### 3.381 $$\int e^{\coth ^{-1}(a x)} (c-\frac{c}{a x})^2 \, dx$$

Optimal. Leaf size=62 $\frac{c^2 x \sqrt{1-\frac{1}{a^2 x^2}} \left (a+\frac{1}{x}\right )}{a}-\frac{c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{c^2 \csc ^{-1}(a x)}{a}$

[Out]

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

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Rubi [A]  time = 0.0960085, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.35, Rules used = {6177, 813, 844, 216, 266, 63, 208} $\frac{c^2 x \sqrt{1-\frac{1}{a^2 x^2}} \left (a+\frac{1}{x}\right )}{a}-\frac{c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{c^2 \csc ^{-1}(a x)}{a}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[a*x]*(c - c/(a*x))^2,x]

[Out]

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

Rule 6177

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

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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]

Rubi steps

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

Mathematica [B]  time = 0.162609, size = 158, normalized size = 2.55 $\frac{c^2 \left (2 a^3 x^3+2 a^2 x^2-2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )+a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{1}{a x}\right )-2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-2 a x-2\right )}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.126, size = 166, normalized size = 2.7 \begin{align*}{\frac{{c}^{2} \left ( ax-1 \right ) }{{a}^{2}x} \left ( \sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+ax\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) - \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}}+\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) x{a}^{2} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \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)^2,x)

[Out]

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

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Maxima [B]  time = 1.52721, size = 169, normalized size = 2.73 \begin{align*} -{\left (\frac{4 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac{2 \, c^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} a \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)^2,x, algorithm="maxima")

[Out]

-(4*c^2*sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)^2*a^2/(a*x + 1)^2 - a^2) + 2*c^2*arctan(sqrt((a*x - 1)/(a*x + 1))
)/a^2 + c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2)*a

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Fricas [B]  time = 1.63475, size = 278, normalized size = 4.48 \begin{align*} -\frac{2 \, a c^{2} x \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + a c^{2} x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - a c^{2} x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (a^{2} c^{2} x^{2} + 2 \, a c^{2} x + c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} 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)^2,x, algorithm="fricas")

[Out]

-(2*a*c^2*x*arctan(sqrt((a*x - 1)/(a*x + 1))) + a*c^2*x*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - a*c^2*x*log(sqrt(
(a*x - 1)/(a*x + 1)) - 1) - (a^2*c^2*x^2 + 2*a*c^2*x + c^2)*sqrt((a*x - 1)/(a*x + 1)))/(a^2*x)

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

[Out]

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

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Giac [B]  time = 1.20774, size = 165, normalized size = 2.66 \begin{align*} -a{\left (\frac{2 \, c^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{c^{2} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac{4 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 1\right )}}\right )} \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)^2,x, algorithm="giac")

[Out]

-a*(2*c^2*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - c^2*log(abs(sqr
t((a*x - 1)/(a*x + 1)) - 1))/a^2 + 4*c^2*sqrt((a*x - 1)/(a*x + 1))/(a^2*((a*x - 1)^2/(a*x + 1)^2 - 1)))