3.3.2 \(\int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx\) [202]

Optimal. Leaf size=28 \[ -\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{c^2 \left (a-\frac {1}{x}\right )} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 = {6310, 6313, 665} \begin {gather*} -\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{c^2 \left (a-\frac {1}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcCoth[a*x]*(c - a*c*x)^2),x]

[Out]

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

Rule 665

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^m*((a + c*x^2)^(p + 1)/
(2*c*d*(p + 1))), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rule 6310

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[u*x^p*(1 + c/(d*
x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !IntegerQ[n/2] && Inte
gerQ[p]

Rule 6313

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[-c^n, Subst[Int[(c +
d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 27, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2 (-1+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^2),x]

[Out]

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

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Maple [A]
time = 0.16, size = 36, normalized size = 1.29

method	result	size
gosper	\(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{\left (a x -1\right ) c^{2} a}\)	\(36\)
default	\(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{\left (a x -1\right ) c^{2} a}\)	\(36\)
trager	\(-\frac {\left (a x +1\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{a \,c^{2} \left (a x -1\right )}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 23, normalized size = 0.82 \begin {gather*} -\frac {1}{a c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a*c^2*sqrt((a*x - 1)/(a*x + 1)))

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Fricas [A]
time = 0.35, size = 39, normalized size = 1.39 \begin {gather*} -\frac {{\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2} x - a c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef

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Mupad [B]
time = 0.03, size = 23, normalized size = 0.82 \begin {gather*} -\frac {1}{a\,c^2\,\sqrt {\frac {a\,x-1}{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 - a*c*x)^2,x)

[Out]

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

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