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

Optimal. Leaf size=289 \[ \frac{2 \left (\frac{1}{a x}+1\right )^{n/2} \left (1-\frac{1}{a x}\right )^{-n/2} \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n+2}{2},\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\right )}{a c^2}+\frac{\left (-n^3-n^2+4 n+6\right ) \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}}}{a c^2 (2-n) n (n+2)}-\frac{\left (n^2+4 n+6\right ) \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \left (1-\frac{1}{a x}\right )^{-n/2}}{a c^2 n (n+2)}-\frac{(n+3) \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1}}{a c^2 (n+2)}+\frac{x \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1}}{c^2} \]

[Out]

-(((3 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2))/(a*c^2*(2 + n))) + ((6 + 4*n - n^2 - n^3)*(1 -
1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2))/(a*c^2*(2 - n)*n*(2 + n)) - ((6 + 4*n + n^2)*(1 + 1/(a*x))^((-
2 + n)/2))/(a*c^2*n*(2 + n)*(1 - 1/(a*x))^(n/2)) + ((1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2)*x)/c^2
+ (2*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, n/2, (2 + n)/2, (a + x^(-1))/(a - x^(-1))])/(a*c^2*(1 - 1/(a*x)
)^(n/2))

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Rubi [A]  time = 0.250866, antiderivative size = 303, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6194, 129, 155, 12, 131} \[ \frac{2 n \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \, _2F_1\left (1,1-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a c^2 (2-n)}+\frac{\left (-n^3-n^2+4 n+6\right ) \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}}}{a c^2 (2-n) n (n+2)}-\frac{\left (n^2+4 n+6\right ) \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \left (1-\frac{1}{a x}\right )^{-n/2}}{a c^2 n (n+2)}-\frac{(n+3) \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1}}{a c^2 (n+2)}+\frac{x \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1}}{c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((3 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2))/(a*c^2*(2 + n))) + ((6 + 4*n - n^2 - n^3)*(1 -
1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2))/(a*c^2*(2 - n)*n*(2 + n)) - ((6 + 4*n + n^2)*(1 + 1/(a*x))^((-
2 + n)/2))/(a*c^2*n*(2 + n)*(1 - 1/(a*x))^(n/2)) + ((1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2)*x)/c^2
+ (2*n*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2)*Hypergeometric2F1[1, 1 - n/2, 2 - n/2, (a - x^(-1))
/(a + x^(-1))])/(a*c^2*(2 - n))

Rule 6194

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

Rule 129

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

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-2-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-2+\frac{n}{2}}}{x^2} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)} x}{c^2}+\frac{\operatorname{Subst}\left (\int \frac{\left (-\frac{n}{a}-\frac{3 x}{a^2}\right ) \left (1-\frac{x}{a}\right )^{-2-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-2+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=-\frac{(3+n) \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)}}{a c^2 (2+n)}+\frac{\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)} x}{c^2}-\frac{a \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-1-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-2+\frac{n}{2}} \left (\frac{n (2+n)}{a^2}+\frac{2 (3+n) x}{a^3}\right )}{x} \, dx,x,\frac{1}{x}\right )}{c^2 (2+n)}\\ &=-\frac{(3+n) \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)}}{a c^2 (2+n)}-\frac{\left (6+4 n+n^2\right ) \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)}}{a c^2 n (2+n)}+\frac{\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)} x}{c^2}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-n/2} \left (1+\frac{x}{a}\right )^{-2+\frac{n}{2}} \left (-\frac{n^2 (2+n)}{a^3}-\frac{\left (6+4 n+n^2\right ) x}{a^4}\right )}{x} \, dx,x,\frac{1}{x}\right )}{c^2 n (2+n)}\\ &=-\frac{(3+n) \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)}}{a c^2 (2+n)}+\frac{\left (6+4 n-n^2-n^3\right ) \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)}}{a c^2 (2-n) n (2+n)}-\frac{\left (6+4 n+n^2\right ) \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)}}{a c^2 n (2+n)}+\frac{\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)} x}{c^2}-\frac{a^3 \operatorname{Subst}\left (\int \frac{n^2 \left (4-n^2\right ) \left (1-\frac{x}{a}\right )^{-n/2} \left (1+\frac{x}{a}\right )^{-1+\frac{n}{2}}}{a^4 x} \, dx,x,\frac{1}{x}\right )}{c^2 n \left (4-n^2\right )}\\ &=-\frac{(3+n) \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)}}{a c^2 (2+n)}+\frac{\left (6+4 n-n^2-n^3\right ) \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)}}{a c^2 (2-n) n (2+n)}-\frac{\left (6+4 n+n^2\right ) \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)}}{a c^2 n (2+n)}+\frac{\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)} x}{c^2}-\frac{n \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-n/2} \left (1+\frac{x}{a}\right )^{-1+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{a c^2}\\ &=-\frac{(3+n) \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)}}{a c^2 (2+n)}+\frac{\left (6+4 n-n^2-n^3\right ) \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)}}{a c^2 (2-n) n (2+n)}-\frac{\left (6+4 n+n^2\right ) \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)}}{a c^2 n (2+n)}+\frac{\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)} x}{c^2}+\frac{2 n \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)} \, _2F_1\left (1,1-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a c^2 (2-n)}\\ \end{align*}

Mathematica [A]  time = 0.431414, size = 142, normalized size = 0.49 \[ \frac{e^{n \coth ^{-1}(a x)} \left (2 (n-2) n^2 e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,e^{2 \coth ^{-1}(a x)}\right )+2 \left (n^2-4\right ) n \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \coth ^{-1}(a x)}\right )+2 a n^3 x-n^2 \cosh \left (2 \coth ^{-1}(a x)\right )-8 a n x+2 n \sinh \left (2 \coth ^{-1}(a x)\right )-3 n^2+12\right )}{2 a c^2 (n-2) n (n+2)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(E^(n*ArcCoth[a*x])*(12 - 3*n^2 - 8*a*n*x + 2*a*n^3*x - n^2*Cosh[2*ArcCoth[a*x]] + 2*E^(2*ArcCoth[a*x])*(-2 +
n)*n^2*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + 2*n*(-4 + n^2)*Hypergeometric2F1[1, n/2, 1
+ n/2, E^(2*ArcCoth[a*x])] + 2*n*Sinh[2*ArcCoth[a*x]]))/(2*a*c^2*(-2 + n)*n*(2 + n))

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( c-{\frac{c}{{a}^{2}{x}^{2}}} \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(exp(n*arccoth(a*x))/(c-c/a^2/x^2)^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^{2} x^{2}}\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{4} x^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{4} \int \frac{x^{4} e^{n \operatorname{acoth}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*Integral(x**4*exp(n*acoth(a*x))/(a**4*x**4 - 2*a**2*x**2 + 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^{2} x^{2}}\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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