3.12.97 \(\int \frac {e^{-\tanh ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx\) [1197]

Optimal. Leaf size=53 \[ -\frac {1-a x}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x}{3 c^2 \sqrt {1-a^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6274, 653, 197} \begin {gather*} \frac {2 x}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {1-a x}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 653

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

Rule 6274

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

Rubi steps

\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=-\frac {1-a x}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=-\frac {1-a x}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x}{3 c^2 \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.81 \begin {gather*} \frac {-1+2 a x+2 a^2 x^2}{3 a c^2 \sqrt {1-a x} (1+a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.06, size = 409, normalized size = 7.72

method	result	size
gosper	\(\frac {2 a^{2} x^{2}+2 a x -1}{3 \sqrt {-a^{2} x^{2}+1}\, \left (a x +1\right ) a \,c^{2}}\)	\(42\)
trager	\(-\frac {\left (2 a^{2} x^{2}+2 a x -1\right ) \sqrt {-a^{2} x^{2}+1}}{3 c^{2} \left (a x +1\right )^{2} a \left (a x -1\right )}\)	\(49\)
default	\(\frac {\frac {\frac {3 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16}+\frac {3 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{16 \sqrt {a^{2}}}}{a}+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}-a \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{4 a^{2}}-\frac {3 \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}\right )}{\sqrt {a^{2}}}\right )}{16 a}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{12 a^{4} \left (x +\frac {1}{a}\right )^{3}}+\frac {\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{a \left (x -\frac {1}{a}\right )^{2}}+a \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}\right )}{\sqrt {a^{2}}}\right )}{8 a^{2}}}{c^{2}}\)	\(409\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

1/c^2*(3/16/a*((-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/
a))^(1/2)))+1/4/a^2*(-1/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-a*((-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+a
/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))))-3/16/a*((-a^2*(x-1/a)^2-2*(x-1/a)*a)^(
1/2)-a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(1/2)))-1/12/a^4/(x+1/a)^3*(-a^2*(x+1/a)^
2+2*a*(x+1/a))^(3/2)+1/8/a^2*(1/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(3/2)+a*((-a^2*(x-1/a)^2-2*(x-1/a)*a)
^(1/2)-a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(1/2)))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.38, size = 87, normalized size = 1.64 \begin {gather*} -\frac {a^{3} x^{3} + a^{2} x^{2} - a x + {\left (2 \, a^{2} x^{2} + 2 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1} - 1}{3 \, {\left (a^{4} c^{2} x^{3} + a^{3} c^{2} x^{2} - a^{2} c^{2} x - a c^{2}\right )}} \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

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

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

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)**2,x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.00, size = 48, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {1-a^2\,x^2}\,\left (2\,a^2\,x^2+2\,a\,x-1\right )}{3\,a\,c^2\,\left (a\,x-1\right )\,{\left (a\,x+1\right )}^2} \end {gather*}

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)^2*(a*x + 1)),x)

[Out]

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

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