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

Optimal. Leaf size=46 \[ \frac {(1-a n x) e^{n \coth ^{-1}(a x)}}{a^2 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6186} \[ \frac {(1-a n x) e^{n \coth ^{-1}(a x)}}{a^2 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(n*ArcCoth[a*x])*(1 - a*n*x))/(a^2*c*(1 - n^2)*Sqrt[c - a^2*c*x^2])

Rule 6186

Int[(E^(ArcCoth[(a_.)*(x_)]*(n_))*(x_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[((1 - a*n*x)*E^(n*ArcC
oth[a*x]))/(a^2*c*(n^2 - 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[
n]

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 43, normalized size = 0.93 \[ \frac {(a n x-1) e^{n \coth ^{-1}(a x)}}{a^2 c \left (n^2-1\right ) \sqrt {c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.20, size = 83, normalized size = 1.80 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a n x + 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{2} n^{2} - a^{2} c^{2} - {\left (a^{4} c^{2} n^{2} - a^{4} c^{2}\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 49, normalized size = 1.07 \[ -\frac {\left (a x -1\right ) \left (a x +1\right ) \left (x a n -1\right ) {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{a^{2} \left (n^{2}-1\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arccoth(a*x))*x/(-a^2*c*x^2+c)^(3/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.43, size = 81, normalized size = 1.76 \[ -\frac {\left (\frac {1}{a^2\,c\,\left (n^2-1\right )}-\frac {n\,x}{a\,c\,\left (n^2-1\right )}\right )\,{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}}{\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(n*acoth(a*x)))/(c - a^2*c*x^2)^(3/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*acoth(a*x))*x/(-a**2*c*x**2+c)**(3/2),x)

[Out]

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

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