} \[ -\frac {e^{-3 \tanh ^{-1}(a x)}}{3 a c} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)),x]

[Out]

-1/(3*a*c*E^(3*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^{-3 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=-\frac {e^{-3 \tanh ^{-1}(a x)}}{3 a c}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.61 \[ -\frac {(1-a x)^{3/2}}{3 a c (a x+1)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)),x]

[Out]

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

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fricas [B]  time = 0.65, size = 55, normalized size = 3.06 \[ -\frac {a^{2} x^{2} + 2 \, a x - \sqrt {-a^{2} x^{2} + 1} {\left (a x - 1\right )} + 1}{3 \, {\left (a^{3} c x^{2} + 2 \, a^{2} c x + a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.38, size = 66, normalized size = 3.67 \[ \frac {2 \, {\left (\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + 1\right )}}{3 \, c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )}^{3} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 28, normalized size = 1.56 \[ -\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a c \left (a x +1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a^{2} c x^{2} - c\right )} {\left (a x + 1\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.96, size = 32, normalized size = 1.78 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (a\,x-1\right )}{3\,a\,c\,{\left (a\,x+1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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