3.1343 \(\int \frac {e^{n \tanh ^{-1}(a x)} x}{(c-a^2 c x^2)^{3/2}} \, dx\)

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

[Out]

exp(n*arctanh(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 = {6144} \[ \frac {(1-a n x) e^{n \tanh ^{-1}(a x)}}{a^2 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6144

Int[(E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((1 - a*n*x)*E^(n*ArcTa
nh[a*x]))/(d*(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 \tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {e^{n \tanh ^{-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 = 0.06, size = 81, normalized size = 1.76 \[ \frac {\sqrt {1-a^2 x^2} (1-a x)^{-\frac {n}{2}-\frac {1}{2}} (a x+1)^{\frac {n-1}{2}} (a n x-1)}{a^2 c (n-1) (n+1) \sqrt {c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.59, size = 82, normalized size = 1.78 \[ \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a n x - 1\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{2} n^{2} - a^{2} c^{2} - {\left (a^{4} c^{2} n^{2} - a^{4} c^{2}\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(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*arctanh(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 = 0.03, size = 49, normalized size = 1.07 \[ -\frac {\left (a x -1\right ) \left (a x +1\right ) \left (n a x -1\right ) {\mathrm e}^{n \arctanh \left (a x \right )}}{a^{2} \left (n^{2}-1\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(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*arctanh(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*arctanh(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 = 1.08, size = 68, normalized size = 1.48 \[ -\frac {{\mathrm {e}}^{\frac {n\,\ln \left (a\,x+1\right )}{2}-\frac {n\,\ln \left (1-a\,x\right )}{2}}\,\left (\frac {1}{a^2\,c\,\left (n^2-1\right )}-\frac {n\,x}{a\,c\,\left (n^2-1\right )}\right )}{\sqrt {c-a^2\,c\,x^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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