### 3.889 $$\int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx$$

Optimal. Leaf size=116 $x \sqrt{c-\frac{c}{a^2 x^2}}-\frac{2 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{\sqrt{1-a x} \sqrt{a x+1}}$

[Out]

Sqrt[c - c/(a^2*x^2)]*x - (2*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (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 = 0.340949, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.375, Rules used = {6167, 6159, 6129, 102, 157, 41, 216, 92, 208} $x \sqrt{c-\frac{c}{a^2 x^2}}-\frac{2 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{\sqrt{1-a x} \sqrt{a x+1}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Sqrt[c - c/(a^2*x^2)]*x - (2*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (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 102

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))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]

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 e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, 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} \, 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 \sqrt{1-a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=\sqrt{c-\frac{c}{a^2 x^2}} x+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{-a-2 a^2 x}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{a \sqrt{1-a x} \sqrt{1+a x}}\\ &=\sqrt{c-\frac{c}{a^2 x^2}} x-\frac{\left (\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 (2 a \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}} x+\frac{\left (a \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}}-\frac{\left (2 a \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}}\\ &=\sqrt{c-\frac{c}{a^2 x^2}} x-\frac{2 \sqrt{c-\frac{c}{a^2 x^2}} x \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{1+a x}}+\frac{\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 = 0.0713889, size = 80, normalized size = 0.69 $\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1}+2 \log \left (\sqrt{a^2 x^2-1}+a x\right )-\tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{\sqrt{a^2 x^2-1}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.195, size = 196, normalized size = 1.7 \begin{align*} -{\frac{x}{{a}^{2}}\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}}}}{a}^{2}-2\,\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}{a}^{2}\sqrt{-{\frac{c}{{a}^{2}}}}-2\,\sqrt{c}\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 ) a\sqrt{-{\frac{c}{{a}^{2}}}}+c\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 ) \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)

[Out]

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

[Out]

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

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Fricas [A]  time = 1.60418, size = 576, normalized size = 4.97 \begin{align*} \left [\frac{2 \, a x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 4 \, \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 )}{2 \, a}, \frac{a x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{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 ) + \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 )}{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, algorithm="fricas")

[Out]

[1/2*(2*a*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 4*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))/a
, (a*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - sqrt(c)*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2
- c)) + 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))/a]

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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 )}{a x - 1}\, 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)

[Out]

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

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Giac [A]  time = 1.15162, size = 207, normalized size = 1.78 \begin{align*}{\left (\frac{2 \, \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^{2}} - \frac{2 \, \sqrt{c} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} - c} \right |}\right ) \mathrm{sgn}\left (x\right )}{a{\left | a \right |}} + \frac{\sqrt{a^{2} c x^{2} - c} \mathrm{sgn}\left (x\right )}{a^{2}} - \frac{{\left (2 \, \sqrt{c}{\left | a \right |} \arctan \left (\frac{\sqrt{-c}}{\sqrt{c}}\right ) - a \sqrt{c} \log \left ({\left | c \right |}\right ) + \sqrt{-c}{\left | a \right |}\right )} \mathrm{sgn}\left (x\right )}{a^{2}{\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, algorithm="giac")

[Out]

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