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

Optimal. Leaf size=29 \[ \frac {2 (a x+1) e^{-\coth ^{-1}(a x)}}{a \sqrt {c-a c x}} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6174} \[ \frac {2 (a x+1) e^{-\coth ^{-1}(a x)}}{a \sqrt {c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6174

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 28, normalized size = 0.97 \[ \frac {2 x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-a c x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.72, size = 44, normalized size = 1.52 \[ -\frac {2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(a*x+1)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.04, size = 35, normalized size = 1.21 \[ \frac {2 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-a c x +c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.33, size = 29, normalized size = 1.00 \[ -\frac {2 \, {\left (a \sqrt {-c} x + \sqrt {-c}\right )}}{\sqrt {a x + 1} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.25, size = 34, normalized size = 1.17 \[ \frac {\left (2\,x+\frac {2}{a}\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{\sqrt {c-a\,c\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\sqrt {- c \left (a x - 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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