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

Optimal. Leaf size=102 \[ \frac {\left (3-n^2\right ) (n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}}-\frac {(n-3 a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

-exp(n*arccoth(a*x))*(-3*a*x+n)/a^3/c/(-n^2+9)/(-a^2*c*x^2+c)^(3/2)+exp(n*arccoth(a*x))*(-n^2+3)*(-a*x+n)/a^3/
c^2/(n^4-10*n^2+9)/(-a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {6189, 6184} \[ \frac {\left (3-n^2\right ) (n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}}-\frac {(n-3 a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a^3*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) + (E^(n*ArcCoth[a*x])*(3 - n^2)*(n
- a*x))/(a^3*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])

Rule 6184

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((n - a*x)*E^(n*ArcCoth[a*x]))
/(a*c*(n^2 - 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n]

Rule 6189

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^2*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((n + 2*(p + 1)*a*x)*(c
+ d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]))/(a^3*c*(n^2 - 4*(p + 1)^2)), x] - Dist[(n^2 + 2*(p + 1))/(a^2*c*(n^2 - 4
*(p + 1)^2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0
] &&  !IntegerQ[n/2] && LeQ[p, -1] && NeQ[n^2 + 2*(p + 1), 0] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] ||
!IntegerQ[n])

Rubi steps

\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=-\frac {e^{n \coth ^{-1}(a x)} (n-3 a x)}{a^3 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {\left (3-n^2\right ) \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{a^2 c \left (9-n^2\right )}\\ &=-\frac {e^{n \coth ^{-1}(a x)} (n-3 a x)}{a^3 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {e^{n \coth ^{-1}(a x)} \left (3-n^2\right ) (n-a x)}{a^3 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.76, size = 109, normalized size = 1.07 \[ \frac {e^{n \coth ^{-1}(a x)} \left (3 a \left (n^2-1\right ) x \sqrt {1-\frac {1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )+a n^2 x-2 \left (n^2-1\right ) n \cosh \left (2 \coth ^{-1}(a x)\right )-9 a x-2 n^3+10 n\right )}{4 a^3 c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(E^(n*ArcCoth[a*x])*(10*n - 2*n^3 - 9*a*x + a*n^2*x - 2*n*(-1 + n^2)*Cosh[2*ArcCoth[a*x]] + 3*a*(-1 + n^2)*Sqr
t[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]]))/(4*a^3*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 179, normalized size = 1.75 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (2 \, a n^{2} x + {\left (a^{3} n^{2} - 3 \, a^{3}\right )} x^{3} + {\left (a^{2} n^{3} - 3 \, a^{2} n\right )} x^{2} + 2 \, n\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a^{3} c^{3} n^{4} - 10 \, a^{3} c^{3} n^{2} + 9 \, a^{3} c^{3} + {\left (a^{7} c^{3} n^{4} - 10 \, a^{7} c^{3} n^{2} + 9 \, a^{7} c^{3}\right )} x^{4} - 2 \, {\left (a^{5} c^{3} n^{4} - 10 \, a^{5} c^{3} n^{2} + 9 \, a^{5} c^{3}\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-a^2*c*x^2 + c)*(2*a*n^2*x + (a^3*n^2 - 3*a^3)*x^3 + (a^2*n^3 - 3*a^2*n)*x^2 + 2*n)*((a*x - 1)/(a*x + 1)
)^(1/2*n)/(a^3*c^3*n^4 - 10*a^3*c^3*n^2 + 9*a^3*c^3 + (a^7*c^3*n^4 - 10*a^7*c^3*n^2 + 9*a^7*c^3)*x^4 - 2*(a^5*
c^3*n^4 - 10*a^5*c^3*n^2 + 9*a^5*c^3)*x^2)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 96, normalized size = 0.94 \[ \frac {\left (a x -1\right ) \left (a x +1\right ) \left (a^{3} n^{2} x^{3}-a^{2} n^{3} x^{2}-3 x^{3} a^{3}+3 a^{2} n \,x^{2}+2 a \,n^{2} x -2 n \right ) {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{\left (n^{4}-10 n^{2}+9\right ) a^{3} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x-1)*(a*x+1)*(a^3*n^2*x^3-a^2*n^3*x^2-3*a^3*x^3+3*a^2*n*x^2+2*a*n^2*x-2*n)*exp(n*arccoth(a*x))/(n^4-10*n^2+
9)/a^3/(-a^2*c*x^2+c)^(5/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.56, size = 175, normalized size = 1.72 \[ \frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {2\,n}{a^5\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {x^3\,\left (n^2-3\right )}{a^2\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {2\,n^2\,x}{a^4\,c^2\,\left (n^4-10\,n^2+9\right )}+\frac {n\,x^2\,\left (n^2-3\right )}{a^3\,c^2\,\left (n^4-10\,n^2+9\right )}\right )}{\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^2}-x^2\,\sqrt {c-a^2\,c\,x^2}\right )\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((a*x + 1)/(a*x))^(n/2)*((2*n)/(a^5*c^2*(n^4 - 10*n^2 + 9)) - (x^3*(n^2 - 3))/(a^2*c^2*(n^4 - 10*n^2 + 9)) -
(2*n^2*x)/(a^4*c^2*(n^4 - 10*n^2 + 9)) + (n*x^2*(n^2 - 3))/(a^3*c^2*(n^4 - 10*n^2 + 9))))/(((c - a^2*c*x^2)^(1
/2)/a^2 - x^2*(c - a^2*c*x^2)^(1/2))*((a*x - 1)/(a*x))^(n/2))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*acoth(a*x))*x**2/(-a**2*c*x**2+c)**(5/2),x)

[Out]

Integral(x**2*exp(n*acoth(a*x))/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)

________________________________________________________________________________________