3.5.17 \(\int \frac {e^{-\coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx\) [417]

Optimal. Leaf size=19 \[ \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6312, 270} \begin {gather*} \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcCoth[a*x]*(c - c/(a*x))),x]

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/c

Rule 270

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

Rule 6312

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]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 19, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^ArcCoth[a*x]*(c - c/(a*x))),x]

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/c

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Maple [A]
time = 0.11, size = 28, normalized size = 1.47

method	result	size
gosper	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{a c}\)	\(28\)
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{a c}\)	\(28\)
risch	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{a c}\)	\(28\)
trager	\(\frac {\left (a x +1\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{a c}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (17) = 34\).
time = 0.26, size = 44, normalized size = 2.32 \begin {gather*} -\frac {2 \, a \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2} c}{a x + 1} - a^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.33, size = 27, normalized size = 1.42 \begin {gather*} \frac {{\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {a \int \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x - 1}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.41, size = 24, normalized size = 1.26 \begin {gather*} \frac {\sqrt {a^{2} x^{2} - 1} \mathrm {sgn}\left (a x + 1\right )}{a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a^2*x^2 - 1)*sgn(a*x + 1)/(a*c)

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Mupad [B]
time = 0.05, size = 39, normalized size = 2.05 \begin {gather*} \frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c-\frac {a\,c\,\left (a\,x-1\right )}{a\,x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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