3.1152 \(\int \frac {e^{3 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx\)

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

[Out]

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

[Out]

E^(3*ArcTanh[a*x])/(3*a*c)

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 \[ \frac {(a x+1)^{3/2}}{3 a c (1-a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 54, normalized size = 3.00 \[ \frac {a^{2} x^{2} - 2 \, a x + \sqrt {-a^{2} x^{2} + 1} {\left (a x + 1\right )} + 1}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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 = 0.34, 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((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 \[ \frac {\left (a x +1\right )^{3}}{3 a c \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.49, size = 329, normalized size = 18.28 \[ \frac {1}{6} \, {\left (\frac {a^{3} c^{3}}{\sqrt {-a^{2} x^{2} + 1} a^{6} c^{4} x + \sqrt {-a^{2} x^{2} + 1} a^{5} c^{4}} - \frac {a^{3} c^{3}}{\sqrt {-a^{2} x^{2} + 1} a^{6} c^{4} x - \sqrt {-a^{2} x^{2} + 1} a^{5} c^{4}} + \frac {3 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{4} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{3} c^{2}} - \frac {3 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{4} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{3} c^{2}} - \frac {4 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{3} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{2} c^{2}} - \frac {4 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{3} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{2} c^{2}} - \frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1} a c} - \frac {6}{\sqrt {-a^{2} x^{2} + 1} a^{2} c}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(a^3*c^3/(sqrt(-a^2*x^2 + 1)*a^6*c^4*x + sqrt(-a^2*x^2 + 1)*a^5*c^4) - a^3*c^3/(sqrt(-a^2*x^2 + 1)*a^6*c^4
*x - sqrt(-a^2*x^2 + 1)*a^5*c^4) + 3*a*c/(sqrt(-a^2*x^2 + 1)*a^4*c^2*x + sqrt(-a^2*x^2 + 1)*a^3*c^2) - 3*a*c/(
sqrt(-a^2*x^2 + 1)*a^4*c^2*x - sqrt(-a^2*x^2 + 1)*a^3*c^2) - 4*c/(sqrt(-a^2*x^2 + 1)*a^3*c^2*x + sqrt(-a^2*x^2
+ 1)*a^2*c^2) - 4*c/(sqrt(-a^2*x^2 + 1)*a^3*c^2*x - sqrt(-a^2*x^2 + 1)*a^2*c^2) - 2*x/(sqrt(-a^2*x^2 + 1)*a*c
) - 6/(sqrt(-a^2*x^2 + 1)*a^2*c))*a

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mupad [B]  time = 0.06, 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((a*x + 1)^3/((c - a^2*c*x^2)*(1 - a^2*x^2)^(3/2)),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 \[ \frac {\int \frac {3 a x}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(3*a*x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)
+ Integral(3*a**2*x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 +
1)), x) + Integral(a**3*x**3/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*
x**2 + 1)), x) + Integral(1/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x*
*2 + 1)), x))/c

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