### 3.397 $$\int e^{3 \coth ^{-1}(a x)} (c-\frac{c}{a x})^3 \, dx$$

Optimal. Leaf size=61 $c^3 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{3 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+\frac{3 c^3 \csc ^{-1}(a x)}{2 a}$

[Out]

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

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Rubi [A]  time = 0.0557759, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.182, Rules used = {6177, 277, 195, 216} $c^3 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{3 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+\frac{3 c^3 \csc ^{-1}(a x)}{2 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*c^3*Sqrt[1 - 1/(a^2*x^2)])/(2*a^2*x) + c^3*(1 - 1/(a^2*x^2))^(3/2)*x + (3*c^3*ArcCsc[a*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 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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]

Rubi steps

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

Mathematica [A]  time = 0.0697137, size = 51, normalized size = 0.84 $\frac{c^3 \left (\sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2+1\right )+3 a x \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{2 a^2 x}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.164, size = 105, normalized size = 1.7 \begin{align*} -{\frac{{c}^{3} \left ( ax-1 \right ) ^{2}}{ \left ( 2\,ax+2 \right ){a}^{3}{x}^{2}} \left ( -3\,{a}^{2}{x}^{2}\sqrt{{a}^{2}{x}^{2}-1}-3\,{a}^{2}{x}^{2}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) + \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{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(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^3,x)

[Out]

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

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Maxima [B]  time = 1.52451, size = 204, normalized size = 3.34 \begin{align*} -{\left (\frac{3 \, c^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} - \frac{3 \, 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}} + 3 \, 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(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^3,x, algorithm="maxima")

[Out]

-(3*c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 - (3*c^3*((a*x - 1)/(a*x + 1))^(5/2) + 2*c^3*((a*x - 1)/(a*x + 1
))^(3/2) + 3*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 = 1.6236, size = 192, normalized size = 3.15 \begin{align*} -\frac{6 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (2 \, a^{3} c^{3} x^{3} + 2 \, a^{2} c^{3} x^{2} + 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(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^3,x, algorithm="fricas")

[Out]

-1/2*(6*a^2*c^3*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) - (2*a^3*c^3*x^3 + 2*a^2*c^3*x^2 + 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 \frac{3 a}{\frac{a x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int - \frac{3 a^{2}}{\frac{a x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int \frac{a^{3}}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int - \frac{1}{\frac{a x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx\right )}{a^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(3*a/(a*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)
)/(a*x + 1)), x) + Integral(-3*a**2/(a*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x*sqrt(a*x/(a*x + 1)
- 1/(a*x + 1))/(a*x + 1)), x) + Integral(a**3/(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - sqrt(a*x/(a*
x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(-1/(a*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**3*
sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x))/a**3

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Giac [B]  time = 1.20123, size = 232, normalized size = 3.8 \begin{align*} -\frac{1}{4} \,{\left (\frac{3 \,{\left (\pi + 2 \, \arctan \left (\frac{\frac{a x - 1}{a x + 1} - 1}{2 \, \sqrt{\frac{a x - 1}{a x + 1}}}\right )\right )} c^{3}}{a^{2}} + \frac{4 \,{\left (3 \, c^{3}{\left (\sqrt{\frac{a x - 1}{a x + 1}} - \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )}^{2} + 8 \, c^{3}\right )}}{{\left ({\left (\sqrt{\frac{a x - 1}{a x + 1}} - \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )}^{3} + 4 \, \sqrt{\frac{a x - 1}{a x + 1}} - \frac{4}{\sqrt{\frac{a x - 1}{a x + 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))^(3/2)*(c-c/a/x)^3,x, algorithm="giac")

[Out]

-1/4*(3*(pi + 2*arctan(1/2*((a*x - 1)/(a*x + 1) - 1)/sqrt((a*x - 1)/(a*x + 1))))*c^3/a^2 + 4*(3*c^3*(sqrt((a*x
- 1)/(a*x + 1)) - 1/sqrt((a*x - 1)/(a*x + 1)))^2 + 8*c^3)/(((sqrt((a*x - 1)/(a*x + 1)) - 1/sqrt((a*x - 1)/(a*
x + 1)))^3 + 4*sqrt((a*x - 1)/(a*x + 1)) - 4/sqrt((a*x - 1)/(a*x + 1)))*a^2))*a