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

Optimal. Leaf size=27 \[ c x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{c \csc ^{-1}(a x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.0290766, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6177, 277, 216} \[ c x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{c \csc ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]*(c - c/(a*x)),x]

[Out]

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

Mathematica [A]  time = 0.0367674, size = 31, normalized size = 1.15 \[ \frac{c \left (a x \sqrt{1-\frac{1}{a^2 x^2}}+\sin ^{-1}\left (\frac{1}{a x}\right )\right )}{a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.116, size = 63, normalized size = 2.3 \begin{align*}{\frac{c \left ( ax-1 \right ) }{a} \left ( \sqrt{{a}^{2}{x}^{2}-1}+\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\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))^(1/2)*(c-c/a/x),x)

[Out]

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

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Maxima [B]  time = 1.57694, size = 89, normalized size = 3.3 \begin{align*} -2 \, a{\left (\frac{c \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} + \frac{c \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}}\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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.58948, size = 113, normalized size = 4.19 \begin{align*} -\frac{2 \, c \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (a c x + c\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{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),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [B]  time = 1.28138, size = 117, normalized size = 4.33 \begin{align*} -\frac{1}{2} \, a{\left (\frac{{\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}{a^{2}} + \frac{4 \, c}{a^{2}{\left (\sqrt{\frac{a x - 1}{a x + 1}} - \frac{1}{\sqrt{\frac{a x - 1}{a x + 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),x, algorithm="giac")

[Out]

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