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

Optimal. Leaf size=139 \[ -\frac{2^{\frac{n}{2}+1} (n+2) (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (-\frac{n}{2},-\frac{n}{2},1-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a c^2 n}+\frac{(n+3) (a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a c^2 (n+2)}-\frac{x (a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-1}}{c^2} \]

[Out]

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

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Rubi [A]  time = 0.172383, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6131, 6129, 90, 79, 69} \[ -\frac{2^{\frac{n}{2}+1} (n+2) (1-a x)^{-n/2} \, _2F_1\left (-\frac{n}{2},-\frac{n}{2};1-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a c^2 n}+\frac{(n+3) (a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a c^2 (n+2)}-\frac{x (a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-1}}{c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6131

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

Rule 6129

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

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 79

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

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]))

Rubi steps

\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^2} \, dx &=\frac{a^2 \int \frac{e^{n \tanh ^{-1}(a x)} x^2}{(1-a x)^2} \, dx}{c^2}\\ &=\frac{a^2 \int x^2 (1-a x)^{-2-\frac{n}{2}} (1+a x)^{n/2} \, dx}{c^2}\\ &=-\frac{x (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{c^2}-\frac{\int (1-a x)^{-2-\frac{n}{2}} (1+a x)^{n/2} (-1-a (2+n) x) \, dx}{c^2}\\ &=\frac{(3+n) (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{a c^2 (2+n)}-\frac{x (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{c^2}-\frac{(2+n) \int (1-a x)^{-1-\frac{n}{2}} (1+a x)^{n/2} \, dx}{c^2}\\ &=\frac{(3+n) (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{a c^2 (2+n)}-\frac{x (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{c^2}-\frac{2^{1+\frac{n}{2}} (2+n) (1-a x)^{-n/2} \, _2F_1\left (-\frac{n}{2},-\frac{n}{2};1-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a c^2 n}\\ \end{align*}

Mathematica [A]  time = 0.442551, size = 194, normalized size = 1.4 \[ \frac{e^{n \tanh ^{-1}(a x)} \left (-2 n (a x-1) e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \tanh ^{-1}(a x)}\right )-4 n e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (2,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \tanh ^{-1}(a x)}\right )+4 a n x e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (2,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \tanh ^{-1}(a x)}\right )+2 (n+2) (a x-1) \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,-e^{2 \tanh ^{-1}(a x)}\right )-3 a n x-4 a x+n+4\right )}{a c^2 n (n+2) (a x-1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(E^(n*ArcTanh[a*x])*(4 + n - 4*a*x - 3*a*n*x - 2*E^(2*ArcTanh[a*x])*n*(-1 + a*x)*Hypergeometric2F1[1, 1 + n/2,
2 + n/2, -E^(2*ArcTanh[a*x])] + 2*(2 + n)*(-1 + a*x)*Hypergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcTanh[a*x])]
- 4*E^(2*ArcTanh[a*x])*n*Hypergeometric2F1[2, 1 + n/2, 2 + n/2, -E^(2*ArcTanh[a*x])] + 4*a*E^(2*ArcTanh[a*x])*
n*x*Hypergeometric2F1[2, 1 + n/2, 2 + n/2, -E^(2*ArcTanh[a*x])]))/(a*c^2*n*(2 + n)*(-1 + a*x))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( c-{\frac{c}{ax}} \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (c - \frac{c}{a x}\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{2} x^{2} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{2} \int \frac{x^{2} e^{n \operatorname{atanh}{\left (a x \right )}}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (c - \frac{c}{a x}\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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