### 3.793 $$\int \frac{e^{3 \coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx$$

Optimal. Leaf size=144 $\frac{x \sqrt{\frac{1}{a x}+1}}{c \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{14 \sqrt{\frac{1}{a x}+1}}{3 a c \sqrt{1-\frac{1}{a x}}}-\frac{5 \sqrt{\frac{1}{a x}+1}}{3 a c \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c}$

[Out]

(-5*Sqrt[1 + 1/(a*x)])/(3*a*c*(1 - 1/(a*x))^(3/2)) - (14*Sqrt[1 + 1/(a*x)])/(3*a*c*Sqrt[1 - 1/(a*x)]) + (Sqrt[
1 + 1/(a*x)]*x)/(c*(1 - 1/(a*x))^(3/2)) + (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c)

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Rubi [A]  time = 0.101411, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.273, Rules used = {6194, 99, 152, 12, 92, 208} $\frac{x \sqrt{\frac{1}{a x}+1}}{c \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{14 \sqrt{\frac{1}{a x}+1}}{3 a c \sqrt{1-\frac{1}{a x}}}-\frac{5 \sqrt{\frac{1}{a x}+1}}{3 a c \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-5*Sqrt[1 + 1/(a*x)])/(3*a*c*(1 - 1/(a*x))^(3/2)) - (14*Sqrt[1 + 1/(a*x)])/(3*a*c*Sqrt[1 - 1/(a*x)]) + (Sqrt[
1 + 1/(a*x)]*x)/(c*(1 - 1/(a*x))^(3/2)) + (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c)

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 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

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^{3 \coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^2 \left (1-\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{\sqrt{1+\frac{1}{a x}} x}{c \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{3}{a}+\frac{2 x}{a^2}}{x \left (1-\frac{x}{a}\right )^{5/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{5 \sqrt{1+\frac{1}{a x}}}{3 a c \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{\sqrt{1+\frac{1}{a x}} x}{c \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{a \operatorname{Subst}\left (\int \frac{-\frac{9}{a^2}-\frac{5 x}{a^3}}{x \left (1-\frac{x}{a}\right )^{3/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{3 c}\\ &=-\frac{5 \sqrt{1+\frac{1}{a x}}}{3 a c \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{14 \sqrt{1+\frac{1}{a x}}}{3 a c \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} x}{c \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{a^2 \operatorname{Subst}\left (\int \frac{9}{a^3 x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{3 c}\\ &=-\frac{5 \sqrt{1+\frac{1}{a x}}}{3 a c \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{14 \sqrt{1+\frac{1}{a x}}}{3 a c \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} x}{c \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{3 \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}\\ &=-\frac{5 \sqrt{1+\frac{1}{a x}}}{3 a c \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{14 \sqrt{1+\frac{1}{a x}}}{3 a c \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} x}{c \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{3 \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}\\ &=-\frac{5 \sqrt{1+\frac{1}{a x}}}{3 a c \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{14 \sqrt{1+\frac{1}{a x}}}{3 a c \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} x}{c \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a c}\\ \end{align*}

Mathematica [A]  time = 0.124484, size = 69, normalized size = 0.48 $\frac{\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (3 a^2 x^2-19 a x+14\right )}{(a x-1)^2}+\frac{9 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a}}{3 c}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((Sqrt[1 - 1/(a^2*x^2)]*x*(14 - 19*a*x + 3*a^2*x^2))/(-1 + a*x)^2 + (9*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a)/
(3*c)

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Maple [B]  time = 0.184, size = 346, normalized size = 2.4 \begin{align*}{\frac{1}{3\, \left ( ax-1 \right ) ac \left ( ax+1 \right ) } \left ( 9\,\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}+9\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}-27\,\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}-6\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-27\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+27\,\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}+5\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}+27\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-9\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -9\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(9*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^3*a^4+9*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/
2)*x^3*a^3-27*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^2*a^3-6*(a^2)^(1/2)*((a*x-1)*(a*x+
1))^(3/2)*x*a-27*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^2*a^2+27*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))
/(a^2)^(1/2))*x*a^2+5*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)+27*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a-9*a*ln((a
^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))-9*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/a/(a^2)^(1/2)/(a
*x-1)/c/((a*x-1)*(a*x+1))^(1/2)/(a*x+1)/((a*x-1)/(a*x+1))^(3/2)

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Maxima [A]  time = 1.06804, size = 180, normalized size = 1.25 \begin{align*} \frac{1}{3} \, a{\left (\frac{\frac{11 \,{\left (a x - 1\right )}}{a x + 1} - \frac{18 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - a^{2} c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}} + \frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*((11*(a*x - 1)/(a*x + 1) - 18*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c*((a*x - 1)/(a*x + 1))^(5/2) - a^2*c*((
a*x - 1)/(a*x + 1))^(3/2)) + 9*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c) - 9*log(sqrt((a*x - 1)/(a*x + 1)) -
1)/(a^2*c))

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Fricas [A]  time = 1.58576, size = 301, normalized size = 2.09 \begin{align*} \frac{9 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 9 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (3 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 5 \, a x + 14\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(9*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 9*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(
a*x + 1)) - 1) + (3*a^3*x^3 - 16*a^2*x^2 - 5*a*x + 14)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c*x^2 - 2*a^2*c*x + a*c
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{2} \int \frac{x^{2}}{\frac{a^{3} x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} + \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx}{c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.195, size = 200, normalized size = 1.39 \begin{align*} \frac{1}{3} \, a{\left (\frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac{9 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c} - \frac{{\left (a x + 1\right )}{\left (\frac{12 \,{\left (a x - 1\right )}}{a x + 1} + 1\right )}}{{\left (a x - 1\right )} a^{2} c \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{6 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c{\left (\frac{a x - 1}{a x + 1} - 1\right )}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*(9*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c) - 9*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a^2*c) - (a*x
+ 1)*(12*(a*x - 1)/(a*x + 1) + 1)/((a*x - 1)*a^2*c*sqrt((a*x - 1)/(a*x + 1))) - 6*sqrt((a*x - 1)/(a*x + 1))/(
a^2*c*((a*x - 1)/(a*x + 1) - 1)))