### 3.890 $$\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x} \, dx$$

Optimal. Leaf size=117 $\sqrt{c-\frac{c}{a^2 x^2}}-\frac{a x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}+\frac{2 a x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{\sqrt{1-a x} \sqrt{a x+1}}$

[Out]

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

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Rubi [A]  time = 0.550538, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {6167, 6159, 6129, 98, 157, 41, 216, 92, 208} $\sqrt{c-\frac{c}{a^2 x^2}}-\frac{a x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}+\frac{2 a x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{\sqrt{1-a x} \sqrt{a x+1}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6159

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

Rule 6129

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

Rule 98

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

Rule 157

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

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 216

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

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

Mathematica [A]  time = 0.0818302, size = 82, normalized size = 0.7 $\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1}+a x \log \left (\sqrt{a^2 x^2-1}+a x\right )-2 a x \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{\sqrt{a^2 x^2-1}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.179, size = 306, normalized size = 2.6 \begin{align*} -{\frac{1}{ac}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \left ( -\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{2}{a}^{3}c+{a}^{3} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{{\frac{3}{2}}}\sqrt{-{\frac{c}{{a}^{2}}}}+{c}^{{\frac{3}{2}}}\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) \sqrt{-{\frac{c}{{a}^{2}}}}xa-2\,{c}^{3/2}\sqrt{-{\frac{c}{{a}^{2}}}}\ln \left ({\frac{1}{\sqrt{c}} \left ( \sqrt{c}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}+cx \right ) } \right ) xa-2\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}x{a}^{2}c+2\,\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}cx{a}^{2}\sqrt{-{\frac{c}{{a}^{2}}}}+2\,\ln \left ( 2\,{\frac{1}{{a}^{2}x} \left ( \sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{a}^{2}-c \right ) } \right ) x{c}^{2} \right ){\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}}}{\frac{1}{\sqrt{-{\frac{c}{{a}^{2}}}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.66008, size = 555, normalized size = 4.74 \begin{align*} \left [-\sqrt{-c} \arctan \left (\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}, -2 \, \sqrt{c} \arctan \left (\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + \frac{1}{2} \, \sqrt{c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + sqrt(-c)*log(-(a^2*c*x^2
+ 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) + sqrt((a^2*c*x^2 - c)/(a^2*x^2)), -2*sqrt(c)*ar
ctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 1/2*sqrt(c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c
)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) + sqrt((a^2*c*x^2 - c)/(a^2*x^2))]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.31417, size = 171, normalized size = 1.46 \begin{align*}{\left (\frac{4 \, \sqrt{c} \arctan \left (-\frac{\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}}{\sqrt{c}}\right ) \mathrm{sgn}\left (x\right )}{a} - \frac{\sqrt{c} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} - c} \right |}\right ) \mathrm{sgn}\left (x\right )}{{\left | a \right |}} + \frac{2 \, c^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{{\left ({\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} + c\right )}{\left | a \right |}}\right )}{\left | a \right |} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*sqrt(c)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x)/a - sqrt(c)*log(abs(-sqrt(a^2*c)*x +
sqrt(a^2*c*x^2 - c)))*sgn(x)/abs(a) + 2*c^(3/2)*sgn(x)/(((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)*abs(a)))
*abs(a)