3.1347 \(\int \frac{e^{n \tanh ^{-1}(a x)}}{x^3 (c-a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=417 \[ \frac{a^2 \left (n^2+3\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{n-1}{2},\frac{n+1}{2},\frac{a x+1}{1-a x}\right )}{c (1-n) \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (n^2+2 n+3\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{2 c (n+1) \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (n^3+2 n^2+5 n+6\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}}}{2 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{a n \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{2 c x \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{2 c x^2 \sqrt{c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.420699, antiderivative size = 422, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6153, 6150, 129, 151, 155, 12, 131} \[ -\frac{a^2 \left (n^2+3\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{3-n}{2}} \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{1-a x}{a x+1}\right )}{c (3-n) \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (n^2+2 n+3\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{2 c (n+1) \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (n^3+2 n^2+5 n+6\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}}}{2 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{a n \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{2 c x \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{2 c x^2 \sqrt{c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Int[E^(n*ArcTanh[a*x])/(x^3*(c - a^2*c*x^2)^(3/2)),x]

[Out]

(a^2*(3 + 2*n + n^2)*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a^2*x^2])/(2*c*(1 + n)*Sqrt[c - a^
2*c*x^2]) - ((1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a^2*x^2])/(2*c*x^2*Sqrt[c - a^2*c*x^2]) -
(a*n*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a^2*x^2])/(2*c*x*Sqrt[c - a^2*c*x^2]) - (a^2*(6 +
5*n + 2*n^2 + n^3)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a^2*x^2])/(2*c*(1 - n^2)*Sqrt[c - a^2
*c*x^2]) - (a^2*(3 + n^2)*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1,
(3 - n)/2, (5 - n)/2, (1 - a*x)/(1 + a*x)])/(c*(3 - n)*Sqrt[c - a^2*c*x^2])

Rule 6153

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

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 151

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] && LtQ[m, -1] && IntegerQ[m]

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^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{n \tanh ^{-1}(a x)}}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{(1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}}}{x^3} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=-\frac{(1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c x^2 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \int \frac{(1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}} \left (-a n-3 a^2 x\right )}{x^2} \, dx}{2 c \sqrt{c-a^2 c x^2}}\\ &=-\frac{(1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c x \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \int \frac{(1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}} \left (a^2 \left (3+n^2\right )+2 a^3 n x\right )}{x} \, dx}{2 c \sqrt{c-a^2 c x^2}}\\ &=\frac{a^2 \left (3+2 n+n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c (1+n) \sqrt{c-a^2 c x^2}}-\frac{(1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c x \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \int \frac{(1-a x)^{-\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}} \left (-a^3 (1+n) \left (3+n^2\right )-a^4 \left (3+2 n+n^2\right ) x\right )}{x} \, dx}{2 a c (1+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{a^2 \left (3+2 n+n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c (1+n) \sqrt{c-a^2 c x^2}}-\frac{(1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c x \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (6+5 n+2 n^2+n^3\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c (1-n) (1+n) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \int \frac{a^4 (1-n) (1+n) \left (3+n^2\right ) (1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}}}{x} \, dx}{2 a^2 c (1-n) (1+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{a^2 \left (3+2 n+n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c (1+n) \sqrt{c-a^2 c x^2}}-\frac{(1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c x \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (6+5 n+2 n^2+n^3\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c (1-n) (1+n) \sqrt{c-a^2 c x^2}}+\frac{\left (a^2 \left (3+n^2\right ) \sqrt{1-a^2 x^2}\right ) \int \frac{(1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}}}{x} \, dx}{2 c \sqrt{c-a^2 c x^2}}\\ &=\frac{a^2 \left (3+2 n+n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c (1+n) \sqrt{c-a^2 c x^2}}-\frac{(1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c x \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (6+5 n+2 n^2+n^3\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2}}{2 c (1-n) (1+n) \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (3+n^2\right ) (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2} \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{1-a x}{1+a x}\right )}{c (3-n) \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.215807, size = 219, normalized size = 0.53 \[ \frac{\sqrt{1-a^2 x^2} (1-a x)^{\frac{1}{2} (-n-1)} (a x+1)^{\frac{n-3}{2}} \left (2 a^2 \left (n^4+2 n^2-3\right ) x^2 (a x-1)^2 \text{Hypergeometric2F1}\left (1,\frac{3}{2}-\frac{n}{2},\frac{5}{2}-\frac{n}{2},\frac{1-a x}{a x+1}\right )-(n-3) (a x+1) \left (a n x \left (5 a^2 x^2-6 a x-1\right )+6 a^3 x^3-3 a^2 x^2+a n^3 x (a x-1)^2+n^2 (a x-1)^2 (2 a x+1)-1\right )\right )}{2 c (n-3) (n-1) (n+1) x^2 \sqrt{c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcTanh[a*x])/(x^3*(c - a^2*c*x^2)^(3/2)),x]

[Out]

((1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2]*(-((-3 + n)*(1 + a*x)*(-1 - 3*a^2*x^2 + 6*a^3
*x^3 + a*n^3*x*(-1 + a*x)^2 + n^2*(-1 + a*x)^2*(1 + 2*a*x) + a*n*x*(-1 - 6*a*x + 5*a^2*x^2))) + 2*a^2*(-3 + 2*
n^2 + n^4)*x^2*(-1 + a*x)^2*Hypergeometric2F1[1, 3/2 - n/2, 5/2 - n/2, (1 - a*x)/(1 + a*x)]))/(2*c*(-3 + n)*(-
1 + n)*(1 + n)*x^2*Sqrt[c - a^2*c*x^2])

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Maple [F]  time = 0.201, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{x}^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(3/2),x)

[Out]

int(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(3/2),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 )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/((-a^2*c*x^2 + c)^(3/2)*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} c x^{2} + c} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{7} - 2 \, a^{2} c^{2} x^{5} + c^{2} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^7 - 2*a^2*c^2*x^5 + c^2*x^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))/x**3/(-a**2*c*x**2+c)**(3/2),x)

[Out]

Timed out

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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 )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/((-a^2*c*x^2 + c)^(3/2)*x^3), x)