3.951 \(\int e^{\tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx\)

Optimal. Leaf size=69 \[ \frac {a x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}+\frac {x \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6143, 6140} \[ \frac {a x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}+\frac {x \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*Sqrt[c - a^2*c*x^2],x]

[Out]

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

Rule 6140

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

Rule 6143

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 40, normalized size = 0.58 \[ \frac {\left (\frac {a x^2}{2}+x\right ) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]*Sqrt[c - a^2*c*x^2],x]

[Out]

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

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fricas [A]  time = 0.89, size = 47, normalized size = 0.68 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (a x^{2} + 2 \, x\right )}}{2 \, {\left (a^{2} x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 34, normalized size = 0.49 \[ \frac {x \left (a x +2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{2 \sqrt {-a^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 0.95, size = 34, normalized size = 0.49 \[ \frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {a\,x^2}{2}+x\right )}{\sqrt {1-a^2\,x^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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