\(\int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx\) [171]

Optimal result

Integrand size = 18, antiderivative size = 20 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=-c^2 x+\frac {1}{3} a^2 c^2 x^3 \]

[Out]

-c^2*x+1/3*a^2*c^2*x^3

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6302, 6264, 41} \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {1}{3} a^2 c^2 x^3-c^2 x \]

[In]

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

[Out]

-(c^2*x) + (a^2*c^2*x^3)/3

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 6264

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 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} (c-a c x)^2 \, dx \\ & = -\left (c^2 \int (1-a x) (1+a x) \, dx\right ) \\ & = -\left (c^2 \int \left (1-a^2 x^2\right ) \, dx\right ) \\ & = -c^2 x+\frac {1}{3} a^2 c^2 x^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=-c^2 \left (x-\frac {a^2 x^3}{3}\right ) \]

[In]

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

[Out]

-(c^2*(x - (a^2*x^3)/3))

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80

method	result	size
gosper	\(\frac {x \left (a^{2} x^{2}-3\right ) c^{2}}{3}\)	\(16\)
default	\(c^{2} \left (\frac {1}{3} a^{2} x^{3}-x \right )\)	\(17\)
norman	\(-c^{2} x +\frac {1}{3} a^{2} c^{2} x^{3}\)	\(19\)
risch	\(-c^{2} x +\frac {1}{3} a^{2} c^{2} x^{3}\)	\(19\)
parallelrisch	\(-c^{2} x +\frac {1}{3} a^{2} c^{2} x^{3}\)	\(19\)
meijerg	\(-\frac {c^{2} \left (-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{12}-\ln \left (-a x +1\right )\right )}{a}-\frac {c^{2} \left (\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{2} \left (-a x -\ln \left (-a x +1\right )\right )}{a}+\frac {c^{2} \ln \left (-a x +1\right )}{a}\)	\(99\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {1}{3} \, a^{2} c^{2} x^{3} - c^{2} x \]

[In]

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

[Out]

1/3*a^2*c^2*x^3 - c^2*x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {a^{2} c^{2} x^{3}}{3} - c^{2} x \]

[In]

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

[Out]

a**2*c**2*x**3/3 - c**2*x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {1}{3} \, a^{2} c^{2} x^{3} - c^{2} x \]

[In]

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

[Out]

1/3*a^2*c^2*x^3 - c^2*x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {1}{3} \, a^{2} c^{2} x^{3} - c^{2} x \]

[In]

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

[Out]

1/3*a^2*c^2*x^3 - c^2*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {c^2\,x\,\left (a^2\,x^2-3\right )}{3} \]

[In]

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

[Out]

(c^2*x*(a^2*x^2 - 3))/3