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

Optimal. Leaf size=33 \[ \frac{1}{2} c x \sqrt{1-a^2 x^2}+\frac{c \sin ^{-1}(a x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0191962, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6127, 195, 216} \[ \frac{1}{2} c x \sqrt{1-a^2 x^2}+\frac{c \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*(c - a*c*x),x]

[Out]

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

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]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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]

Rubi steps

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

Mathematica [A]  time = 0.0120922, size = 30, normalized size = 0.91 \[ \frac{c \left (a x \sqrt{1-a^2 x^2}+\sin ^{-1}(a x)\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.032, size = 46, normalized size = 1.4 \begin{align*}{\frac{cx}{2}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{c}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.43426, size = 49, normalized size = 1.48 \begin{align*} \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} c x + \frac{c \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.62335, size = 107, normalized size = 3.24 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} a c x - 2 \, c \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.66111, size = 46, normalized size = 1.39 \begin{align*} \begin{cases} - \frac{c \left (\begin{cases} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (a x \right )}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) - c \operatorname{asin}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\c x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(c*Piecewise((-a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -1) & (a*x < 1))) - c*asin(a*x))/a
, Ne(a, 0)), (c*x, True))

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Giac [A]  time = 1.19223, size = 41, normalized size = 1.24 \begin{align*} \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} c x + \frac{c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \,{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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