3.2.72 \(\int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx\) [172]

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

[Out]

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

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Rubi [C] Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
time = 0.01, 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 = {2326} \begin {gather*} \frac {c \left (1-a^2 x^2\right ) e^{2 \coth ^{-1}(a x)}}{2 a} \end {gather*}

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 2326

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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
time = 0.01, size = 26, normalized size = 1.86 \begin {gather*} \frac {c e^{2 \coth ^{-1}(a x)} \left (1-a^2 x^2\right )}{2 a} \end {gather*}

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 = 0.09, size = 13, normalized size = 0.93

method	result	size
gosper	\(-\frac {c x \left (a x +2\right )}{2}\)	\(10\)
default	\(c \left (-\frac {1}{2} a \,x^{2}-x \right )\)	\(13\)
norman	\(-c x -\frac {1}{2} a c \,x^{2}\)	\(13\)
risch	\(-c x -\frac {1}{2} a c \,x^{2}\)	\(13\)
meijerg	\(-\frac {c \left (\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )\right )}{a}+\frac {c \ln \left (-a x +1\right )}{a}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{2} \, a c x^{2} - c x \end {gather*}

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 = 0.35, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{2} \, a c x^{2} - c x \end {gather*}

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 = 0.01, size = 12, normalized size = 0.86 \begin {gather*} - \frac {a c x^{2}}{2} - c x \end {gather*}

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 = 0.40, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{2} \, a c x^{2} - c x \end {gather*}

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

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Mupad [B]
time = 0.02, size = 9, normalized size = 0.64 \begin {gather*} -\frac {c\,x\,\left (a\,x+2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c*x*(a*x + 2))/2

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