3.1324 \(\int \frac{e^{n \tanh ^{-1}(a x)}}{x^2 (c-a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=239 \[ -\frac{2 a (a x+1)^{n/2} (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n+2}{2},\frac{a x+1}{1-a x}\right )}{c^2}-\frac{a \left (-n^3-n^2+4 n+6\right ) (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}}}{c^2 n \left (4-n^2\right )}+\frac{a \left (n^2+4 n+6\right ) (a x+1)^{\frac{n-2}{2}} (1-a x)^{-n/2}}{c^2 n (n+2)}+\frac{a (n+3) (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{c^2 (n+2)}-\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{c^2 x} \]

[Out]

(a*(3 + n)*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((-2 + n)/2))/(c^2*(2 + n)) - ((1 - a*x)^(-1 - n/2)*(1 + a*x)^((-2 +
n)/2))/(c^2*x) - (a*(6 + 4*n - n^2 - n^3)*(1 - a*x)^(1 - n/2)*(1 + a*x)^((-2 + n)/2))/(c^2*n*(4 - n^2)) + (a*
(6 + 4*n + n^2)*(1 + a*x)^((-2 + n)/2))/(c^2*n*(2 + n)*(1 - a*x)^(n/2)) - (2*a*(1 + a*x)^(n/2)*Hypergeometric2
F1[1, n/2, (2 + n)/2, (1 + a*x)/(1 - a*x)])/(c^2*(1 - a*x)^(n/2))

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Rubi [A]  time = 0.251409, antiderivative size = 253, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6150, 129, 155, 12, 131} \[ -\frac{2 a n (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (1,1-\frac{n}{2};2-\frac{n}{2};\frac{1-a x}{a x+1}\right )}{c^2 (2-n)}-\frac{a \left (-n^3-n^2+4 n+6\right ) (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}}}{c^2 n \left (4-n^2\right )}+\frac{a \left (n^2+4 n+6\right ) (a x+1)^{\frac{n-2}{2}} (1-a x)^{-n/2}}{c^2 n (n+2)}+\frac{a (n+3) (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{c^2 (n+2)}-\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{c^2 x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 6150

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

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

Mathematica [A]  time = 0.0881545, size = 163, normalized size = 0.68 \[ -\frac{(1-a x)^{-\frac{n}{2}-1} (a x+1)^{\frac{n}{2}-1} \left (-2 a (n+2) n^2 x (a x-1)^2 \text{Hypergeometric2F1}\left (1,1-\frac{n}{2},2-\frac{n}{2},\frac{1-a x}{a x+1}\right )+a n^2 x \left (a^2 x^2-2\right )+n \left (-4 a^3 x^3+6 a^2 x^2+4 a x-4\right )-6 a^3 x^3+n^3 (a x-1)^2 (a x+1)+6 a x\right )}{c^2 (n-2) n (n+2) x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.187, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{x}^{2} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{6} - 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^6 - 2*a^2*c^2*x^4 + c^2*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{e^{n \operatorname{atanh}{\left (a x \right )}}}{a^{4} x^{6} - 2 a^{2} x^{4} + x^{2}}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(n*atanh(a*x))/(a**4*x**6 - 2*a**2*x**4 + x**2), 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 (a^{2} c x^{2} - c\right )}^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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