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]

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

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Rubi [A]  time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 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 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 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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.48, size = 47, normalized size = 1.62 \[ \frac {2 \, a c x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} c}{x} \]

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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giac [B]  time = 0.20, size = 74, normalized size = 2.55 \[ \frac {a^{4} c x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {a^{2} c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c}{2 \, x {\left | a \right |}} \]

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))

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maple [A]  time = 0.04, size = 51, normalized size = 1.76 \[ -\frac {c \,a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {c \sqrt {-a^{2} x^{2}+1}}{x} \]

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 = 0.47, size = 27, normalized size = 0.93 \[ -a c \arcsin \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} c}{x} \]

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

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mupad [B]  time = 0.05, size = 38, normalized size = 1.31 \[ c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}-\frac {c\,\sqrt {1-a^2\,x^2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.43, size = 88, normalized size = 3.03 \[ c \left (\begin {cases} - \frac {i a^{2} x}{\sqrt {a^{2} x^{2} - 1}} + i a \operatorname {acosh}{\left (a x \right )} + \frac {i}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} x}{\sqrt {- a^{2} x^{2} + 1}} - a \operatorname {asin}{\left (a x \right )} - \frac {1}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right ) \]

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]

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

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