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

Optimal. Leaf size=373 \[ -\frac{2^{n/2} n (1-a x)^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (\frac{2-n}{2},1-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a c^2 (2-n)}-\frac{a^2 x^3 (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{c^2}-\frac{(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-\frac{n}{2}-1}}{a c^2 \left (4-n^2\right )}-\frac{(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-n/2}}{a c^2 n \left (4-n^2\right )}+\frac{(1-n) (n+3) (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a c^2 (2-n)}+\frac{(n+3) x (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{c^2}+\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}}}{a c^2 (2-n)}-\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{-n/2}}{a c^2} \]

[Out]

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

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Rubi [A]  time = 0.423005, antiderivative size = 373, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {6157, 6150, 100, 159, 89, 79, 69, 90, 45, 37} \[ -\frac{a^2 x^3 (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{c^2}-\frac{2^{n/2} n (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (\frac{2-n}{2},1-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a c^2 (2-n)}-\frac{(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-\frac{n}{2}-1}}{a c^2 \left (4-n^2\right )}-\frac{(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-n/2}}{a c^2 n \left (4-n^2\right )}+\frac{(1-n) (n+3) (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a c^2 (2-n)}+\frac{(n+3) x (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{c^2}+\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}}}{a c^2 (2-n)}-\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{-n/2}}{a c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6157

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

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 100

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))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 159

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

Rule 89

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

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]))

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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

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

Rubi steps

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

Mathematica [A]  time = 0.156713, size = 178, normalized size = 0.48 \[ -\frac{(1-a x)^{-\frac{n}{2}-1} \left ((a x+1)^{n/2} \left (n^2 \left (1-2 a^2 x^2\right )+n \left (-4 a^3 x^3+4 a^2 x^2+6 a x-4\right )+6 a^2 x^2+n^3 (a x-1)^2 (a x+1)-6\right )-2^{n/2} n^2 (n+2) (a x-1)^2 (a x+1) \text{Hypergeometric2F1}\left (1-\frac{n}{2},1-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (1-a x)\right )\right )}{a c^2 (n-2) n (n+2) (a x+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\it Artanh} \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*arctanh(a*x))/(c-c/a^2/x^2)^2,x)

[Out]

int(exp(n*arctanh(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*arctanh(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*arctanh(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{atanh}{\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*atanh(a*x))/(c-c/a**2/x**2)**2,x)

[Out]

a**4*Integral(x**4*exp(n*atanh(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*arctanh(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)