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

Optimal. Leaf size=26 \[ \frac{1}{2} a c x^2+\frac{4 c \log (a x+1)}{a}-3 c x \]

[Out]

-3*c*x + (a*c*x^2)/2 + (4*c*Log[1 + a*x])/a

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Rubi [A]  time = 0.0230231, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6129, 43} \[ \frac{1}{2} a c x^2+\frac{4 c \log (a x+1)}{a}-3 c x \]

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)/E^(2*ArcTanh[a*x]),x]

[Out]

-3*c*x + (a*c*x^2)/2 + (4*c*Log[1 + a*x])/a

Rule 6129

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} (c-a c x) \, dx &=c \int \frac{(1-a x)^2}{1+a x} \, dx\\ &=c \int \left (-3+a x+\frac{4}{1+a x}\right ) \, dx\\ &=-3 c x+\frac{1}{2} a c x^2+\frac{4 c \log (1+a x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0134928, size = 25, normalized size = 0.96 \[ c \left (\frac{a x^2}{2}+\frac{4 \log (a x+1)}{a}-3 x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c*(-3*x + (a*x^2)/2 + (4*Log[1 + a*x])/a)

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Maple [A]  time = 0.036, size = 25, normalized size = 1. \begin{align*} -3\,cx+{\frac{ac{x}^{2}}{2}}+4\,{\frac{c\ln \left ( ax+1 \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*c*x+1/2*a*c*x^2+4*c*ln(a*x+1)/a

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Maxima [A]  time = 0.94396, size = 32, normalized size = 1.23 \begin{align*} \frac{1}{2} \, a c x^{2} - 3 \, c x + \frac{4 \, c \log \left (a x + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*c*x^2 - 3*c*x + 4*c*log(a*x + 1)/a

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Fricas [A]  time = 1.60623, size = 65, normalized size = 2.5 \begin{align*} \frac{a^{2} c x^{2} - 6 \, a c x + 8 \, c \log \left (a x + 1\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^2*c*x^2 - 6*a*c*x + 8*c*log(a*x + 1))/a

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Sympy [A]  time = 0.286534, size = 24, normalized size = 0.92 \begin{align*} \frac{a c x^{2}}{2} - 3 c x + \frac{4 c \log{\left (a x + 1 \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x**2/2 - 3*c*x + 4*c*log(a*x + 1)/a

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Giac [B]  time = 1.23473, size = 68, normalized size = 2.62 \begin{align*} \frac{{\left (a x + 1\right )}^{2}{\left (c - \frac{8 \, c}{a x + 1}\right )}}{2 \, a} - \frac{4 \, c \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*x + 1)^2*(c - 8*c/(a*x + 1))/a - 4*c*log(abs(a*x + 1)/((a*x + 1)^2*abs(a)))/a