### 3.444 $$\int \frac{e^{\coth ^{-1}(a x)}}{(c-\frac{c}{a x})^{3/2}} \, dx$$

Optimal. Leaf size=215 $\frac{a x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}}{\left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}}{\left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{7 \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{\sqrt{2} a \left (c-\frac{c}{a x}\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.1548, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.364, Rules used = {6182, 6179, 99, 151, 156, 63, 208, 206} $\frac{a x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}}{\left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}}{\left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{7 \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{\sqrt{2} a \left (c-\frac{c}{a x}\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6182

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

Rule 6179

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

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 151

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] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.09067, size = 122, normalized size = 0.57 $\frac{\sqrt{1-\frac{1}{a x}} \left (2 a x \sqrt{\frac{1}{a x}+1} (a x-2)+10 (a x-1) \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )-7 \sqrt{2} (a x-1) \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )\right )}{2 a c (a x-1) \sqrt{c-\frac{c}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(-2 + a*x) + 10*(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)]] - 7*Sqrt[2]*
(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(2*a*c*Sqrt[c - c/(a*x)]*(-1 + a*x))

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Maple [A]  time = 0.187, size = 259, normalized size = 1.2 \begin{align*}{\frac{x}{ \left ( 4\,ax-4 \right ){c}^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 4\,{a}^{5/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}x-7\,{a}^{3/2}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ) x+10\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){a}^{2}\sqrt{{a}^{-1}}x-8\,\sqrt{ \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}-10\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}}+7\,\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ) \sqrt{a} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/((a*x-1)/(a*x+1))^(1/2)/(a*x-1)*(c*(a*x-1)/a/x)^(1/2)*x*(4*a^(5/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*x-7*a^(3/
2)*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*a+3*a*x+1)/(a*x-1))*x+10*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^
(1/2)+2*a*x+1)/a^(1/2))*a^2*(1/a)^(1/2)*x-8*((a*x+1)*x)^(1/2)*a^(3/2)*(1/a)^(1/2)-10*ln(1/2*(2*((a*x+1)*x)^(1/
2)*a^(1/2)+2*a*x+1)/a^(1/2))*a*(1/a)^(1/2)+7*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*a+3*a*x+1)/(a
*x-1))*a^(1/2))/a^(3/2)/c^2/((a*x+1)*x)^(1/2)/(1/a)^(1/2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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