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

Optimal. Leaf size=393 \[ \frac {1}{9} a^8 c^4 x^9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{9/2}-\frac {1}{8} a^7 c^4 x^8 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {1}{8} a^6 c^4 x^7 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{9/2}-\frac {5}{48} a^5 c^4 x^6 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {1}{16} a^4 c^4 x^5 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}-\frac {1}{64} a^3 c^4 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {7}{192} a^2 c^4 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {35}{384} a c^4 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {35}{128} c^4 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}-\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{128 a} \]

[Out]

-5/48*a^5*c^4*(1-1/a/x)^(3/2)*(1+1/a/x)^(9/2)*x^6+1/8*a^6*c^4*(1-1/a/x)^(5/2)*(1+1/a/x)^(9/2)*x^7-1/8*a^7*c^4*
(1-1/a/x)^(7/2)*(1+1/a/x)^(9/2)*x^8+1/9*a^8*c^4*(1-1/a/x)^(9/2)*(1+1/a/x)^(9/2)*x^9-35/128*c^4*arctanh((1-1/a/
x)^(1/2)*(1+1/a/x)^(1/2))/a-35/384*a*c^4*(1+1/a/x)^(3/2)*x^2*(1-1/a/x)^(1/2)-7/192*a^2*c^4*(1+1/a/x)^(5/2)*x^3
*(1-1/a/x)^(1/2)-1/64*a^3*c^4*(1+1/a/x)^(7/2)*x^4*(1-1/a/x)^(1/2)+1/16*a^4*c^4*(1+1/a/x)^(9/2)*x^5*(1-1/a/x)^(
1/2)-35/128*c^4*x*(1-1/a/x)^(1/2)*(1+1/a/x)^(1/2)

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Rubi [A]  time = 0.32, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6191, 6195, 94, 92, 208} \[ \frac {1}{9} a^8 c^4 x^9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{9/2}-\frac {1}{8} a^7 c^4 x^8 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {1}{8} a^6 c^4 x^7 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{9/2}-\frac {5}{48} a^5 c^4 x^6 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {1}{16} a^4 c^4 x^5 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}-\frac {1}{64} a^3 c^4 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {7}{192} a^2 c^4 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {35}{384} a c^4 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {35}{128} c^4 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}-\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{128 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-35*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/128 - (35*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2)/384
- (7*a^2*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3)/192 - (a^3*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*
x^4)/64 + (a^4*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)*x^5)/16 - (5*a^5*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x)
)^(9/2)*x^6)/48 + (a^6*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2)*x^7)/8 - (a^7*c^4*(1 - 1/(a*x))^(7/2)*(1 +
1/(a*x))^(9/2)*x^8)/8 + (a^8*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(9/2)*x^9)/9 - (35*c^4*ArcTanh[Sqrt[1 - 1/(
a*x)]*Sqrt[1 + 1/(a*x)]])/(128*a)

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 6191

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p, Int[u*x^(2*p)*(1 -
1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2] &
& IntegerQ[p]

Rule 6195

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^p, Subst[Int[((
1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2
*d, 0] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegersQ[2*p, p + n/2] && IntegerQ[m]

Rubi steps

\begin {align*} \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx &=\left (a^8 c^4\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^4 x^8 \, dx\\ &=-\left (\left (a^8 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{9/2} \left (1+\frac {x}{a}\right )^{7/2}}{x^{10}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{9/2} x^9+\left (a^7 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{7/2}}{x^9} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{8} a^7 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{9/2} x^9-\frac {1}{8} \left (7 a^6 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{7/2}}{x^8} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{8} a^6 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{8} a^7 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{9/2} x^9+\frac {1}{8} \left (5 a^5 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{7/2}}{x^7} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {5}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6+\frac {1}{8} a^6 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{8} a^7 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{9/2} x^9-\frac {1}{16} \left (5 a^4 c^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}}{x^6} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{16} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6+\frac {1}{8} a^6 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{8} a^7 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{9/2} x^9+\frac {1}{16} \left (a^3 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{7/2}}{x^5 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{16} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6+\frac {1}{8} a^6 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{8} a^7 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{9/2} x^9+\frac {1}{64} \left (7 a^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{5/2}}{x^4 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {7}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{16} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6+\frac {1}{8} a^6 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{8} a^7 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{9/2} x^9+\frac {1}{192} \left (35 a c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^3 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {35}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {7}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{16} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6+\frac {1}{8} a^6 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{8} a^7 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{9/2} x^9+\frac {1}{128} \left (35 c^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^2 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {35}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {35}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {7}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{16} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6+\frac {1}{8} a^6 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{8} a^7 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{9/2} x^9+\frac {\left (35 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{128 a}\\ &=-\frac {35}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {35}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {7}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{16} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6+\frac {1}{8} a^6 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{8} a^7 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{9/2} x^9-\frac {\left (35 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{128 a^2}\\ &=-\frac {35}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {35}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {7}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{16} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6+\frac {1}{8} a^6 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{8} a^7 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{9/2} x^9-\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{128 a}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 111, normalized size = 0.28 \[ \frac {c^4 \left (a x \sqrt {1-\frac {1}{a^2 x^2}} \left (128 a^8 x^8-144 a^7 x^7-512 a^6 x^6+600 a^5 x^5+768 a^4 x^4-978 a^3 x^3-512 a^2 x^2+837 a x+128\right )-315 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )\right )}{1152 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^4*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(128 + 837*a*x - 512*a^2*x^2 - 978*a^3*x^3 + 768*a^4*x^4 + 600*a^5*x^5 - 512*a
^6*x^6 - 144*a^7*x^7 + 128*a^8*x^8) - 315*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(1152*a)

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fricas [A]  time = 0.57, size = 170, normalized size = 0.43 \[ -\frac {315 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 315 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (128 \, a^{9} c^{4} x^{9} - 16 \, a^{8} c^{4} x^{8} - 656 \, a^{7} c^{4} x^{7} + 88 \, a^{6} c^{4} x^{6} + 1368 \, a^{5} c^{4} x^{5} - 210 \, a^{4} c^{4} x^{4} - 1490 \, a^{3} c^{4} x^{3} + 325 \, a^{2} c^{4} x^{2} + 965 \, a c^{4} x + 128 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{1152 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1152*(315*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 315*c^4*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (128*a^9*c^
4*x^9 - 16*a^8*c^4*x^8 - 656*a^7*c^4*x^7 + 88*a^6*c^4*x^6 + 1368*a^5*c^4*x^5 - 210*a^4*c^4*x^4 - 1490*a^3*c^4*
x^3 + 325*a^2*c^4*x^2 + 965*a*c^4*x + 128*c^4)*sqrt((a*x - 1)/(a*x + 1)))/a

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giac [A]  time = 0.18, size = 196, normalized size = 0.50 \[ \frac {35 \, c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{128 \, {\left | a \right |}} + \frac {1}{1152} \, \sqrt {a^{2} x^{2} - 1} {\left (\frac {128 \, c^{4} \mathrm {sgn}\left (a x + 1\right )}{a} + {\left (837 \, c^{4} \mathrm {sgn}\left (a x + 1\right ) - 2 \, {\left (256 \, a c^{4} \mathrm {sgn}\left (a x + 1\right ) + {\left (489 \, a^{2} c^{4} \mathrm {sgn}\left (a x + 1\right ) - 4 \, {\left (96 \, a^{3} c^{4} \mathrm {sgn}\left (a x + 1\right ) + {\left (75 \, a^{4} c^{4} \mathrm {sgn}\left (a x + 1\right ) - 2 \, {\left (32 \, a^{5} c^{4} \mathrm {sgn}\left (a x + 1\right ) - {\left (8 \, a^{7} c^{4} x \mathrm {sgn}\left (a x + 1\right ) - 9 \, a^{6} c^{4} \mathrm {sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/128*c^4*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + 1/1152*sqrt(a^2*x^2 - 1)*(128*c^4*sgn
(a*x + 1)/a + (837*c^4*sgn(a*x + 1) - 2*(256*a*c^4*sgn(a*x + 1) + (489*a^2*c^4*sgn(a*x + 1) - 4*(96*a^3*c^4*sg
n(a*x + 1) + (75*a^4*c^4*sgn(a*x + 1) - 2*(32*a^5*c^4*sgn(a*x + 1) - (8*a^7*c^4*x*sgn(a*x + 1) - 9*a^6*c^4*sgn
(a*x + 1))*x)*x)*x)*x)*x)*x)*x)

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maple [A]  time = 0.07, size = 279, normalized size = 0.71 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{4} \left (-128 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{6} a^{6}+144 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}+384 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}-456 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}-384 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}+522 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +384 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-256 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-315 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a +315 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{1152 a \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1152*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*c^4/a*(-128*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^6*a^6+144*(a^2*x^2-1)^(3/2
)*(a^2)^(1/2)*x^5*a^5+384*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^4*a^4-456*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^3*a^3-384*
(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^2*a^2+522*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x*a+384*((a*x-1)*(a*x+1))^(3/2)*(a^2)^
(1/2)-256*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)-315*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x*a+315*ln((a^2*x+(a^2*x^2-1)^(1/2)*
(a^2)^(1/2))/(a^2)^(1/2))*a)/((a*x-1)*(a*x+1))^(1/2)/(a^2)^(1/2)

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maxima [A]  time = 0.33, size = 415, normalized size = 1.06 \[ -\frac {1}{1152} \, {\left (\frac {315 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {315 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (315 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{2}} - 2730 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 10458 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} - 23202 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} - 32768 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 23202 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 10458 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2730 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 315 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {9 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {36 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {84 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {126 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {126 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {84 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac {36 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {9 \, {\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + \frac {{\left (a x - 1\right )}^{9} a^{2}}{{\left (a x + 1\right )}^{9}} - a^{2}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1152*(315*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 315*c^4*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(
315*c^4*((a*x - 1)/(a*x + 1))^(17/2) - 2730*c^4*((a*x - 1)/(a*x + 1))^(15/2) + 10458*c^4*((a*x - 1)/(a*x + 1))
^(13/2) - 23202*c^4*((a*x - 1)/(a*x + 1))^(11/2) - 32768*c^4*((a*x - 1)/(a*x + 1))^(9/2) + 23202*c^4*((a*x - 1
)/(a*x + 1))^(7/2) - 10458*c^4*((a*x - 1)/(a*x + 1))^(5/2) + 2730*c^4*((a*x - 1)/(a*x + 1))^(3/2) - 315*c^4*sq
rt((a*x - 1)/(a*x + 1)))/(9*(a*x - 1)*a^2/(a*x + 1) - 36*(a*x - 1)^2*a^2/(a*x + 1)^2 + 84*(a*x - 1)^3*a^2/(a*x
+ 1)^3 - 126*(a*x - 1)^4*a^2/(a*x + 1)^4 + 126*(a*x - 1)^5*a^2/(a*x + 1)^5 - 84*(a*x - 1)^6*a^2/(a*x + 1)^6 +
36*(a*x - 1)^7*a^2/(a*x + 1)^7 - 9*(a*x - 1)^8*a^2/(a*x + 1)^8 + (a*x - 1)^9*a^2/(a*x + 1)^9 - a^2))*a

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mupad [B]  time = 1.34, size = 362, normalized size = 0.92 \[ \frac {\frac {35\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64}-\frac {455\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{96}+\frac {581\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{32}-\frac {1289\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{32}+\frac {512\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{9}+\frac {1289\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{32}-\frac {581\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{32}+\frac {455\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{96}-\frac {35\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/2}}{64}}{a-\frac {9\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {36\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {84\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {126\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {126\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {84\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {36\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}+\frac {9\,a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}-\frac {a\,{\left (a\,x-1\right )}^9}{{\left (a\,x+1\right )}^9}}-\frac {35\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{64\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((35*c^4*((a*x - 1)/(a*x + 1))^(1/2))/64 - (455*c^4*((a*x - 1)/(a*x + 1))^(3/2))/96 + (581*c^4*((a*x - 1)/(a*x
+ 1))^(5/2))/32 - (1289*c^4*((a*x - 1)/(a*x + 1))^(7/2))/32 + (512*c^4*((a*x - 1)/(a*x + 1))^(9/2))/9 + (1289
*c^4*((a*x - 1)/(a*x + 1))^(11/2))/32 - (581*c^4*((a*x - 1)/(a*x + 1))^(13/2))/32 + (455*c^4*((a*x - 1)/(a*x +
1))^(15/2))/96 - (35*c^4*((a*x - 1)/(a*x + 1))^(17/2))/64)/(a - (9*a*(a*x - 1))/(a*x + 1) + (36*a*(a*x - 1)^2
)/(a*x + 1)^2 - (84*a*(a*x - 1)^3)/(a*x + 1)^3 + (126*a*(a*x - 1)^4)/(a*x + 1)^4 - (126*a*(a*x - 1)^5)/(a*x +
1)^5 + (84*a*(a*x - 1)^6)/(a*x + 1)^6 - (36*a*(a*x - 1)^7)/(a*x + 1)^7 + (9*a*(a*x - 1)^8)/(a*x + 1)^8 - (a*(a
*x - 1)^9)/(a*x + 1)^9) - (35*c^4*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(64*a)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{4} \left (\int \left (- 4 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx + \int 6 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- 4 a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx + \int a^{8} x^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**4*(Integral(-4*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(6*a**4*x**4*sqrt(a*x/(a*x + 1) -
1/(a*x + 1)), x) + Integral(-4*a**6*x**6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(a**8*x**8*sqrt(a*x/(
a*x + 1) - 1/(a*x + 1)), x) + Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x))

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