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

Optimal. Leaf size=63 \[ \frac{c (1-a x) \sqrt{1-a^2 x^2}}{2 a}+\frac{3 c \sqrt{1-a^2 x^2}}{2 a}+\frac{3 c \sin ^{-1}(a x)}{2 a} \]

[Out]

(3*c*Sqrt[1 - a^2*x^2])/(2*a) + (c*(1 - a*x)*Sqrt[1 - a^2*x^2])/(2*a) + (3*c*ArcSin[a*x])/(2*a)

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Rubi [A]  time = 0.0428937, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6127, 671, 641, 216} \[ \frac{c (1-a x) \sqrt{1-a^2 x^2}}{2 a}+\frac{3 c \sqrt{1-a^2 x^2}}{2 a}+\frac{3 c \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)/E^ArcTanh[a*x],x]

[Out]

(3*c*Sqrt[1 - a^2*x^2])/(2*a) + (c*(1 - a*x)*Sqrt[1 - a^2*x^2])/(2*a) + (3*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 671

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*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rule 641

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

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 &=\frac{\int \frac{(c-a c x)^2}{\sqrt{1-a^2 x^2}} \, dx}{c}\\ &=\frac{c (1-a x) \sqrt{1-a^2 x^2}}{2 a}+\frac{3}{2} \int \frac{c-a c x}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3 c \sqrt{1-a^2 x^2}}{2 a}+\frac{c (1-a x) \sqrt{1-a^2 x^2}}{2 a}+\frac{1}{2} (3 c) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3 c \sqrt{1-a^2 x^2}}{2 a}+\frac{c (1-a x) \sqrt{1-a^2 x^2}}{2 a}+\frac{3 c \sin ^{-1}(a x)}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0494209, size = 61, normalized size = 0.97 \[ \frac{c \left (\frac{\sqrt{a x+1} \left (a^2 x^2-5 a x+4\right )}{\sqrt{1-a x}}-6 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{2 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*((Sqrt[1 + a*x]*(4 - 5*a*x + a^2*x^2))/Sqrt[1 - a*x] - 6*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(2*a)

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Maple [B]  time = 0.036, size = 114, normalized size = 1.8 \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}}}}}+2\,{\frac{c\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}{a}}+2\,{\frac{c}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)/(a*x+1)*(-a^2*x^2+1)^(1/2),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))+2*c/a*(-a^2*(x+1/a)^2+2
*a*(x+1/a))^(1/2)+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 = 1.43728, size = 61, normalized size = 0.97 \begin{align*} -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} c x + \frac{3 \, c \arcsin \left (a x\right )}{2 \, a} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} c}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-a^2*x^2 + 1)*c*x + 3/2*c*arcsin(a*x)/a + 2*sqrt(-a^2*x^2 + 1)*c/a

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Fricas [A]  time = 1.65818, size = 119, normalized size = 1.89 \begin{align*} -\frac{6 \, c \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt{-a^{2} x^{2} + 1}{\left (a c x - 4 \, c\right )}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(6*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*c*x - 4*c))/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - c \left (\int - \frac{\sqrt{- a^{2} x^{2} + 1}}{a x + 1}\, dx + \int \frac{a x \sqrt{- a^{2} x^{2} + 1}}{a x + 1}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20465, size = 51, normalized size = 0.81 \begin{align*} \frac{3 \, c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \,{\left | a \right |}} - \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1}{\left (c x - \frac{4 \, c}{a}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*c*arcsin(a*x)*sgn(a)/abs(a) - 1/2*sqrt(-a^2*x^2 + 1)*(c*x - 4*c/a)