3.151 \(\int e^{n \tanh ^{-1}(a x)} x \, dx\)

Optimal. Leaf size=99 \[ -\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^2 (2-n)}-\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{1-\frac {n}{2}}}{2 a^2} \]

[Out]

-1/2*(-a*x+1)^(1-1/2*n)*(a*x+1)^(1+1/2*n)/a^2-2^(1/2*n)*n*(-a*x+1)^(1-1/2*n)*hypergeom([-1/2*n, 1-1/2*n],[2-1/
2*n],-1/2*a*x+1/2)/a^2/(2-n)

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Rubi [A]  time = 0.04, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6126, 80, 69} \[ -\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^2 (2-n)}-\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{1-\frac {n}{2}}}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTanh[a*x])*x,x]

[Out]

-((1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/(2*a^2) - (2^(n/2)*n*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/
2, -n/2, 2 - n/2, (1 - a*x)/2])/(a^2*(2 - n))

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 6126

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[(x^m*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x] /; Fre
eQ[{a, m, n}, x] &&  !IntegerQ[(n - 1)/2]

Rubi steps

\begin {align*} \int e^{n \tanh ^{-1}(a x)} x \, dx &=\int x (1-a x)^{-n/2} (1+a x)^{n/2} \, dx\\ &=-\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{2 a^2}+\frac {n \int (1-a x)^{-n/2} (1+a x)^{n/2} \, dx}{2 a}\\ &=-\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{2 a^2}-\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^2 (2-n)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 86, normalized size = 0.87 \[ -\frac {(1-a x)^{1-\frac {n}{2}} \left ((n-2) (a x+1)^{\frac {n}{2}+1}-2^{\frac {n}{2}+1} n \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )\right )}{2 a^2 (n-2)} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*ArcTanh[a*x])*x,x]

[Out]

-1/2*((1 - a*x)^(1 - n/2)*((-2 + n)*(1 + a*x)^(1 + n/2) - 2^(1 + n/2)*n*Hypergeometric2F1[1 - n/2, -1/2*n, 2 -
n/2, (1 - a*x)/2]))/(a^2*(-2 + n))

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fricas [F]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x,x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))*x,x)

[Out]

int(exp(n*arctanh(a*x))*x,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*exp(n*atanh(a*x)),x)

[Out]

int(x*exp(n*atanh(a*x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))*x,x)

[Out]

Integral(x*exp(n*atanh(a*x)), x)

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