### 3.861 $$\int \frac{e^{-\coth ^{-1}(a x)}}{(c-\frac{c}{a^2 x^2})^{5/2}} \, dx$$

Optimal. Leaf size=263 $\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{a c^2 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{7 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{23 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}$

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/(c^2*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(
1 - a*x)) + Sqrt[1 - 1/(a^2*x^2)]/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2) - Sqrt[1 - 1/(a^2*x^2)]/(a*c^2*S
qrt[c - c/(a^2*x^2)]*(1 + a*x)) + (7*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)]) - (2
3*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)])

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Rubi [A]  time = 0.169411, antiderivative size = 263, 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} $\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{a c^2 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{7 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{23 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(5/2)),x]

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/(c^2*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(
1 - a*x)) + Sqrt[1 - 1/(a^2*x^2)]/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2) - Sqrt[1 - 1/(a^2*x^2)]/(a*c^2*S
qrt[c - c/(a^2*x^2)]*(1 + a*x)) + (7*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)]) - (2
3*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(16*a*c^2*Sqrt[c - c/(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 \frac{e^{-\coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \, dx &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} \int \frac{e^{-\coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{5/2}} \, dx}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a^5 \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \frac{x^5}{(-1+a x)^2 (1+a x)^3} \, dx}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a^5 \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \left (\frac{1}{a^5}+\frac{1}{8 a^5 (-1+a x)^2}+\frac{7}{16 a^5 (-1+a x)}-\frac{1}{4 a^5 (1+a x)^3}+\frac{1}{a^5 (1+a x)^2}-\frac{23}{16 a^5 (1+a x)}\right ) \, dx}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)^2}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{a c^2 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)}+\frac{7 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{23 \sqrt{1-\frac{1}{a^2 x^2}} \log (1+a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.118547, size = 85, normalized size = 0.32 $\frac{\left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (2 \left (8 a x+\frac{1}{1-a x}-\frac{8}{a x+1}+\frac{1}{(a x+1)^2}\right )+7 \log (1-a x)-23 \log (a x+1)\right )}{16 a \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(5/2)),x]

[Out]

((1 - 1/(a^2*x^2))^(5/2)*(2*(8*a*x + (1 - a*x)^(-1) + (1 + a*x)^(-2) - 8/(1 + a*x)) + 7*Log[1 - a*x] - 23*Log[
1 + a*x]))/(16*a*(c - c/(a^2*x^2))^(5/2))

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Maple [A]  time = 0.241, size = 175, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) \left ( -16\,{x}^{4}{a}^{4}+23\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -7\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}-16\,{x}^{3}{a}^{3}+23\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-7\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+34\,{a}^{2}{x}^{2}-23\,ax\ln \left ( ax+1 \right ) +7\,\ln \left ( ax-1 \right ) xa+18\,ax-23\,\ln \left ( ax+1 \right ) +7\,\ln \left ( ax-1 \right ) -12 \right ) }{16\,{a}^{6}{x}^{5}}\sqrt{{\frac{ax-1}{ax+1}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(a*x-1)*(-16*x^4*a^4+23*a^3*x^3*ln(a*x+1)-7*ln(a*x-1)*x^3*a^3-16*x^3*a^3
+23*ln(a*x+1)*a^2*x^2-7*ln(a*x-1)*a^2*x^2+34*a^2*x^2-23*a*x*ln(a*x+1)+7*ln(a*x-1)*x*a+18*a*x-23*ln(a*x+1)+7*ln
(a*x-1)-12)/a^6/x^5/(c*(a^2*x^2-1)/a^2/x^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.81731, size = 292, normalized size = 1.11 \begin{align*} \frac{{\left (16 \, a^{4} x^{4} + 16 \, a^{3} x^{3} - 34 \, a^{2} x^{2} - 18 \, a x - 23 \,{\left (a^{3} x^{3} + a^{2} x^{2} - a x - 1\right )} \log \left (a x + 1\right ) + 7 \,{\left (a^{3} x^{3} + a^{2} x^{2} - a x - 1\right )} \log \left (a x - 1\right ) + 12\right )} \sqrt{a^{2} c}}{16 \,{\left (a^{5} c^{3} x^{3} + a^{4} c^{3} x^{2} - a^{3} c^{3} x - a^{2} c^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^(5/2),x, algorithm="fricas")

[Out]

1/16*(16*a^4*x^4 + 16*a^3*x^3 - 34*a^2*x^2 - 18*a*x - 23*(a^3*x^3 + a^2*x^2 - a*x - 1)*log(a*x + 1) + 7*(a^3*x
^3 + a^2*x^2 - a*x - 1)*log(a*x - 1) + 12)*sqrt(a^2*c)/(a^5*c^3*x^3 + a^4*c^3*x^2 - a^3*c^3*x - a^2*c^3)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^(5/2),x, algorithm="giac")

[Out]

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