### 3.757 $$\int \frac{e^{n \coth ^{-1}(a x)} x^3}{(c-a^2 c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=330 $-\frac{3 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{(n+3) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}$

[Out]

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

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Rubi [A]  time = 0.366695, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.148, Rules used = {6192, 6195, 45, 37} $-\frac{3 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{(n+3) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(E^(n*ArcCoth[a*x])*x^3)/(c - a^2*c*x^2)^(5/2),x]

[Out]

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

Rule 6192

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

Rule 6195

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

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]

Rubi steps

\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^2} \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{-\frac{5}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1-n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (9-10 n^2+n^4\right ) \left (c-a^2 c x^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.654351, size = 110, normalized size = 0.33 $-\frac{e^{n \coth ^{-1}(a x)} \left (a n \left (n^2-1\right ) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )+3 \left (a n^3 x-9 a n x-2 n^2+10\right )-6 \left (n^2-1\right ) \cosh \left (2 \coth ^{-1}(a x)\right )\right )}{4 a^4 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^(n*ArcCoth[a*x])*x^3)/(c - a^2*c*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.043, size = 93, normalized size = 0.3 \begin{align*} -{\frac{ \left ({a}^{3}{n}^{3}{x}^{3}-7\,n{x}^{3}{a}^{3}-3\,{a}^{2}{n}^{2}{x}^{2}+9\,{a}^{2}{x}^{2}+6\,nax-6 \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{{a}^{4} \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))*x^3/(-a^2*c*x^2+c)^(5/2),x)

[Out]

-(a*x-1)*(a*x+1)*(a^3*n^3*x^3-7*a^3*n*x^3-3*a^2*n^2*x^2+9*a^2*x^2+6*a*n*x-6)*exp(n*arccoth(a*x))/a^4/(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{x^{3} \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))*x^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(x^3*((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 = 1.63409, size = 354, normalized size = 1.07 \begin{align*} -\frac{\sqrt{-a^{2} c x^{2} + c}{\left ({\left (a^{3} n^{3} - 7 \, a^{3} n\right )} x^{3} + 6 \, a n x + 3 \,{\left (a^{2} n^{2} - 3 \, a^{2}\right )} x^{2} + 6\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3} +{\left (a^{8} c^{3} n^{4} - 10 \, a^{8} c^{3} n^{2} + 9 \, a^{8} c^{3}\right )} x^{4} - 2 \,{\left (a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right )} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))*x^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

-sqrt(-a^2*c*x^2 + c)*((a^3*n^3 - 7*a^3*n)*x^3 + 6*a*n*x + 3*(a^2*n^2 - 3*a^2)*x^2 + 6)*((a*x - 1)/(a*x + 1))^
(1/2*n)/(a^4*c^3*n^4 - 10*a^4*c^3*n^2 + 9*a^4*c^3 + (a^8*c^3*n^4 - 10*a^8*c^3*n^2 + 9*a^8*c^3)*x^4 - 2*(a^6*c^
3*n^4 - 10*a^6*c^3*n^2 + 9*a^6*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))*x**3/(-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{x^{3} \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))*x^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(x^3*((a*x - 1)/(a*x + 1))^(1/2*n)/(-a^2*c*x^2 + c)^(5/2), x)