3.414 \(\int e^{-\coth ^{-1}(a x)} (c-\frac{c}{a x})^3 \, dx\)

Optimal. Leaf size=106 \[ c^3 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{4 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}-\frac{4 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}-\frac{13 c^3 \csc ^{-1}(a x)}{2 a} \]

[Out]

(-4*c^3*Sqrt[1 - 1/(a^2*x^2)])/a + (c^3*Sqrt[1 - 1/(a^2*x^2)])/(2*a^2*x) + c^3*Sqrt[1 - 1/(a^2*x^2)]*x - (13*c
^3*ArcCsc[a*x])/(2*a) - (4*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a

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Rubi [A]  time = 0.3312, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ c^3 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{4 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}-\frac{4 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}-\frac{13 c^3 \csc ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - c/(a*x))^3/E^ArcCoth[a*x],x]

[Out]

(-4*c^3*Sqrt[1 - 1/(a^2*x^2)])/a + (c^3*Sqrt[1 - 1/(a^2*x^2)])/(2*a^2*x) + c^3*Sqrt[1 - 1/(a^2*x^2)]*x - (13*c
^3*ArcCsc[a*x])/(2*a) - (4*c^3*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 )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^4}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=c^3 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{\operatorname{Subst}\left (\int \frac{\frac{4 c^4}{a}-\frac{6 c^4 x}{a^2}+\frac{4 c^4 x^2}{a^3}-\frac{c^4 x^3}{a^4}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{a^2 \operatorname{Subst}\left (\int \frac{-\frac{8 c^4}{a^3}+\frac{13 c^4 x}{a^4}-\frac{8 c^4 x^2}{a^5}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=-\frac{4 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{a^4 \operatorname{Subst}\left (\int \frac{\frac{8 c^4}{a^5}-\frac{13 c^4 x}{a^6}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=-\frac{4 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{\left (13 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}+\frac{\left (4 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{4 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{13 c^3 \csc ^{-1}(a x)}{2 a}+\frac{\left (2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a}\\ &=-\frac{4 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{13 c^3 \csc ^{-1}(a x)}{2 a}-\left (4 a c^3\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{4 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{13 c^3 \csc ^{-1}(a x)}{2 a}-\frac{4 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.121667, size = 167, normalized size = 1.58 \[ \frac{c^3 \left (2 a^4 x^4-8 a^3 x^3-a^2 x^2+10 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-8 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{1}{a x}\right )-8 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+8 a x-1\right )}{2 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(-1 + 8*a*x - a^2*x^2 - 8*a^3*x^3 + 2*a^4*x^4 + 10*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcSin[Sqrt[1 - 1/(a*x)]
/Sqrt[2]] - 8*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcSin[1/(a*x)] - 8*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcTanh[Sqrt[1 -
1/(a^2*x^2)]]))/(2*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3)

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Maple [B]  time = 0.135, size = 266, normalized size = 2.5 \begin{align*}{\frac{ \left ( ax+1 \right ){c}^{3}}{2\,{x}^{2}{a}^{3}}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -8\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+8\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-13\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+8\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-13\,{a}^{2}{x}^{2}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +16\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}-16\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}- \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{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)^3*((a*x-1)/(a*x+1))^(1/2),x)

[Out]

1/2*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*c^3*(-8*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^3*a^3+8*(a^2)^(1/2)*(a^2*x^2-1)^(3
/2)*x*a-13*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^2*a^2+8*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a
^3-13*a^2*x^2*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))+16*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^2*a^2-16*ln((a^
2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^2*a^3-(a^2*x^2-1)^(3/2)*(a^2)^(1/2))/((a*x-1)*(a*x+1))
^(1/2)/a^3/x^2/(a^2)^(1/2)

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Maxima [B]  time = 1.57952, size = 271, normalized size = 2.56 \begin{align*}{\left (\frac{13 \, c^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} - \frac{4 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac{4 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac{11 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 2 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 5 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac{{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + a^{2}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^3*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")

[Out]

(13*c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 - 4*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 + 4*c^3*log(sqrt(
(a*x - 1)/(a*x + 1)) - 1)/a^2 + (11*c^3*((a*x - 1)/(a*x + 1))^(5/2) + 2*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 5*c^
3*sqrt((a*x - 1)/(a*x + 1)))/((a*x - 1)*a^2/(a*x + 1) - (a*x - 1)^2*a^2/(a*x + 1)^2 - (a*x - 1)^3*a^2/(a*x + 1
)^3 + a^2))*a

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Fricas [A]  time = 2.02462, size = 332, normalized size = 3.13 \begin{align*} \frac{26 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - 8 \, a^{2} c^{3} x^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) + 8 \, a^{2} c^{3} x^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (2 \, a^{3} c^{3} x^{3} - 6 \, a^{2} c^{3} x^{2} - 7 \, a c^{3} x + c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^3*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")

[Out]

1/2*(26*a^2*c^3*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) - 8*a^2*c^3*x^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) + 8*a
^2*c^3*x^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (2*a^3*c^3*x^3 - 6*a^2*c^3*x^2 - 7*a*c^3*x + c^3)*sqrt((a*x -
1)/(a*x + 1)))/(a^3*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{c^{3} \left (\int a^{3} \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^{3}}\, dx + \int \frac{3 a \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{x^{2}}\, dx + \int - \frac{3 a^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{x}\, dx\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.23225, size = 313, normalized size = 2.95 \begin{align*} \frac{13 \, c^{3} \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right )}{a} + \frac{4 \, c^{3} \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^{3} \mathrm{sgn}\left (a x + 1\right )}{a} - \frac{{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{3} c^{3}{\left | a \right |} \mathrm{sgn}\left (a x + 1\right ) + 8 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} a c^{3} \mathrm{sgn}\left (a x + 1\right ) -{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )} c^{3}{\left | a \right |} \mathrm{sgn}\left (a x + 1\right ) + 8 \, a c^{3} \mathrm{sgn}\left (a x + 1\right )}{{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{2} a{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^3*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")

[Out]

13*c^3*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1)/a + 4*c^3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sg
n(a*x + 1)/abs(a) + sqrt(a^2*x^2 - 1)*c^3*sgn(a*x + 1)/a - ((x*abs(a) - sqrt(a^2*x^2 - 1))^3*c^3*abs(a)*sgn(a*
x + 1) + 8*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a*c^3*sgn(a*x + 1) - (x*abs(a) - sqrt(a^2*x^2 - 1))*c^3*abs(a)*sgn
(a*x + 1) + 8*a*c^3*sgn(a*x + 1))/(((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^2*a*abs(a))