3.1321 \(\int \frac {e^{n \tanh ^{-1}(a x)} x}{(c-a^2 c x^2)^2} \, dx\)

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

[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)

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Rubi [A]  time = 0.15, 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 = {6145, 6136, 6137} \[ \frac {n (n-2 a x) e^{n \tanh ^{-1}(a x)}}{2 a^2 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}-\frac {e^{n \tanh ^{-1}(a x)}}{a^2 c^2 \left (4-n^2\right )}+\frac {e^{n \tanh ^{-1}(a x)}}{2 a^2 c^2 \left (1-a^2 x^2\right )} \]

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 6136

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 6137

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 6145

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*}

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Mathematica [A]  time = 0.04, size = 56, normalized size = 0.81 \[ -\frac {(1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n}{2}-1} \left (a^2 x^2-a n x+1\right )}{a^2 c^2 \left (n^2-4\right )} \]

Warning: Unable to verify antiderivative.

[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)))

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fricas [A]  time = 0.46, size = 78, normalized size = 1.13 \[ -\frac {{\left (a^{2} x^{2} - a n x + 1\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{2} n^{2} - 4 \, a^{2} c^{2} - {\left (a^{4} c^{2} n^{2} - 4 \, a^{4} c^{2}\right )} x^{2}} \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 47, normalized size = 0.68 \[ \frac {{\mathrm e}^{n \arctanh \left (a x \right )} \left (a^{2} x^{2}-n a x +1\right )}{\left (a^{2} x^{2}-1\right ) c^{2} a^{2} \left (n^{2}-4\right )} \]

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)

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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))

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

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((zoo*x**2*exp(-oo*n), Eq(a, -1/x)), (zoo*x**2*exp(oo*n), Eq(a, 1/x)), (zoo*Integral(x*exp(n*atanh(a*
x)), x), Eq(c, 0)), (-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*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
) - 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*at
anh(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))

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