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

Optimal. Leaf size=29 \[ -\frac {2 (a x+1) e^{-3 \coth ^{-1}(a x)}}{a (c-a c x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6174} \[ -\frac {2 (a x+1) e^{-3 \coth ^{-1}(a x)}}{a (c-a c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6174

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 41, normalized size = 1.41 \[ -\frac {2 x \left (1-\frac {1}{a x}\right )^{3/2}}{\sqrt {\frac {1}{a x}+1} (c-a c x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 43, normalized size = 1.48 \[ -\frac {2 \, \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2} x - a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 41, normalized size = 1.41 \[ \frac {{\left (\frac {\sqrt {2}}{a \sqrt {-c}} - \frac {2}{\sqrt {-a c x - c} a}\right )} {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 35, normalized size = 1.21 \[ -\frac {2 \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{a \left (-a c x +c \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.34, size = 45, normalized size = 1.55 \[ -\frac {2 \, {\left (a \sqrt {-c} x + \sqrt {-c}\right )} {\left (a x - 1\right )}}{{\left (a^{2} c^{2} x - a c^{2}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.35, size = 32, normalized size = 1.10 \[ \frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c\,\sqrt {c-a\,c\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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