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

Optimal. Leaf size=11 \[ \frac {\sin ^{-1}(a x)}{a c} \]

[Out]

arcsin(a*x)/a/c

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Rubi [A]  time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6127, 216} \[ \frac {\sin ^{-1}(a x)}{a c} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcTanh[a*x]*(c - a*c*x)),x]

[Out]

ArcSin[a*x]/(a*c)

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

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

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Mathematica [A]  time = 0.01, size = 11, normalized size = 1.00 \[ \frac {\sin ^{-1}(a x)}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[a*x]/(a*c)

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fricas [B]  time = 1.32, size = 30, normalized size = 2.73 \[ -\frac {2 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{a c} \]

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),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.22, size = 14, normalized size = 1.27 \[ \frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c {\left | a \right |}} \]

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),x, algorithm="giac")

[Out]

arcsin(a*x)*sgn(a)/(c*abs(a))

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maple [B]  time = 0.04, size = 154, normalized size = 14.00 \[ -\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{2 c a}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{2 c \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{2 c a}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 c \sqrt {a^{2}}} \]

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),x)

[Out]

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

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maxima [A]  time = 0.43, size = 11, normalized size = 1.00 \[ \frac {\arcsin \left (a x\right )}{a c} \]

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),x, algorithm="maxima")

[Out]

arcsin(a*x)/(a*c)

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mupad [B]  time = 0.04, size = 21, normalized size = 1.91 \[ \frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 4.56, size = 44, normalized size = 4.00 \[ \frac {\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}}{c} \]

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),x)

[Out]

Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0))/c

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