### 3.871 $$\int e^{-3 \coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^{9/2} \, dx$$

Optimal. Leaf size=322 $\frac{c^4 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{8 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{7 a^8 x^7 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{8 a^9 x^8 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^4 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}$

[Out]

-(c^4*Sqrt[c - c/(a^2*x^2)])/(8*a^9*Sqrt[1 - 1/(a^2*x^2)]*x^8) + (3*c^4*Sqrt[c - c/(a^2*x^2)])/(7*a^8*Sqrt[1 -
1/(a^2*x^2)]*x^7) - (8*c^4*Sqrt[c - c/(a^2*x^2)])/(5*a^6*Sqrt[1 - 1/(a^2*x^2)]*x^5) + (3*c^4*Sqrt[c - c/(a^2*
x^2)])/(2*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) + (2*c^4*Sqrt[c - c/(a^2*x^2)])/(a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) - (4*
c^4*Sqrt[c - c/(a^2*x^2)])/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (c^4*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2
)] - (3*c^4*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])

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Rubi [A]  time = 0.155947, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {6197, 6193, 88} $\frac{c^4 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{8 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{7 a^8 x^7 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{8 a^9 x^8 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^4 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c - c/(a^2*x^2))^(9/2)/E^(3*ArcCoth[a*x]),x]

[Out]

-(c^4*Sqrt[c - c/(a^2*x^2)])/(8*a^9*Sqrt[1 - 1/(a^2*x^2)]*x^8) + (3*c^4*Sqrt[c - c/(a^2*x^2)])/(7*a^8*Sqrt[1 -
1/(a^2*x^2)]*x^7) - (8*c^4*Sqrt[c - c/(a^2*x^2)])/(5*a^6*Sqrt[1 - 1/(a^2*x^2)]*x^5) + (3*c^4*Sqrt[c - c/(a^2*
x^2)])/(2*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) + (2*c^4*Sqrt[c - c/(a^2*x^2)])/(a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) - (4*
c^4*Sqrt[c - c/(a^2*x^2)])/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (c^4*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2
)] - (3*c^4*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])

Rule 6197

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

Rule 6193

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[c^p/a^(2*p), Int[(u*(-1
+ a*x)^(p - n/2)*(1 + a*x)^(p + n/2))/x^(2*p), 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]

Rule 88

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, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{9/2} \, dx &=\frac{\left (c^4 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c^4 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \frac{(-1+a x)^6 (1+a x)^3}{x^9} \, dx}{a^9 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c^4 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \left (a^9+\frac{1}{x^9}-\frac{3 a}{x^8}+\frac{8 a^3}{x^6}-\frac{6 a^4}{x^5}-\frac{6 a^5}{x^4}+\frac{8 a^6}{x^3}-\frac{3 a^8}{x}\right ) \, dx}{a^9 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=-\frac{c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{8 a^9 \sqrt{1-\frac{1}{a^2 x^2}} x^8}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{7 a^8 \sqrt{1-\frac{1}{a^2 x^2}} x^7}-\frac{8 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 \sqrt{1-\frac{1}{a^2 x^2}} x^5}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^5 \sqrt{1-\frac{1}{a^2 x^2}} x^4}+\frac{2 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 \sqrt{1-\frac{1}{a^2 x^2}} x^3}-\frac{4 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}+\frac{c^4 \sqrt{c-\frac{c}{a^2 x^2}} x}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}} \log (x)}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.0809334, size = 97, normalized size = 0.3 $\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} \left (-\frac{4 a^6}{x^2}+\frac{2 a^5}{x^3}+\frac{3 a^4}{2 x^4}-\frac{8 a^3}{5 x^5}+a^9 x-3 a^8 \log (x)+\frac{3 a}{7 x^7}-\frac{1}{8 x^8}\right )}{a^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c - c/(a^2*x^2))^(9/2)/E^(3*ArcCoth[a*x]),x]

[Out]

((c - c/(a^2*x^2))^(9/2)*(-1/(8*x^8) + (3*a)/(7*x^7) - (8*a^3)/(5*x^5) + (3*a^4)/(2*x^4) + (2*a^5)/x^3 - (4*a^
6)/x^2 + a^9*x - 3*a^8*Log[x]))/(a^9*(1 - 1/(a^2*x^2))^(9/2))

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Maple [A]  time = 0.243, size = 112, normalized size = 0.4 \begin{align*} -{\frac{ \left ( -280\,{a}^{9}{x}^{9}+840\,{a}^{8}\ln \left ( x \right ){x}^{8}+1120\,{x}^{6}{a}^{6}-560\,{x}^{5}{a}^{5}-420\,{x}^{4}{a}^{4}+448\,{x}^{3}{a}^{3}-120\,ax+35 \right ) x}{280\, \left ( ax-1 \right ) ^{3} \left ({a}^{2}{x}^{2}-1 \right ) ^{3}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{9}{2}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c-c/a^2/x^2)^(9/2)*((a*x-1)/(a*x+1))^(3/2),x)

[Out]

-1/280*(-280*a^9*x^9+840*a^8*ln(x)*x^8+1120*x^6*a^6-560*x^5*a^5-420*x^4*a^4+448*x^3*a^3-120*a*x+35)*x*(c*(a^2*
x^2-1)/a^2/x^2)^(9/2)*((a*x-1)/(a*x+1))^(3/2)/(a*x-1)^3/(a^2*x^2-1)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{9}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(9/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")

[Out]

integrate((c - c/(a^2*x^2))^(9/2)*((a*x - 1)/(a*x + 1))^(3/2), x)

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Fricas [A]  time = 1.56452, size = 228, normalized size = 0.71 \begin{align*} \frac{{\left (280 \, a^{9} c^{4} x^{9} - 840 \, a^{8} c^{4} x^{8} \log \left (x\right ) - 1120 \, a^{6} c^{4} x^{6} + 560 \, a^{5} c^{4} x^{5} + 420 \, a^{4} c^{4} x^{4} - 448 \, a^{3} c^{4} x^{3} + 120 \, a c^{4} x - 35 \, c^{4}\right )} \sqrt{a^{2} c}}{280 \, a^{10} x^{8}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(9/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")

[Out]

1/280*(280*a^9*c^4*x^9 - 840*a^8*c^4*x^8*log(x) - 1120*a^6*c^4*x^6 + 560*a^5*c^4*x^5 + 420*a^4*c^4*x^4 - 448*a
^3*c^4*x^3 + 120*a*c^4*x - 35*c^4)*sqrt(a^2*c)/(a^10*x^8)

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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((c-c/a**2/x**2)**(9/2)*((a*x-1)/(a*x+1))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{9}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(9/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")

[Out]

integrate((c - c/(a^2*x^2))^(9/2)*((a*x - 1)/(a*x + 1))^(3/2), x)