3.13.49 \(\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^{3/2}} \, dx\) [1249]

Optimal. Leaf size=52 \[ -\frac {2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}+\frac {x}{3 c \sqrt {c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6277, 667, 197} \begin {gather*} \frac {x}{3 c \sqrt {c-a^2 c x^2}}-\frac {2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - a*x))/(3*a*(c - a^2*c*x^2)^(3/2)) + x/(3*c*Sqrt[c - a^2*c*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 667

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

Rule 6277

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/c^(n/2), Int[(c + d*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] || GtQ[c, 0]) && I
LtQ[n/2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 63, normalized size = 1.21 \begin {gather*} -\frac {\sqrt {1-a x} (2+a x) \sqrt {1-a^2 x^2}}{3 a c (1+a x)^{3/2} \sqrt {c-a^2 c x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(115\) vs. \(2(44)=88\).
time = 0.06, size = 116, normalized size = 2.23

method	result	size
gosper	\(-\frac {\left (a x -1\right )^{2} \left (a x +2\right )}{3 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} a}\)	\(31\)
trager	\(-\frac {\left (a x +2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{3 c^{2} \left (a x +1\right )^{2} a}\)	\(34\)
default	\(-\frac {x}{c \sqrt {-a^{2} c \,x^{2}+c}}+\frac {-\frac {2}{3 a c \left (x +\frac {1}{a}\right ) \sqrt {-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )}}-\frac {2 \left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right )}{3 a \,c^{2} \sqrt {-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )}}}{a}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.25, size = 60, normalized size = 1.15 \begin {gather*} \frac {x}{3 \, \sqrt {-a^{2} c x^{2} + c} c} - \frac {2}{3 \, {\left (\sqrt {-a^{2} c x^{2} + c} a^{2} c x + \sqrt {-a^{2} c x^{2} + c} a c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.34, size = 47, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 2\right )}}{3 \, {\left (a^{3} c^{2} x^{2} + 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)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

-1/3*sqrt(-a^2*c*x^2 + c)*(a*x + 2)/(a^3*c^2*x^2 + 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*} - \int \frac {a x}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} - a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} + c \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {1}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} - a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} + c \sqrt {- a^{2} c x^{2} + c}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.44, size = 82, normalized size = 1.58 \begin {gather*} \frac {\frac {2 \, \sqrt {-c} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )}{c^{2}} - \frac {3 \, c \sqrt {-c + \frac {2 \, c}{a x + 1}} + {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}}}{c^{3} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )}}{6 \, {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*sqrt(-c)*sgn(1/(a*x + 1))*sgn(a)/c^2 - (3*c*sqrt(-c + 2*c/(a*x + 1)) + (-c + 2*c/(a*x + 1))^(3/2))/(c^3
*sgn(1/(a*x + 1))*sgn(a)))/abs(a)

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Mupad [B]
time = 0.98, size = 33, normalized size = 0.63 \begin {gather*} -\frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+2\right )}{3\,a\,c^2\,{\left (a\,x+1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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