\(\int e^{2 \coth ^{-1}(a x)} (c-a^2 c x^2) \, dx\) [568]

Optimal result

Integrand size = 20, antiderivative size = 15 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {c (1+a x)^3}{3 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6302, 6275, 32} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {c (a x+1)^3}{3 a} \]

[In]

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

[Out]

-1/3*(c*(1 + a*x)^3)/a

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 6275

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 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)} \left (c-a^2 c x^2\right ) \, dx \\ & = -\left (c \int (1+a x)^2 \, dx\right ) \\ & = -\frac {c (1+a x)^3}{3 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-c \left (x+a x^2+\frac {a^2 x^3}{3}\right ) \]

[In]

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

[Out]

-(c*(x + a*x^2 + (a^2*x^3)/3))

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
default	\(-\frac {c \left (a x +1\right )^{3}}{3 a}\)	\(14\)
gosper	\(-\frac {c x \left (a^{2} x^{2}+3 a x +3\right )}{3}\)	\(18\)
norman	\(-c x -a c \,x^{2}-\frac {1}{3} a^{2} c \,x^{3}\)	\(22\)
parallelrisch	\(-c x -a c \,x^{2}-\frac {1}{3} a^{2} c \,x^{3}\)	\(22\)
risch	\(-\frac {a^{2} c \,x^{3}}{3}-a c \,x^{2}-c x -\frac {c}{3 a}\)	\(28\)
meijerg	\(\frac {c \left (-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{12}-\ln \left (-a x +1\right )\right )}{a}-\frac {c \left (-a x -\ln \left (-a x +1\right )\right )}{a}-\frac {c \left (\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )\right )}{a}+\frac {c \ln \left (-a x +1\right )}{a}\)	\(91\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.40 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {1}{3} \, a^{2} c x^{3} - a c x^{2} - c x \]

[In]

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

[Out]

-1/3*a^2*c*x^3 - a*c*x^2 - c*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=- \frac {a^{2} c x^{3}}{3} - a c x^{2} - c x \]

[In]

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

[Out]

-a**2*c*x**3/3 - a*c*x**2 - c*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.40 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {1}{3} \, a^{2} c x^{3} - a c x^{2} - c x \]

[In]

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

[Out]

-1/3*a^2*c*x^3 - a*c*x^2 - c*x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.40 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {1}{3} \, a^{2} c x^{3} - a c x^{2} - c x \]

[In]

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

[Out]

-1/3*a^2*c*x^3 - a*c*x^2 - c*x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {c\,x\,\left (a^2\,x^2+3\,a\,x+3\right )}{3} \]

[In]

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

[Out]

-(c*x*(3*a*x + a^2*x^2 + 3))/3