3.1319 \(\int \frac{e^{n \tanh ^{-1}(a x)} x^3}{(c-a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=311 \[ \frac{2^{\frac{n}{2}+2} (1-a x)^{-\frac{n}{2}-1} \text{Hypergeometric2F1}\left (-\frac{n}{2}-1,-\frac{n}{2}-1,-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a^4 c^2 (n+2)}+\frac{2 (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}}}{a^4 c^2 n \left (4-n^2\right )}-\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a^4 c^2 (n+2)}+\frac{3 (a x+1)^{n/2} (1-a x)^{-\frac{n}{2}-1}}{a^4 c^2 (n+2)}-\frac{3 (a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a^4 c^2 (n+2)}-\frac{2 (a x+1)^{\frac{n-2}{2}} (1-a x)^{-n/2}}{a^4 c^2 n (n+2)}+\frac{3 (a x+1)^{n/2} (1-a x)^{-n/2}}{a^4 c^2 n (n+2)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.252191, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6150, 128, 45, 37, 69} \[ \frac{2^{\frac{n}{2}+2} (1-a x)^{-\frac{n}{2}-1} \, _2F_1\left (-\frac{n}{2}-1,-\frac{n}{2}-1;-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a^4 c^2 (n+2)}+\frac{2 (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}}}{a^4 c^2 n \left (4-n^2\right )}-\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a^4 c^2 (n+2)}+\frac{3 (a x+1)^{n/2} (1-a x)^{-\frac{n}{2}-1}}{a^4 c^2 (n+2)}-\frac{3 (a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a^4 c^2 (n+2)}-\frac{2 (a x+1)^{\frac{n-2}{2}} (1-a x)^{-n/2}}{a^4 c^2 n (n+2)}+\frac{3 (a x+1)^{n/2} (1-a x)^{-n/2}}{a^4 c^2 n (n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((1 - a*x)^(-1 - n/2)*(1 + a*x)^((-2 + n)/2))/(a^4*c^2*(2 + n))) + (2*(1 - a*x)^(1 - n/2)*(1 + a*x)^((-2 + n
)/2))/(a^4*c^2*n*(4 - n^2)) - (2*(1 + a*x)^((-2 + n)/2))/(a^4*c^2*n*(2 + n)*(1 - a*x)^(n/2)) + (3*(1 - a*x)^(-
1 - n/2)*(1 + a*x)^(n/2))/(a^4*c^2*(2 + n)) + (3*(1 + a*x)^(n/2))/(a^4*c^2*n*(2 + n)*(1 - a*x)^(n/2)) - (3*(1
- a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/(a^4*c^2*(2 + n)) + (2^(2 + n/2)*(1 - a*x)^(-1 - n/2)*Hypergeometric2
F1[-1 - n/2, -1 - n/2, -n/2, (1 - a*x)/2])/(a^4*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 128

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

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]))

Rubi steps

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

Mathematica [A]  time = 5.19257, size = 122, normalized size = 0.39 \[ -\frac{e^{n \tanh ^{-1}(a x)} \left (-2 \left (n^2-4\right ) \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,-e^{2 \tanh ^{-1}(a x)}\right )+2 (n-2) n e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \tanh ^{-1}(a x)}\right )+n \left (2 \cosh \left (2 \tanh ^{-1}(a x)\right )-n \sinh \left (2 \tanh ^{-1}(a x)\right )\right )\right )}{2 a^4 c^2 n \left (n^2-4\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(E^(n*ArcTanh[a*x])*(2*E^(2*ArcTanh[a*x])*(-2 + n)*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, -E^(2*ArcTanh[a*x
])] - 2*(-4 + n^2)*Hypergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcTanh[a*x])] + n*(2*Cosh[2*ArcTanh[a*x]] - n*Sin
h[2*ArcTanh[a*x]])))/(2*a^4*c^2*n*(-4 + n^2))

________________________________________________________________________________________

Maple [F]  time = 0.192, 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 ) ^{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)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3} \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))*x^3/(-a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

integral(x^3*((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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{3} 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))*x**3/(-a**2*c*x**2+c)**2,x)

[Out]

Integral(x**3*exp(n*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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