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

Optimal. Leaf size=29 \[ -\frac{c \sqrt{1-a^2 x^2}}{x}-a c \sin ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.0429045, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {6128, 277, 216} \[ -\frac{c \sqrt{1-a^2 x^2}}{x}-a c \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*(c - a*c*x))/x^2,x]

[Out]

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

Rule 6128

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

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

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

Mathematica [A]  time = 0.0285526, size = 28, normalized size = 0.97 \[ -\frac{c \left (\sqrt{1-a^2 x^2}+a x \sin ^{-1}(a x)\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*(c - a*c*x))/x^2,x]

[Out]

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

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Maple [A]  time = 0.039, size = 51, normalized size = 1.8 \begin{align*} -{{a}^{2}c\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{c}{x}\sqrt{-{a}^{2}{x}^{2}+1}} \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^2,x)

[Out]

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

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Maxima [A]  time = 1.44362, size = 55, normalized size = 1.9 \begin{align*} -\frac{a^{2} c \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - \frac{\sqrt{-a^{2} x^{2} + 1} c}{x} \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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.68843, size = 101, normalized size = 3.48 \begin{align*} \frac{2 \, a c x \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt{-a^{2} x^{2} + 1} c}{x} \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^2,x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 2.96254, size = 88, normalized size = 3.03 \begin{align*} - a^{2} c \left (\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}\right ) + c \left (\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\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**2,x)

[Out]

-a**2*c*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*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))

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Giac [B]  time = 1.15102, size = 100, normalized size = 3.45 \begin{align*} \frac{a^{4} c x}{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} - \frac{a^{2} c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c}{2 \, x{\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^2,x, algorithm="giac")

[Out]

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