### 3.760 $$\int \frac{e^{n \coth ^{-1}(a x)}}{(c-a^2 c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=102 $-\frac{6 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (1-n^2\right ) \left (9-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{(n-3 a x) e^{n \coth ^{-1}(a x)}}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.106421, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {6185, 6184} $-\frac{6 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (1-n^2\right ) \left (9-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{(n-3 a x) e^{n \coth ^{-1}(a x)}}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6185

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

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]

Rubi steps

\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{6 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{c \left (9-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{6 e^{n \coth ^{-1}(a x)} (n-a x)}{a c^2 \left (1-n^2\right ) \left (9-n^2\right ) \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.199781, size = 110, normalized size = 1.08 $\frac{e^{n \coth ^{-1}(a x)} \left (3 a \left (n^2-1\right ) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )-3 a n^2 x-2 \left (n^2-1\right ) n \cosh \left (2 \coth ^{-1}(a x)\right )+27 a x+2 n^3-26 n\right )}{4 a c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(E^(n*ArcCoth[a*x])*(-26*n + 2*n^3 + 27*a*x - 3*a*n^2*x - 2*n*(-1 + n^2)*Cosh[2*ArcCoth[a*x]] + 3*a*(-1 + n^2)
*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]]))/(4*a*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])

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Maple [A]  time = 0.041, size = 84, normalized size = 0.8 \begin{align*}{\frac{ \left ( 6\,{x}^{3}{a}^{3}-6\,n{a}^{2}{x}^{2}+3\,a{n}^{2}x-{n}^{3}-9\,ax+7\,n \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{a \left ({n}^{4}-10\,{n}^{2}+9 \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x-1)*(a*x+1)*(6*a^3*x^3-6*a^2*n*x^2+3*a*n^2*x-n^3-9*a*x+7*n)*exp(n*arccoth(a*x))/a/(n^4-10*n^2+9)/(-a^2*c*x
^2+c)^(5/2)

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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}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*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)

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Fricas [A]  time = 1.51349, size = 335, normalized size = 3.28 \begin{align*} -\frac{{\left (6 \, a^{3} x^{3} + 6 \, a^{2} n x^{2} + n^{3} + 3 \,{\left (a n^{2} - 3 \, a\right )} x - 7 \, n\right )} \sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{3} n^{4} - 10 \, a c^{3} n^{2} +{\left (a^{5} c^{3} n^{4} - 10 \, a^{5} c^{3} n^{2} + 9 \, a^{5} c^{3}\right )} x^{4} + 9 \, a c^{3} - 2 \,{\left (a^{3} c^{3} n^{4} - 10 \, a^{3} c^{3} n^{2} + 9 \, a^{3} c^{3}\right )} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6*a^3*x^3 + 6*a^2*n*x^2 + n^3 + 3*(a*n^2 - 3*a)*x - 7*n)*sqrt(-a^2*c*x^2 + c)*((a*x - 1)/(a*x + 1))^(1/2*n)/
(a*c^3*n^4 - 10*a*c^3*n^2 + (a^5*c^3*n^4 - 10*a^5*c^3*n^2 + 9*a^5*c^3)*x^4 + 9*a*c^3 - 2*(a^3*c^3*n^4 - 10*a^3
*c^3*n^2 + 9*a^3*c^3)*x^2)

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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*acoth(a*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}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*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)