3.811 \(\int \frac{e^{-\coth ^{-1}(a x)}}{(c-\frac{c}{a^2 x^2})^2} \, dx\)

Optimal. Leaf size=179 \[ \frac{x}{c^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{\frac{1}{a x}+1}}+\frac{5 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{2}{a c^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^2} \]

[Out]

-2/(a*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) + (5*Sqrt[1 - 1/(a*x)])/(3*a*c^2*(1 + 1/(a*x))^(3/2)) + (8*Sq
rt[1 - 1/(a*x)])/(3*a*c^2*Sqrt[1 + 1/(a*x)]) + x/(c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) - ArcTanh[Sqrt[1
- 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^2)

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Rubi [A]  time = 0.11761, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6194, 103, 152, 12, 92, 208} \[ \frac{x}{c^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{\frac{1}{a x}+1}}+\frac{5 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{2}{a c^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(a*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) + (5*Sqrt[1 - 1/(a*x)])/(3*a*c^2*(1 + 1/(a*x))^(3/2)) + (8*Sq
rt[1 - 1/(a*x)])/(3*a*c^2*Sqrt[1 + 1/(a*x)]) + x/(c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) - ArcTanh[Sqrt[1
- 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^2)

Rule 6194

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

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.152771, size = 85, normalized size = 0.47 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (3 a^3 x^3+7 a^2 x^2-5 a x-8\right )}{3 (a x-1) (a x+1)^2}-\log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.143, size = 530, normalized size = 3. \begin{align*} -{\frac{1}{24\,a \left ( ax-1 \right ) ^{2} \left ( ax+1 \right ) ^{2}{c}^{2}} \left ( -45\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{5}{a}^{5}+24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{5}{a}^{6}+21\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{3}{a}^{3}-45\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{4}{a}^{4}+24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{4}{a}^{5}-11\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}+90\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}-48\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}-5\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa+90\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}-48\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}+19\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-45\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}-45\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+24\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ) \sqrt{{\frac{ax-1}{ax+1}}}{\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \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)^2,x)

[Out]

-1/24*(-45*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^5*a^5+24*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)
^(1/2))*x^5*a^6+21*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3-45*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4+
24*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^4*a^5-11*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*
x^2*a^2+90*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3-48*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)
^(1/2))*x^3*a^4-5*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a+90*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^2*a^2-48*ln
((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^2*a^3+19*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-45*(a
^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a+24*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x*a^2-45*
(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)+24*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2)))/a*((a*x-
1)/(a*x+1))^(1/2)/(a*x-1)^2/(a^2)^(1/2)/(a*x+1)^2/c^2/((a*x-1)*(a*x+1))^(1/2)

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Maxima [A]  time = 1.02282, size = 220, normalized size = 1.23 \begin{align*} -\frac{1}{12} \, a{\left (\frac{3 \,{\left (\frac{9 \,{\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - a^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 18 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}} + \frac{12 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac{12 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\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)^2,x, algorithm="maxima")

[Out]

-1/12*a*(3*(9*(a*x - 1)/(a*x + 1) - 1)/(a^2*c^2*((a*x - 1)/(a*x + 1))^(3/2) - a^2*c^2*sqrt((a*x - 1)/(a*x + 1)
)) - (((a*x - 1)/(a*x + 1))^(3/2) + 18*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^2) + 12*log(sqrt((a*x - 1)/(a*x + 1))
+ 1)/(a^2*c^2) - 12*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^2))

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Fricas [A]  time = 1.45643, size = 267, normalized size = 1.49 \begin{align*} -\frac{3 \,{\left (a^{2} x^{2} - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 3 \,{\left (a^{2} x^{2} - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (3 \, a^{3} x^{3} + 7 \, a^{2} x^{2} - 5 \, a x - 8\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c^{2} x^{2} - a c^{2}\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)^2,x, algorithm="fricas")

[Out]

-1/3*(3*(a^2*x^2 - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 3*(a^2*x^2 - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1)
- (3*a^3*x^3 + 7*a^2*x^2 - 5*a*x - 8)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c^2*x^2 - a*c^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{4} \int \frac{x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \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)**2,x)

[Out]

a**4*Integral(x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2

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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 )}^{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)^2,x, algorithm="giac")

[Out]

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