3.3.57 \(\int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx\) [257]

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

[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 = 0.02, 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 = {6309} \begin {gather*} \frac {2 (a x+1) e^{-\coth ^{-1}(a x)}}{a \sqrt {c-a c x}} \end {gather*}

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 6309

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 = 0.02, size = 28, normalized size = 0.97 \begin {gather*} \frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-a c x}} \end {gather*}

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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Maple [A]
time = 0.08, size = 46, normalized size = 1.59

method	result	size
gosper	\(\frac {2 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-a c x +c}}\)	\(35\)
risch	\(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{\sqrt {-c \left (a x -1\right )}\, a}\)	\(36\)
default	\(-\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}}{\left (a x -1\right ) c a}\)	\(46\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 29, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (a \sqrt {-c} x + \sqrt {-c}\right )}}{\sqrt {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)/(-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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Fricas [A]
time = 0.34, size = 44, normalized size = 1.52 \begin {gather*} -\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} \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)^(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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\sqrt {- c \left (a x - 1\right )}}\, dx \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)**(1/2),x)

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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