### 3.415 $$\int e^{-\coth ^{-1}(a x)} (c-\frac{c}{a x})^2 \, dx$$

Optimal. Leaf size=77 $c^2 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}-\frac{3 c^2 \csc ^{-1}(a x)}{a}$

[Out]

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

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Rubi [A]  time = 0.238034, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.364, Rules used = {6177, 1807, 1809, 844, 216, 266, 63, 208} $c^2 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}-\frac{3 c^2 \csc ^{-1}(a x)}{a}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c - c/(a*x))^2/E^ArcCoth[a*x],x]

[Out]

-((c^2*Sqrt[1 - 1/(a^2*x^2)])/a) + c^2*Sqrt[1 - 1/(a^2*x^2)]*x - (3*c^2*ArcCsc[a*x])/a - (3*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 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

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

Mathematica [A]  time = 0.197717, size = 55, normalized size = 0.71 $\frac{c^2 \left (\sqrt{1-\frac{1}{a^2 x^2}} (a x-1)-3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-3 \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{a}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.131, size = 227, normalized size = 3. \begin{align*}{\frac{ \left ( ax+1 \right ){c}^{2}}{{a}^{2}x}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+ \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}}-3\,\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}-3\,ax\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +4\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-4\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2} \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*c^2*(-(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^2*a^2+(a^2*x^2-1)^(3/2)*(a^2)^(1/2)-3*(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^2-3*a*x*(a^2)^(1/2)*
arctan(1/(a^2*x^2-1)^(1/2))+4*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a-4*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1)
)^(1/2))/(a^2)^(1/2))*x*a^2)/((a*x-1)*(a*x+1))^(1/2)/a^2/x/(a^2)^(1/2)

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Maxima [A]  time = 1.56537, size = 170, normalized size = 2.21 \begin{align*} -{\left (\frac{4 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{\frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} - \frac{6 \, c^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{3 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{3 \, 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((c-c/a/x)^2*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")

[Out]

-(4*c^2*((a*x - 1)/(a*x + 1))^(3/2)/((a*x - 1)^2*a^2/(a*x + 1)^2 - a^2) - 6*c^2*arctan(sqrt((a*x - 1)/(a*x + 1
)))/a^2 + 3*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 3*c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2)*a

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Fricas [A]  time = 1.91077, size = 266, normalized size = 3.45 \begin{align*} \frac{6 \, a c^{2} x \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - 3 \, a c^{2} x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) + 3 \, a c^{2} x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (a^{2} c^{2} x^{2} - 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((c-c/a/x)^2*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")

[Out]

(6*a*c^2*x*arctan(sqrt((a*x - 1)/(a*x + 1))) - 3*a*c^2*x*log(sqrt((a*x - 1)/(a*x + 1)) + 1) + 3*a*c^2*x*log(sq
rt((a*x - 1)/(a*x + 1)) - 1) + (a^2*c^2*x^2 - 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 a^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{x^{2}}\, dx + \int - \frac{2 a \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{x}\, dx\right )}{a^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20112, size = 176, normalized size = 2.29 \begin{align*} \frac{6 \, c^{2} \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right )}{a} + \frac{3 \, c^{2} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac{\sqrt{a^{2} x^{2} - 1} c^{2} \mathrm{sgn}\left (a x + 1\right )}{a} - \frac{2 \, c^{2} \mathrm{sgn}\left (a x + 1\right )}{{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1\right )}{\left | a \right |}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*c^2*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1)/a + 3*c^2*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn
(a*x + 1)/abs(a) + sqrt(a^2*x^2 - 1)*c^2*sgn(a*x + 1)/a - 2*c^2*sgn(a*x + 1)/(((x*abs(a) - sqrt(a^2*x^2 - 1))^
2 + 1)*abs(a))