### 3.172 $$\int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx$$

Optimal. Leaf size=14 $-\frac{1}{2} a c x^2-c x$

[Out]

-(c*x) - (a*c*x^2)/2

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Rubi [C]  time = 0.0120579, antiderivative size = 26, normalized size of antiderivative = 1.86, number of steps used = 1, number of rules used = 1, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.062, Rules used = {2288} $\frac{c \left (1-a^2 x^2\right ) e^{2 \coth ^{-1}(a x)}}{2 a}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(2*ArcCoth[a*x])*(c - a*c*x),x]

[Out]

(c*E^(2*ArcCoth[a*x])*(1 - a^2*x^2))/(2*a)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

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

Mathematica [C]  time = 0.0080872, size = 26, normalized size = 1.86 $\frac{c \left (1-a^2 x^2\right ) e^{2 \coth ^{-1}(a x)}}{2 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c*E^(2*ArcCoth[a*x])*(1 - a^2*x^2))/(2*a)

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Maple [A]  time = 0.039, size = 13, normalized size = 0.9 \begin{align*} c \left ( -{\frac{a{x}^{2}}{2}}-x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03664, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2} \, a c x^{2} - c x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*c*x^2 - c*x

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Fricas [A]  time = 1.43824, size = 27, normalized size = 1.93 \begin{align*} -\frac{1}{2} \, a c x^{2} - c x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*c*x^2 - c*x

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Sympy [A]  time = 0.071802, size = 12, normalized size = 0.86 \begin{align*} - \frac{a c x^{2}}{2} - c x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*c*x**2/2 - c*x

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Giac [A]  time = 1.15182, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2} \, a c x^{2} - c x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*c*x^2 - c*x