\(\int e^{-2 \coth ^{-1}(a x)} (c-a^2 c x^2)^4 \, dx\) [598]

Optimal result

Integrand size = 22, antiderivative size = 73 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {4 c^4 (1-a x)^6}{3 a}-\frac {12 c^4 (1-a x)^7}{7 a}+\frac {3 c^4 (1-a x)^8}{4 a}-\frac {c^4 (1-a x)^9}{9 a} \]

[Out]

4/3*c^4*(-a*x+1)^6/a-12/7*c^4*(-a*x+1)^7/a+3/4*c^4*(-a*x+1)^8/a-1/9*c^4*(-a*x+1)^9/a

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6302, 6275, 45} \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {c^4 (1-a x)^9}{9 a}+\frac {3 c^4 (1-a x)^8}{4 a}-\frac {12 c^4 (1-a x)^7}{7 a}+\frac {4 c^4 (1-a x)^6}{3 a} \]

[In]

Int[(c - a^2*c*x^2)^4/E^(2*ArcCoth[a*x]),x]

[Out]

(4*c^4*(1 - a*x)^6)/(3*a) - (12*c^4*(1 - a*x)^7)/(7*a) + (3*c^4*(1 - a*x)^8)/(4*a) - (c^4*(1 - a*x)^9)/(9*a)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 )^4 \, dx \\ & = -\left (c^4 \int (1-a x)^5 (1+a x)^3 \, dx\right ) \\ & = -\left (c^4 \int \left (8 (1-a x)^5-12 (1-a x)^6+6 (1-a x)^7-(1-a x)^8\right ) \, dx\right ) \\ & = \frac {4 c^4 (1-a x)^6}{3 a}-\frac {12 c^4 (1-a x)^7}{7 a}+\frac {3 c^4 (1-a x)^8}{4 a}-\frac {c^4 (1-a x)^9}{9 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.53 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^4 (-1+a x)^6 \left (65+138 a x+105 a^2 x^2+28 a^3 x^3\right )}{252 a} \]

[In]

Integrate[(c - a^2*c*x^2)^4/E^(2*ArcCoth[a*x]),x]

[Out]

(c^4*(-1 + a*x)^6*(65 + 138*a*x + 105*a^2*x^2 + 28*a^3*x^3))/(252*a)

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84

method	result	size
gosper	\(\frac {c^{4} x \left (28 a^{8} x^{8}-63 a^{7} x^{7}-72 a^{6} x^{6}+252 a^{5} x^{5}-378 a^{3} x^{3}+168 a^{2} x^{2}+252 a x -252\right )}{252}\)	\(61\)
default	\(c^{4} \left (\frac {1}{9} a^{8} x^{9}-\frac {1}{4} a^{7} x^{8}-\frac {2}{7} a^{6} x^{7}+a^{5} x^{6}-\frac {3}{2} a^{3} x^{4}+\frac {2}{3} a^{2} x^{3}+a \,x^{2}-x \right )\)	\(61\)
norman	\(a \,c^{4} x^{2}+a^{5} c^{4} x^{6}-c^{4} x +\frac {2}{3} a^{2} c^{4} x^{3}-\frac {3}{2} a^{3} c^{4} x^{4}-\frac {2}{7} a^{6} c^{4} x^{7}-\frac {1}{4} a^{7} c^{4} x^{8}+\frac {1}{9} a^{8} c^{4} x^{9}\)	\(81\)
risch	\(a \,c^{4} x^{2}+a^{5} c^{4} x^{6}-c^{4} x +\frac {2}{3} a^{2} c^{4} x^{3}-\frac {3}{2} a^{3} c^{4} x^{4}-\frac {2}{7} a^{6} c^{4} x^{7}-\frac {1}{4} a^{7} c^{4} x^{8}+\frac {1}{9} a^{8} c^{4} x^{9}\)	\(81\)
parallelrisch	\(a \,c^{4} x^{2}+a^{5} c^{4} x^{6}-c^{4} x +\frac {2}{3} a^{2} c^{4} x^{3}-\frac {3}{2} a^{3} c^{4} x^{4}-\frac {2}{7} a^{6} c^{4} x^{7}-\frac {1}{4} a^{7} c^{4} x^{8}+\frac {1}{9} a^{8} c^{4} x^{9}\)	\(81\)
meijerg	\(\frac {c^{4} \left (\frac {x a \left (280 a^{8} x^{8}-315 a^{7} x^{7}+360 a^{6} x^{6}-420 a^{5} x^{5}+504 a^{4} x^{4}-630 a^{3} x^{3}+840 a^{2} x^{2}-1260 a x +2520\right )}{2520}-\ln \left (a x +1\right )\right )}{a}-\frac {c^{4} \left (-\frac {a x \left (-315 a^{7} x^{7}+360 a^{6} x^{6}-420 a^{5} x^{5}+504 a^{4} x^{4}-630 a^{3} x^{3}+840 a^{2} x^{2}-1260 a x +2520\right )}{2520}+\ln \left (a x +1\right )\right )}{a}-\frac {4 c^{4} \left (\frac {a x \left (120 a^{6} x^{6}-140 a^{5} x^{5}+168 a^{4} x^{4}-210 a^{3} x^{3}+280 a^{2} x^{2}-420 a x +840\right )}{840}-\ln \left (a x +1\right )\right )}{a}+\frac {4 c^{4} \left (-\frac {a x \left (-70 a^{5} x^{5}+84 a^{4} x^{4}-105 a^{3} x^{3}+140 a^{2} x^{2}-210 a x +420\right )}{420}+\ln \left (a x +1\right )\right )}{a}+\frac {6 c^{4} \left (\frac {a x \left (12 a^{4} x^{4}-15 a^{3} x^{3}+20 a^{2} x^{2}-30 a x +60\right )}{60}-\ln \left (a x +1\right )\right )}{a}-\frac {6 c^{4} \left (-\frac {a x \left (-15 a^{3} x^{3}+20 a^{2} x^{2}-30 a x +60\right )}{60}+\ln \left (a x +1\right )\right )}{a}-\frac {4 c^{4} \left (\frac {a x \left (4 a^{2} x^{2}-6 a x +12\right )}{12}-\ln \left (a x +1\right )\right )}{a}+\frac {4 c^{4} \left (-\frac {a x \left (-3 a x +6\right )}{6}+\ln \left (a x +1\right )\right )}{a}+\frac {c^{4} \left (a x -\ln \left (a x +1\right )\right )}{a}-\frac {c^{4} \ln \left (a x +1\right )}{a}\)	\(466\)

[In]

int((-a^2*c*x^2+c)^4*(a*x-1)/(a*x+1),x,method=_RETURNVERBOSE)

[Out]

1/252*c^4*x*(28*a^8*x^8-63*a^7*x^7-72*a^6*x^6+252*a^5*x^5-378*a^3*x^3+168*a^2*x^2+252*a*x-252)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {1}{9} \, a^{8} c^{4} x^{9} - \frac {1}{4} \, a^{7} c^{4} x^{8} - \frac {2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac {3}{2} \, a^{3} c^{4} x^{4} + \frac {2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} - c^{4} x \]

[In]

integrate((-a^2*c*x^2+c)^4*(a*x-1)/(a*x+1),x, algorithm="fricas")

[Out]

1/9*a^8*c^4*x^9 - 1/4*a^7*c^4*x^8 - 2/7*a^6*c^4*x^7 + a^5*c^4*x^6 - 3/2*a^3*c^4*x^4 + 2/3*a^2*c^4*x^3 + a*c^4*
x^2 - c^4*x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {a^{8} c^{4} x^{9}}{9} - \frac {a^{7} c^{4} x^{8}}{4} - \frac {2 a^{6} c^{4} x^{7}}{7} + a^{5} c^{4} x^{6} - \frac {3 a^{3} c^{4} x^{4}}{2} + \frac {2 a^{2} c^{4} x^{3}}{3} + a c^{4} x^{2} - c^{4} x \]

[In]

integrate((-a**2*c*x**2+c)**4*(a*x-1)/(a*x+1),x)

[Out]

a**8*c**4*x**9/9 - a**7*c**4*x**8/4 - 2*a**6*c**4*x**7/7 + a**5*c**4*x**6 - 3*a**3*c**4*x**4/2 + 2*a**2*c**4*x
**3/3 + a*c**4*x**2 - c**4*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {1}{9} \, a^{8} c^{4} x^{9} - \frac {1}{4} \, a^{7} c^{4} x^{8} - \frac {2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac {3}{2} \, a^{3} c^{4} x^{4} + \frac {2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} - c^{4} x \]

[In]

integrate((-a^2*c*x^2+c)^4*(a*x-1)/(a*x+1),x, algorithm="maxima")

[Out]

1/9*a^8*c^4*x^9 - 1/4*a^7*c^4*x^8 - 2/7*a^6*c^4*x^7 + a^5*c^4*x^6 - 3/2*a^3*c^4*x^4 + 2/3*a^2*c^4*x^3 + a*c^4*
x^2 - c^4*x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {1}{9} \, a^{8} c^{4} x^{9} - \frac {1}{4} \, a^{7} c^{4} x^{8} - \frac {2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac {3}{2} \, a^{3} c^{4} x^{4} + \frac {2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} - c^{4} x \]

[In]

integrate((-a^2*c*x^2+c)^4*(a*x-1)/(a*x+1),x, algorithm="giac")

[Out]

1/9*a^8*c^4*x^9 - 1/4*a^7*c^4*x^8 - 2/7*a^6*c^4*x^7 + a^5*c^4*x^6 - 3/2*a^3*c^4*x^4 + 2/3*a^2*c^4*x^3 + a*c^4*
x^2 - c^4*x

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {a^8\,c^4\,x^9}{9}-\frac {a^7\,c^4\,x^8}{4}-\frac {2\,a^6\,c^4\,x^7}{7}+a^5\,c^4\,x^6-\frac {3\,a^3\,c^4\,x^4}{2}+\frac {2\,a^2\,c^4\,x^3}{3}+a\,c^4\,x^2-c^4\,x \]

[In]

int(((c - a^2*c*x^2)^4*(a*x - 1))/(a*x + 1),x)

[Out]

a*c^4*x^2 - c^4*x + (2*a^2*c^4*x^3)/3 - (3*a^3*c^4*x^4)/2 + a^5*c^4*x^6 - (2*a^6*c^4*x^7)/7 - (a^7*c^4*x^8)/4
+ (a^8*c^4*x^9)/9