3.14.21 \(\int \frac {e^{n \tanh ^{-1}(a x)} x}{(c-a^2 c x^2)^2} \, dx\) [1321]

Optimal. Leaf size=69 \[ -\frac {e^{n \tanh ^{-1}(a x)}}{a^2 c^2 \left (4-n^2\right )}+\frac {e^{n \tanh ^{-1}(a x)} (2-a n x)}{a^2 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 102, normalized size of antiderivative = 1.48, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6280, 6271, 6272} \begin {gather*} \frac {n (n-2 a x) e^{n \tanh ^{-1}(a x)}}{2 a^2 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}-\frac {e^{n \tanh ^{-1}(a x)}}{a^2 c^2 \left (4-n^2\right )}+\frac {e^{n \tanh ^{-1}(a x)}}{2 a^2 c^2 \left (1-a^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(n*ArcTanh[a*x])/(a^2*c^2*(4 - n^2))) + E^(n*ArcTanh[a*x])/(2*a^2*c^2*(1 - a^2*x^2)) + (E^(n*ArcTanh[a*x])
*n*(n - 2*a*x))/(2*a^2*c^2*(4 - n^2)*(1 - a^2*x^2))

Rule 6271

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(n + 2*a*(p + 1)*x)*(c + d*x^2)
^(p + 1)*(E^(n*ArcTanh[a*x])/(a*c*(n^2 - 4*(p + 1)^2))), x] - Dist[2*(p + 1)*((2*p + 3)/(c*(n^2 - 4*(p + 1)^2)
)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p
, -1] &&  !IntegerQ[n] && NeQ[n^2 - 4*(p + 1)^2, 0] && IntegerQ[2*p]

Rule 6272

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

Rule 6280

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

Rubi steps

\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {e^{n \tanh ^{-1}(a x)}}{2 a^2 c^2 \left (1-a^2 x^2\right )}-\frac {n \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{2 a}\\ &=\frac {e^{n \tanh ^{-1}(a x)}}{2 a^2 c^2 \left (1-a^2 x^2\right )}+\frac {e^{n \tanh ^{-1}(a x)} n (n-2 a x)}{2 a^2 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}-\frac {n \int \frac {e^{n \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{a c \left (4-n^2\right )}\\ &=-\frac {e^{n \tanh ^{-1}(a x)}}{a^2 c^2 \left (4-n^2\right )}+\frac {e^{n \tanh ^{-1}(a x)}}{2 a^2 c^2 \left (1-a^2 x^2\right )}+\frac {e^{n \tanh ^{-1}(a x)} n (n-2 a x)}{2 a^2 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 56, normalized size = 0.81 \begin {gather*} -\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \left (1-a n x+a^2 x^2\right )}{a^2 c^2 \left (-4+n^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 47, normalized size = 0.68

method	result	size
gosper	\(\frac {{\mathrm e}^{n \arctanh \left (a x \right )} \left (a^{2} x^{2}-n a x +1\right )}{\left (a^{2} x^{2}-1\right ) c^{2} a^{2} \left (n^{2}-4\right )}\)	\(47\)
risch	\(\frac {\left (a^{2} x^{2}-n a x +1\right ) \left (-a x +1\right )^{-\frac {n}{2}} \left (a x +1\right )^{\frac {n}{2}}}{\left (a^{2} x^{2}-1\right ) c^{2} a^{2} \left (n^{2}-4\right )}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 79, normalized size = 1.14 \begin {gather*} -\frac {{\left (a^{2} x^{2} - a n x + 1\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{2} n^{2} - 4 \, a^{2} c^{2} - {\left (a^{4} c^{2} n^{2} - 4 \, a^{4} c^{2}\right )} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {x^{2}}{2 c^{2}} & \text {for}\: a = 0 \\\tilde {\infty } x^{2} e^{- \infty n} & \text {for}\: a = - \frac {1}{x} \\\tilde {\infty } x^{2} e^{\infty n} & \text {for}\: a = \frac {1}{x} \\- \frac {a^{2} x^{2} \operatorname {atanh}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{2} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {2 a x \operatorname {atanh}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{2} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} + \frac {a x}{4 a^{4} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{2} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {\operatorname {atanh}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{2} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} & \text {for}\: n = -2 \\\frac {\int \frac {x e^{2 \operatorname {atanh}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} & \text {for}\: n = 2 \\\frac {a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} c^{2} n^{2} x^{2} - 4 a^{4} c^{2} x^{2} - a^{2} c^{2} n^{2} + 4 a^{2} c^{2}} - \frac {a n x e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} c^{2} n^{2} x^{2} - 4 a^{4} c^{2} x^{2} - a^{2} c^{2} n^{2} + 4 a^{2} c^{2}} + \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} c^{2} n^{2} x^{2} - 4 a^{4} c^{2} x^{2} - a^{2} c^{2} n^{2} + 4 a^{2} c^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**2/(2*c**2), Eq(a, 0)), (zoo*x**2*exp(-oo*n), Eq(a, -1/x)), (zoo*x**2*exp(oo*n), Eq(a, 1/x)), (-a
**2*x**2*atanh(a*x)/(4*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x))) - 2*a*x*atanh(a*x)/(4
*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x))) + a*x/(4*a**4*c**2*x**2*exp(2*atanh(a*x)) -
4*a**2*c**2*exp(2*atanh(a*x))) - atanh(a*x)/(4*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x
))), Eq(n, -2)), (Integral(x*exp(2*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2, Eq(n, 2)), (a**2*x**2*e
xp(n*atanh(a*x))/(a**4*c**2*n**2*x**2 - 4*a**4*c**2*x**2 - a**2*c**2*n**2 + 4*a**2*c**2) - a*n*x*exp(n*atanh(a
*x))/(a**4*c**2*n**2*x**2 - 4*a**4*c**2*x**2 - a**2*c**2*n**2 + 4*a**2*c**2) + exp(n*atanh(a*x))/(a**4*c**2*n*
*2*x**2 - 4*a**4*c**2*x**2 - a**2*c**2*n**2 + 4*a**2*c**2), True))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 1.42, size = 46, normalized size = 0.67 \begin {gather*} \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\left (a^2\,x^2-n\,a\,x+1\right )}{a^2\,c^2\,\left (n^2-4\right )\,\left (a^2\,x^2-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________