3.12.98 \(\int \frac {e^{-\tanh ^{-1}(a x)}}{(c-a^2 c x^2)^3} \, dx\) [1198]

Optimal. Leaf size=75 \[ -\frac {1-a x}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x}{15 c^3 \sqrt {1-a^2 x^2}} \]

[Out]

1/5*(a*x-1)/a/c^3/(-a^2*x^2+1)^(5/2)+4/15*x/c^3/(-a^2*x^2+1)^(3/2)+8/15*x/c^3/(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6274, 653, 198, 197} \begin {gather*} \frac {8 x}{15 c^3 \sqrt {1-a^2 x^2}}+\frac {4 x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1-a x}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(1 - a*x)/(a*c^3*(1 - a^2*x^2)^(5/2)) + (4*x)/(15*c^3*(1 - a^2*x^2)^(3/2)) + (8*x)/(15*c^3*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 198

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

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 )^3} \, dx &=\frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=-\frac {1-a x}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^3}\\ &=-\frac {1-a x}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^3}\\ &=-\frac {1-a x}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x}{15 c^3 \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 59, normalized size = 0.79 \begin {gather*} -\frac {3-12 a x-12 a^2 x^2+8 a^3 x^3+8 a^4 x^4}{15 a c^3 (1-a x)^{3/2} (1+a x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*(3 - 12*a*x - 12*a^2*x^2 + 8*a^3*x^3 + 8*a^4*x^4)/(a*c^3*(1 - a*x)^(3/2)*(1 + a*x)^(5/2))

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.09, size = 524, normalized size = 6.99

method	result	size
gosper	\(-\frac {8 a^{4} x^{4}+8 a^{3} x^{3}-12 a^{2} x^{2}-12 a x +3}{15 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a x +1\right ) a \,c^{3}}\)	\(58\)
trager	\(-\frac {\left (8 a^{4} x^{4}+8 a^{3} x^{3}-12 a^{2} x^{2}-12 a x +3\right ) \sqrt {-a^{2} x^{2}+1}}{15 c^{3} \left (a x +1\right )^{3} \left (a x -1\right )^{2} a}\)	\(65\)
default	\(-\frac {-\frac {5 \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 )}{32 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}}}{5 a \left (x +\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x +\frac {1}{a}\right )^{3}}}{8 a^{4}}-\frac {3 \left (-\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 )\right )}{16 a^{2}}+\frac {\frac {5 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{32}-\frac {5 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 )}{32 \sqrt {a^{2}}}}{a}+\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{48 a^{4} \left (x -\frac {1}{a}\right )^{3}}+\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{16 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^{3}}\)	\(524\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

-1/c^3*(-5/32/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)))-1/8/a^4*(-1/5/a/(x+1/a)^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-1/15/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*
(x+1/a))^(3/2))-3/16/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))))+5/32/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)))+1/48/a^4/(x-1/a)^3*(-a^2*(x-
1/a)^2-2*(x-1/a)*a)^(3/2)+1/16/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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (62) = 124\).
time = 0.34, size = 141, normalized size = 1.88 \begin {gather*} -\frac {3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 3 \, a x + {\left (8 \, a^{4} x^{4} + 8 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 12 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1} + 3}{15 \, {\left (a^{6} c^{3} x^{5} + a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x + a c^{3}\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)^3,x, algorithm="fricas")

[Out]

-1/15*(3*a^5*x^5 + 3*a^4*x^4 - 6*a^3*x^3 - 6*a^2*x^2 + 3*a*x + (8*a^4*x^4 + 8*a^3*x^3 - 12*a^2*x^2 - 12*a*x +
3)*sqrt(-a^2*x^2 + 1) + 3)/(a^6*c^3*x^5 + a^5*c^3*x^4 - 2*a^4*c^3*x^3 - 2*a^3*c^3*x^2 + a^2*c^3*x + a*c^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 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^{3}} \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)**3,x)

[Out]

Integral(1/(a**5*x**5*sqrt(-a**2*x**2 + 1) + a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1)
- 2*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**3

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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)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.08, size = 64, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {1-a^2\,x^2}\,\left (8\,a^4\,x^4+8\,a^3\,x^3-12\,a^2\,x^2-12\,a\,x+3\right )}{15\,a\,c^3\,{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^3} \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)^3*(a*x + 1)),x)

[Out]

-((1 - a^2*x^2)^(1/2)*(8*a^3*x^3 - 12*a^2*x^2 - 12*a*x + 8*a^4*x^4 + 3))/(15*a*c^3*(a*x - 1)^2*(a*x + 1)^3)

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