3.1367 \(\int e^{n \tanh ^{-1}(a x)} x^2 (c-a^2 c x^2)^{-1-\frac {n^2}{2}} \, dx\)

Optimal. Leaf size=53 \[ \frac {(1-a n x) \left (c-a^2 c x^2\right )^{-\frac {n^2}{2}} e^{n \tanh ^{-1}(a x)}}{a^3 c n \left (1-n^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6146} \[ \frac {(1-a n x) \left (c-a^2 c x^2\right )^{-\frac {n^2}{2}} e^{n \tanh ^{-1}(a x)}}{a^3 c n \left (1-n^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6146

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

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 92, normalized size = 1.74 \[ \frac {(1-a x)^{-\frac {1}{2} n (n+1)} (a x+1)^{-\frac {1}{2} (n-1) n} (a n x-1) \left (1-a^2 x^2\right )^{\frac {n^2}{2}} \left (c-a^2 c x^2\right )^{-\frac {n^2}{2}}}{a^3 c (n-1) n (n+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 77, normalized size = 1.45 \[ -\frac {{\left (a^{3} n x^{3} - a^{2} x^{2} - a n x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{-\frac {1}{2} \, n^{2} - 1} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{3} n^{3} - a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 58, normalized size = 1.09 \[ -\frac {\left (a x -1\right ) \left (a x +1\right ) \left (n a x -1\right ) {\mathrm e}^{n \arctanh \left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}}}{a^{3} n \left (n^{2}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))*x^2*(-a^2*c*x^2+c)^(-1-1/2*n^2),x)

[Out]

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

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maxima [A]  time = 0.36, size = 75, normalized size = 1.42 \[ \frac {{\left (a n x - 1\right )} e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + 1\right ) - \frac {1}{2} \, n^{2} \log \left (a x - 1\right ) + \frac {1}{2} \, n \log \left (a x + 1\right ) - \frac {1}{2} \, n \log \left (a x - 1\right )\right )}}{{\left (n^{3} - n\right )} a^{3} \left (-c\right )^{\frac {1}{2} \, n^{2}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.09, size = 101, normalized size = 1.91 \[ \frac {{\mathrm {e}}^{\frac {n\,\ln \left (a\,x+1\right )}{2}-\frac {n\,\ln \left (1-a\,x\right )}{2}}-a\,n\,x\,{\mathrm {e}}^{\frac {n\,\ln \left (a\,x+1\right )}{2}-\frac {n\,\ln \left (1-a\,x\right )}{2}}}{a^3\,c\,n\,{\left (c-a^2\,c\,x^2\right )}^{\frac {n^2}{2}}-a^3\,c\,n^3\,{\left (c-a^2\,c\,x^2\right )}^{\frac {n^2}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(n*atanh(a*x)))/(c - a^2*c*x^2)^(n^2/2 + 1),x)

[Out]

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

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

[Out]

Timed out

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