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

Optimal. Leaf size=30 \[ \frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6127, 649} \[ \frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 649

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

Rule 6127

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 30, normalized size = 1.00 \[ \frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.59, size = 36, normalized size = 1.20 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{a^{2} c x - a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(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(1/(a*x+1)*(-a^2*x^2+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:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.03, size = 27, normalized size = 0.90 \[ \frac {2 \sqrt {-a^{2} x^{2}+1}}{a \sqrt {-a c x +c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.37, size = 15, normalized size = 0.50 \[ \frac {2 \, \sqrt {a x + 1}}{a \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a*x + 1)/(a*sqrt(c))

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mupad [B]  time = 0.96, size = 26, normalized size = 0.87 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{a\,\sqrt {c-a\,c\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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