### 3.511 $$\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x} \, dx$$

Optimal. Leaf size=146 $\frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{\sqrt{1-\frac{1}{a x}}}$

[Out]

(2*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])/Sqrt[1 - 1/(a*x)] + (2*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/S
qrt[1 - 1/(a*x)] - (4*Sqrt[2]*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/Sqrt[1 - 1/(a*x)]

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Rubi [A]  time = 0.272623, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.259, Rules used = {6182, 6180, 84, 156, 63, 208, 206} $\frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{\sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])/Sqrt[1 - 1/(a*x)] + (2*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/S
qrt[1 - 1/(a*x)] - (4*Sqrt[2]*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/Sqrt[1 - 1/(a*x)]

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 6180

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^p, Subst[Int[((1
+ (d*x)/c)^p*(1 + x/a)^(n/2))/(x^(m + 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]) && IntegerQ[m]

Rule 84

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

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

Mathematica [A]  time = 0.369214, size = 218, normalized size = 1.49 $\frac{2 a x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}}{a x-1}+\sqrt{c} \log \left (2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )-2 \sqrt{2} \sqrt{c} \log \left (2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (3 a^2 x^2-2 a x-1\right )\right )-\sqrt{c} \log (1-a x)+2 \sqrt{2} \sqrt{c} \log \left ((a x-1)^2\right )$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.18, size = 161, normalized size = 1.1 \begin{align*} -{\frac{ax-1}{ax+1}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,\sqrt{a}\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-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1 \right ){\frac{1}{\sqrt{a}}}} \right ) \sqrt{{a}^{-1}}xa-2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}\sqrt{{a}^{-1}} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{a}}}{\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))^(3/2)*(c-c/a/x)^(1/2)/x,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}}}{x \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(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(1/2)/x,x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}}}{x \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(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(1/2)/x,x, algorithm="giac")

[Out]

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