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

Optimal result

Integrand size = 16, antiderivative size = 14 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx=-c x-\frac {1}{2} a c x^2 \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2326} \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx=\frac {c \left (1-a^2 x^2\right ) e^{2 \coth ^{-1}(a x)}}{2 a} \]

[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*} \text {integral}& = \frac {c e^{2 \coth ^{-1}(a x)} \left (1-a^2 x^2\right )}{2 a} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx=\frac {c e^{2 \coth ^{-1}(a x)} \left (1-a^2 x^2\right )}{2 a} \]

[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)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71

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	\(-c x -\frac {1}{2} a c \,x^{2}\)	\(13\)
parallelrisch	\(-c x -\frac {1}{2} a c \,x^{2}\)	\(13\)
meijerg	\(-\frac {c \left (\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )\right )}{a}+\frac {c \ln \left (-a x +1\right )}{a}\)	\(38\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {1}{2} \, a c x^{2} - c x \]

[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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx=- \frac {a c x^{2}}{2} - c x \]

[In]

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

[Out]

-a*c*x**2/2 - c*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {1}{2} \, a c x^{2} - c x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {1}{2} \, a c x^{2} - c x \]

[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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {c\,x\,\left (a\,x+2\right )}{2} \]

[In]

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

[Out]

-(c*x*(a*x + 2))/2