\(\int e^{4 \coth ^{-1}(a x)} (c-a^2 c x^2)^4 \, dx\) [582]

Optimal result

Integrand size = 22, antiderivative size = 52 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {4 c^4 (1+a x)^7}{7 a}-\frac {c^4 (1+a x)^8}{2 a}+\frac {c^4 (1+a x)^9}{9 a} \]

[Out]

4/7*c^4*(a*x+1)^7/a-1/2*c^4*(a*x+1)^8/a+1/9*c^4*(a*x+1)^9/a

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 52, 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} \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^4 (a x+1)^9}{9 a}-\frac {c^4 (a x+1)^8}{2 a}+\frac {4 c^4 (a x+1)^7}{7 a} \]

[In]

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

[Out]

(4*c^4*(1 + a*x)^7)/(7*a) - (c^4*(1 + a*x)^8)/(2*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^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx \\ & = c^4 \int (1-a x)^2 (1+a x)^6 \, dx \\ & = c^4 \int \left (4 (1+a x)^6-4 (1+a x)^7+(1+a x)^8\right ) \, dx \\ & = \frac {4 c^4 (1+a x)^7}{7 a}-\frac {c^4 (1+a x)^8}{2 a}+\frac {c^4 (1+a x)^9}{9 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.60 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^4 (1+a x)^7 \left (23-35 a x+14 a^2 x^2\right )}{126 a} \]

[In]

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

[Out]

(c^4*(1 + a*x)^7*(23 - 35*a*x + 14*a^2*x^2))/(126*a)

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.33

method	result	size
gosper	\(\frac {c^{4} x \left (14 a^{8} x^{8}+63 a^{7} x^{7}+72 a^{6} x^{6}-84 a^{5} x^{5}-252 a^{4} x^{4}-126 a^{3} x^{3}+168 a^{2} x^{2}+252 a x +126\right )}{126}\)	\(69\)
default	\(c^{4} \left (\frac {1}{9} a^{8} x^{9}+\frac {1}{2} a^{7} x^{8}+\frac {4}{7} a^{6} x^{7}-\frac {2}{3} a^{5} x^{6}-2 a^{4} x^{5}-a^{3} x^{4}+\frac {4}{3} a^{2} x^{3}+2 a \,x^{2}+x \right )\)	\(69\)
risch	\(\frac {1}{9} a^{8} c^{4} x^{9}+\frac {1}{2} a^{7} c^{4} x^{8}+\frac {4}{7} a^{6} c^{4} x^{7}-\frac {2}{3} a^{5} c^{4} x^{6}-2 a^{4} c^{4} x^{5}-a^{3} c^{4} x^{4}+\frac {4}{3} a^{2} c^{4} x^{3}+2 a \,c^{4} x^{2}+c^{4} x\)	\(93\)
parallelrisch	\(\frac {1}{9} a^{8} c^{4} x^{9}+\frac {1}{2} a^{7} c^{4} x^{8}+\frac {4}{7} a^{6} c^{4} x^{7}-\frac {2}{3} a^{5} c^{4} x^{6}-2 a^{4} c^{4} x^{5}-a^{3} c^{4} x^{4}+\frac {4}{3} a^{2} c^{4} x^{3}+2 a \,c^{4} x^{2}+c^{4} x\)	\(93\)
norman	\(\frac {-c^{4} x +a^{4} c^{4} x^{5}-a \,c^{4} x^{2}+\frac {2}{3} a^{2} c^{4} x^{3}+\frac {7}{3} a^{3} c^{4} x^{4}-\frac {4}{3} a^{5} c^{4} x^{6}-\frac {26}{21} a^{6} c^{4} x^{7}+\frac {1}{14} a^{7} c^{4} x^{8}+\frac {7}{18} a^{8} c^{4} x^{9}+\frac {1}{9} a^{9} c^{4} x^{10}}{a x -1}\)	\(112\)
meijerg	\(-\frac {c^{4} \left (-\frac {x a \left (-308 a^{9} x^{9}-385 a^{8} x^{8}-495 a^{7} x^{7}-660 a^{6} x^{6}-924 a^{5} x^{5}-1386 a^{4} x^{4}-2310 a^{3} x^{3}-4620 a^{2} x^{2}-13860 a x +27720\right )}{2772 \left (-a x +1\right )}-10 \ln \left (-a x +1\right )\right )}{a}+\frac {3 c^{4} \left (-\frac {a x \left (-45 a^{7} x^{7}-60 a^{6} x^{6}-84 a^{5} x^{5}-126 a^{4} x^{4}-210 a^{3} x^{3}-420 a^{2} x^{2}-1260 a x +2520\right )}{315 \left (-a x +1\right )}-8 \ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{4} \left (-\frac {a x \left (-14 a^{5} x^{5}-21 a^{4} x^{4}-35 a^{3} x^{3}-70 a^{2} x^{2}-210 a x +420\right )}{70 \left (-a x +1\right )}-6 \ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{4} \left (-\frac {a x \left (-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{15 \left (-a x +1\right )}-4 \ln \left (-a x +1\right )\right )}{a}+\frac {3 c^{4} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{4} \left (\frac {a x \left (-35 a^{8} x^{8}-45 a^{7} x^{7}-60 a^{6} x^{6}-84 a^{5} x^{5}-126 a^{4} x^{4}-210 a^{3} x^{3}-420 a^{2} x^{2}-1260 a x +2520\right )}{-280 a x +280}+9 \ln \left (-a x +1\right )\right )}{a}-\frac {8 c^{4} \left (\frac {a x \left (-20 a^{6} x^{6}-28 a^{5} x^{5}-42 a^{4} x^{4}-70 a^{3} x^{3}-140 a^{2} x^{2}-420 a x +840\right )}{-120 a x +120}+7 \ln \left (-a x +1\right )\right )}{a}+\frac {12 c^{4} \left (\frac {a x \left (-3 a^{4} x^{4}-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{-12 a x +12}+5 \ln \left (-a x +1\right )\right )}{a}-\frac {8 c^{4} \left (\frac {a x \left (-2 a^{2} x^{2}-6 a x +12\right )}{-4 a x +4}+3 \ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{4} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{4} x}{-a x +1}\)	\(654\)

[In]

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

[Out]

1/126*c^4*x*(14*a^8*x^8+63*a^7*x^7+72*a^6*x^6-84*a^5*x^5-252*a^4*x^4-126*a^3*x^3+168*a^2*x^2+252*a*x+126)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.77 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{2} \, a^{7} c^{4} x^{8} + \frac {4}{7} \, a^{6} c^{4} x^{7} - \frac {2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4}{3} \, a^{2} c^{4} x^{3} + 2 \, a c^{4} x^{2} + c^{4} x \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (41) = 82\).

Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.92 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {a^{8} c^{4} x^{9}}{9} + \frac {a^{7} c^{4} x^{8}}{2} + \frac {4 a^{6} c^{4} x^{7}}{7} - \frac {2 a^{5} c^{4} x^{6}}{3} - 2 a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4 a^{2} c^{4} x^{3}}{3} + 2 a c^{4} x^{2} + c^{4} x \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.77 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{2} \, a^{7} c^{4} x^{8} + \frac {4}{7} \, a^{6} c^{4} x^{7} - \frac {2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4}{3} \, a^{2} c^{4} x^{3} + 2 \, a c^{4} x^{2} + c^{4} x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.73 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {{\left (14 \, c^{4} + \frac {189 \, c^{4}}{a x - 1} + \frac {1080 \, c^{4}}{{\left (a x - 1\right )}^{2}} + \frac {3360 \, c^{4}}{{\left (a x - 1\right )}^{3}} + \frac {6048 \, c^{4}}{{\left (a x - 1\right )}^{4}} + \frac {6048 \, c^{4}}{{\left (a x - 1\right )}^{5}} + \frac {2688 \, c^{4}}{{\left (a x - 1\right )}^{6}}\right )} {\left (a x - 1\right )}^{9}}{126 \, a} \]

[In]

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

[Out]

1/126*(14*c^4 + 189*c^4/(a*x - 1) + 1080*c^4/(a*x - 1)^2 + 3360*c^4/(a*x - 1)^3 + 6048*c^4/(a*x - 1)^4 + 6048*
c^4/(a*x - 1)^5 + 2688*c^4/(a*x - 1)^6)*(a*x - 1)^9/a

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.77 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {a^8\,c^4\,x^9}{9}+\frac {a^7\,c^4\,x^8}{2}+\frac {4\,a^6\,c^4\,x^7}{7}-\frac {2\,a^5\,c^4\,x^6}{3}-2\,a^4\,c^4\,x^5-a^3\,c^4\,x^4+\frac {4\,a^2\,c^4\,x^3}{3}+2\,a\,c^4\,x^2+c^4\,x \]

[In]

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

[Out]

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