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

Optimal. Leaf size=13 \[ \frac{1}{2} a c x^2+c x \]

[Out]

c*x + (a*c*x^2)/2

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Rubi [C]  time = 0.011943, antiderivative size = 26, normalized size of antiderivative = 2., 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} \[ \frac{c \left (1-a^2 x^2\right ) e^{2 \tanh ^{-1}(a x)}}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*(c - a*c*x),x]

[Out]

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

Mathematica [C]  time = 0.0077832, size = 26, normalized size = 2. \[ \frac{c \left (1-a^2 x^2\right ) e^{2 \tanh ^{-1}(a x)}}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.027, size = 11, 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)^2/(-a^2*x^2+1)*(-a*c*x+c),x)

[Out]

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

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Maxima [A]  time = 0.943945, size = 15, normalized size = 1.15 \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((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c),x, algorithm="maxima")

[Out]

1/2*a*c*x^2 + c*x

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Fricas [A]  time = 1.51341, size = 26, normalized size = 2. \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((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c),x, algorithm="fricas")

[Out]

1/2*a*c*x^2 + c*x

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Sympy [A]  time = 0.078766, size = 10, normalized size = 0.77 \begin{align*} \frac{a c x^{2}}{2} + c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x**2/2 + c*x

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Giac [A]  time = 1.17431, size = 15, normalized size = 1.15 \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((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c),x, algorithm="giac")

[Out]

1/2*a*c*x^2 + c*x