3.7.55 \(\int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a^2 c x^2)^{3/2}} \, dx\) [655]

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 = 0.07, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6302, 6277, 667, 197} \begin {gather*} \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 {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcCoth[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]

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps

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

[Out]

(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])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(114\) vs. \(2(44)=88\).
time = 0.19, size = 115, normalized size = 2.21

method	result	size
gosper	\(\frac {\left (a x -1\right )^{2} \left (a x +2\right )}{3 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\)	\(31\)
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	\(\frac {x}{c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {2 \left (-\frac {1}{3 a c \left (x +\frac {1}{a}\right ) \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}-\frac {-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c}{3 a \,c^{2} \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{a}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]
time = 0.26, 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((a*x-1)/(a*x+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 = 0.35, 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((a*x-1)/(a*x+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 - 1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (a x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (43) = 86\).
time = 0.43, size = 148, normalized size = 2.85 \begin {gather*} -\frac {{\left (a c + 3 \, \sqrt {-a^{2} c} \sqrt {c}\right )} \mathrm {sgn}\left (x\right )}{3 \, {\left (a^{2} c^{\frac {5}{2}} + \sqrt {-a^{2} c} a c^{2}\right )}} + \frac {2 \, {\left (2 \, a^{2} c - 3 \, a \sqrt {c} {\left (\sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )} + 3 \, {\left (\sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )}^{2}\right )}}{3 \, {\left (a \sqrt {c} - \sqrt {-a^{2} c + \frac {c}{x^{2}}} + \frac {\sqrt {c}}{x}\right )}^{3} c \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(a*c + 3*sqrt(-a^2*c)*sqrt(c))*sgn(x)/(a^2*c^(5/2) + sqrt(-a^2*c)*a*c^2) + 2/3*(2*a^2*c - 3*a*sqrt(c)*(sq
rt(-a^2*c + c/x^2) - sqrt(c)/x) + 3*(sqrt(-a^2*c + c/x^2) - sqrt(c)/x)^2)/((a*sqrt(c) - sqrt(-a^2*c + c/x^2) +
sqrt(c)/x)^3*c*sgn(x))

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Mupad [B]
time = 1.28, 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*x - 1)/((c - a^2*c*x^2)^(3/2)*(a*x + 1)),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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