3.1352 \(\int \frac{e^{n \tanh ^{-1}(a x)}}{x (c-a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=417 \[ \frac{2 \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{n-1}{2},\frac{n+1}{2},\frac{a x+1}{1-a x}\right )}{c^2 (1-n) \sqrt{c-a^2 c x^2}}-\frac{\left (n^2+6 n+15\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1-n}{2}}}{c^2 (n+3) \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}+\frac{\left (-n^3-2 n^2+7 n+18\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{3-n}{2}}}{c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{c^2 (n+3) \sqrt{c-a^2 c x^2}}+\frac{(n+6) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{c^2 (n+1) (n+3) \sqrt{c-a^2 c x^2}} \]

[Out]

((1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) + ((6 + n)
*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(c^2*(1 + n)*(3 + n)*Sqrt[c - a^2*c*x^2]) -
((15 + 6*n + n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(c^2*(3 + n)*(1 - n^2)*Sqrt[
c - a^2*c*x^2]) + ((18 + 7*n - 2*n^2 - n^3)*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(c
^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2]) + (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^2*(1 - n)*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.457503, antiderivative size = 421, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6153, 6150, 129, 155, 12, 131} \[ -\frac{2 \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{3-n}{2}} \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{1-a x}{a x+1}\right )}{c^2 (3-n) \sqrt{c-a^2 c x^2}}-\frac{\left (n^2+6 n+15\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1-n}{2}}}{c^2 (n+3) \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}+\frac{\left (-n^3-2 n^2+7 n+18\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{3-n}{2}}}{c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{c^2 (n+3) \sqrt{c-a^2 c x^2}}+\frac{(n+6) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{c^2 (n+1) (n+3) \sqrt{c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Int[E^(n*ArcTanh[a*x])/(x*(c - a^2*c*x^2)^(5/2)),x]

[Out]

((1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) + ((6 + n)
*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(c^2*(1 + n)*(3 + n)*Sqrt[c - a^2*c*x^2]) -
((15 + 6*n + n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(c^2*(3 + n)*(1 - n^2)*Sqrt[
c - a^2*c*x^2]) + ((18 + 7*n - 2*n^2 - n^3)*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(c
^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2]) - (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^2*(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 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 \left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{n \tanh ^{-1}(a x)}}{x \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{(1-a x)^{-\frac{5}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}}}{x} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (3+n) \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{5}{2}+\frac{n}{2}} \left (-a (3+n)-3 a^2 x\right )}{x} \, dx}{a c^2 (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (3+n) \sqrt{c-a^2 c x^2}}+\frac{(6+n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1+n) (3+n) \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{5}{2}+\frac{n}{2}} \left (a^2 (1+n) (3+n)+2 a^3 (6+n) x\right )}{x} \, dx}{a^2 c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (3+n) \sqrt{c-a^2 c x^2}}+\frac{(6+n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{\left (15+6 n+n^2\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1-n) (1+n) (3+n) \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{5}{2}+\frac{n}{2}} \left (a^3 (1-n) (1+n) (3+n)-a^4 \left (15+6 n+n^2\right ) x\right )}{x} \, dx}{a^3 c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (3+n) \sqrt{c-a^2 c x^2}}+\frac{(6+n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{\left (15+6 n+n^2\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}+\frac{\left (18+7 n-2 n^2-n^3\right ) (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 \left (9-10 n^2+n^4\right ) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \int \frac{a^4 (1-n) (3-n) (1+n) (3+n) (1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}}}{x} \, dx}{a^4 c^2 (1-n) (3-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (3+n) \sqrt{c-a^2 c x^2}}+\frac{(6+n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{\left (15+6 n+n^2\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}+\frac{\left (18+7 n-2 n^2-n^3\right ) (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 \left (9-10 n^2+n^4\right ) \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}}}{x} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (3+n) \sqrt{c-a^2 c x^2}}+\frac{(6+n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{\left (15+6 n+n^2\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}+\frac{\left (18+7 n-2 n^2-n^3\right ) (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 \left (9-10 n^2+n^4\right ) \sqrt{c-a^2 c x^2}}-\frac{2 (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^2 (3-n) \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.242677, size = 222, normalized size = 0.53 \[ -\frac{\sqrt{1-a^2 x^2} (1-a x)^{\frac{1}{2} (-n-3)} (a x+1)^{\frac{n-3}{2}} \left (2 \left (n^3+3 n^2-n-3\right ) (a x-1)^3 \text{Hypergeometric2F1}\left (1,\frac{3}{2}-\frac{n}{2},\frac{5}{2}-\frac{n}{2},\frac{1-a x}{a x+1}\right )+n^3 \left (-\left (a^3 x^3-2 a^2 x^2+2\right )\right )+a n^2 x \left (-2 a^2 x^2+3 a x+2\right )+n \left (7 a^3 x^3-18 a^2 x^2-6 a x+18\right )+3 \left (6 a^3 x^3-3 a^2 x^2-6 a x+2\right )\right )}{c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcTanh[a*x])/(x*(c - a^2*c*x^2)^(5/2)),x]

[Out]

-(((1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2]*(a*n^2*x*(2 + 3*a*x - 2*a^2*x^2) - n^3*(2 -
2*a^2*x^2 + a^3*x^3) + 3*(2 - 6*a*x - 3*a^2*x^2 + 6*a^3*x^3) + n*(18 - 6*a*x - 18*a^2*x^2 + 7*a^3*x^3) + 2*(-
3 - n + 3*n^2 + n^3)*(-1 + a*x)^3*Hypergeometric2F1[1, 3/2 - n/2, 5/2 - n/2, (1 - a*x)/(1 + a*x)]))/(c^2*(9 -
10*n^2 + n^4)*Sqrt[c - a^2*c*x^2]))

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Maple [F]  time = 0.211, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{x} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))/x/(-a^2*c*x^2+c)^(5/2),x)

[Out]

int(exp(n*arctanh(a*x))/x/(-a^2*c*x^2+c)^(5/2),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{5}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/x/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/((-a^2*c*x^2 + c)^(5/2)*x), 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^{6} c^{3} x^{7} - 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} - c^{3} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/x/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^6*c^3*x^7 - 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 - c^
3*x), 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/(-a**2*c*x**2+c)**(5/2),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{5}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/x/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/((-a^2*c*x^2 + c)^(5/2)*x), x)