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

Optimal. Leaf size=97 \[ \frac {2 n (n-a x) e^{n \coth ^{-1}(a x)}}{a^2 c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}}+\frac {(3-a n x) e^{n \coth ^{-1}(a x)}}{a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6187, 6184} \[ \frac {2 n (n-a x) e^{n \coth ^{-1}(a x)}}{a^2 c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}}+\frac {(3-a n x) e^{n \coth ^{-1}(a x)}}{a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(n*ArcCoth[a*x])*(3 - a*n*x))/(a^2*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2)) + (2*E^(n*ArcCoth[a*x])*n*(n - a*x))/
(a^2*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 6187

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((2*(p + 1) + a*n*x)*(c +
d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]))/(a^2*c*(n^2 - 4*(p + 1)^2)), x] - Dist[(n*(2*p + 3))/(a*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] &&  !I
ntegerQ[n/2] && LeQ[p, -1] && NeQ[p, -3/2] && 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}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {e^{n \coth ^{-1}(a x)} (3-a n x)}{a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {(2 n) \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{a c \left (9-n^2\right )}\\ &=\frac {e^{n \coth ^{-1}(a x)} (3-a n x)}{a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 e^{n \coth ^{-1}(a x)} n (n-a x)}{a^2 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.66, size = 108, normalized size = 1.11 \[ \frac {e^{n \coth ^{-1}(a x)} \left (-a \left (n^2-1\right ) n x \sqrt {1-\frac {1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )+a n^3 x+6 \left (n^2-1\right ) \cosh \left (2 \coth ^{-1}(a x)\right )-9 a n x+2 n^2+6\right )}{4 a^2 c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(E^(n*ArcCoth[a*x])*(6 + 2*n^2 - 9*a*n*x + a*n^3*x + 6*(-1 + n^2)*Cosh[2*ArcCoth[a*x]] - a*n*(-1 + n^2)*Sqrt[1
- 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]]))/(4*a^2*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])

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fricas [A]  time = 0.53, size = 170, normalized size = 1.75 \[ -\frac {{\left (2 \, a^{3} n x^{3} + 2 \, a^{2} n^{2} x^{2} + n^{2} + {\left (a n^{3} - 3 \, a n\right )} x - 3\right )} \sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{3} n^{4} - 10 \, a^{2} c^{3} n^{2} + 9 \, a^{2} c^{3} + {\left (a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right )} x^{4} - 2 \, {\left (a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3}\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a^3*n*x^3 + 2*a^2*n^2*x^2 + n^2 + (a*n^3 - 3*a*n)*x - 3)*sqrt(-a^2*c*x^2 + c)*((a*x - 1)/(a*x + 1))^(1/2*n
)/(a^2*c^3*n^4 - 10*a^2*c^3*n^2 + 9*a^2*c^3 + (a^6*c^3*n^4 - 10*a^6*c^3*n^2 + 9*a^6*c^3)*x^4 - 2*(a^4*c^3*n^4
- 10*a^4*c^3*n^2 + 9*a^4*c^3)*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 86, normalized size = 0.89 \[ -\frac {\left (a x -1\right ) \left (a x +1\right ) \left (2 x^{3} a^{3} n -2 a^{2} n^{2} x^{2}+a \,n^{3} x -3 x a n -n^{2}+3\right ) {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{a^{2} \left (n^{4}-10 n^{2}+9\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.55, size = 176, normalized size = 1.81 \[ -\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {n^2-3}{a^4\,c^2\,\left (n^4-10\,n^2+9\right )}+\frac {2\,n^2\,x^2}{a^2\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {2\,n\,x^3}{a\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {n\,x\,\left (n^2-3\right )}{a^3\,c^2\,\left (n^4-10\,n^2+9\right )}\right )}{\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^2}-x^2\,\sqrt {c-a^2\,c\,x^2}\right )\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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