3.591 \(\int e^{-\coth ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\)

Optimal. Leaf size=313 \[ -\frac{1}{7} a^6 c^3 x^7 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{1}{6} a^5 c^3 x^6 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{1}{6} a^4 c^3 x^5 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{1}{8} a^3 c^3 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{1}{24} a^2 c^3 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{5}{48} a c^3 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{5}{16} c^3 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{5 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{16 a} \]

[Out]

(-5*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/16 - (5*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2)/48 - (
a^2*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3)/24 + (a^3*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x^4)/8
- (a^4*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2)*x^5)/6 + (a^5*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)*x
^6)/6 - (a^6*c^3*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(7/2)*x^7)/7 - (5*c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/
(a*x)]])/(16*a)

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Rubi [A]  time = 0.254862, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ -\frac{1}{7} a^6 c^3 x^7 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{1}{6} a^5 c^3 x^6 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{1}{6} a^4 c^3 x^5 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{1}{8} a^3 c^3 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{1}{24} a^2 c^3 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{5}{48} a c^3 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{5}{16} c^3 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{5 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{16 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^3/E^ArcCoth[a*x],x]

[Out]

(-5*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/16 - (5*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2)/48 - (
a^2*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3)/24 + (a^3*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x^4)/8
- (a^4*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2)*x^5)/6 + (a^5*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)*x
^6)/6 - (a^6*c^3*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(7/2)*x^7)/7 - (5*c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/
(a*x)]])/(16*a)

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]

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 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 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]

Rubi steps

\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=-\left (\left (a^6 c^3\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^3 x^6 \, dx\right )\\ &=\left (a^6 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{7/2} \left (1+\frac{x}{a}\right )^{5/2}}{x^8} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\left (a^5 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{5/2}}{x^7} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{6} a^5 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7+\frac{1}{6} \left (5 a^4 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{5/2}}{x^6} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{6} a^4 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5+\frac{1}{6} a^5 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\frac{1}{2} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{5/2}}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{8} a^3 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{1}{6} a^4 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5+\frac{1}{6} a^5 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7+\frac{1}{8} \left (a^2 c^3\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{1}{24} a^2 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{8} a^3 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{1}{6} a^4 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5+\frac{1}{6} a^5 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7+\frac{1}{24} \left (5 a c^3\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{5}{48} a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{1}{24} a^2 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{8} a^3 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{1}{6} a^4 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5+\frac{1}{6} a^5 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7+\frac{1}{16} \left (5 c^3\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{5}{16} c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{5}{48} a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{1}{24} a^2 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{8} a^3 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{1}{6} a^4 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5+\frac{1}{6} a^5 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7+\frac{\left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 a}\\ &=-\frac{5}{16} c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{5}{48} a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{1}{24} a^2 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{8} a^3 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{1}{6} a^4 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5+\frac{1}{6} a^5 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\frac{\left (5 c^3\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 )}{16 a^2}\\ &=-\frac{5}{16} c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{5}{48} a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{1}{24} a^2 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{8} a^3 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{1}{6} a^4 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5+\frac{1}{6} a^5 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\frac{5 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{16 a}\\ \end{align*}

Mathematica [A]  time = 0.196621, size = 95, normalized size = 0.3 \[ \frac{c^3 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (-48 a^6 x^6+56 a^5 x^5+144 a^4 x^4-182 a^3 x^3-144 a^2 x^2+231 a x+48\right )-105 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{336 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - a^2*c*x^2)^3/E^ArcCoth[a*x],x]

[Out]

(c^3*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(48 + 231*a*x - 144*a^2*x^2 - 182*a^3*x^3 + 144*a^4*x^4 + 56*a^5*x^5 - 48*a^6*
x^6) - 105*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(336*a)

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Maple [A]  time = 0.138, size = 231, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax+1 \right ){c}^{3}}{336\,a}\sqrt{{\frac{ax-1}{ax+1}}} \left ( 48\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}{x}^{4}{a}^{4}-56\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}-96\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+126\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-64\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}-105\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+112\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}+105\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/336*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*c^3/a*(48*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*x^4*a^4-56*(a^2*x^2-1)^(3/2)*(a
^2)^(1/2)*x^3*a^3-96*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^2*a^2+126*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*x*a-64*(a^2*x^2-1
)^(3/2)*(a^2)^(1/2)-105*(a^2)^(1/2)*(a^2*x^2-1)^(1/2)*x*a+112*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)+105*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 = 1.07542, size = 455, normalized size = 1.45 \begin{align*} -\frac{1}{336} \,{\left (\frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{2 \,{\left (105 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{2}} - 700 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{2}} + 1981 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} + 3072 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 1981 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 700 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 105 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{7 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{21 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{35 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{35 \,{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac{21 \,{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac{7 \,{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac{{\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - a^{2}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^3*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")

[Out]

-1/336*(105*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 105*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(1
05*c^3*((a*x - 1)/(a*x + 1))^(13/2) - 700*c^3*((a*x - 1)/(a*x + 1))^(11/2) + 1981*c^3*((a*x - 1)/(a*x + 1))^(9
/2) + 3072*c^3*((a*x - 1)/(a*x + 1))^(7/2) - 1981*c^3*((a*x - 1)/(a*x + 1))^(5/2) + 700*c^3*((a*x - 1)/(a*x +
1))^(3/2) - 105*c^3*sqrt((a*x - 1)/(a*x + 1)))/(7*(a*x - 1)*a^2/(a*x + 1) - 21*(a*x - 1)^2*a^2/(a*x + 1)^2 + 3
5*(a*x - 1)^3*a^2/(a*x + 1)^3 - 35*(a*x - 1)^4*a^2/(a*x + 1)^4 + 21*(a*x - 1)^5*a^2/(a*x + 1)^5 - 7*(a*x - 1)^
6*a^2/(a*x + 1)^6 + (a*x - 1)^7*a^2/(a*x + 1)^7 - a^2))*a

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Fricas [A]  time = 2.01974, size = 343, normalized size = 1.1 \begin{align*} -\frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (48 \, a^{7} c^{3} x^{7} - 8 \, a^{6} c^{3} x^{6} - 200 \, a^{5} c^{3} x^{5} + 38 \, a^{4} c^{3} x^{4} + 326 \, a^{3} c^{3} x^{3} - 87 \, a^{2} c^{3} x^{2} - 279 \, a c^{3} x - 48 \, c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{336 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^3*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")

[Out]

-1/336*(105*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 105*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (48*a^7*c^3*
x^7 - 8*a^6*c^3*x^6 - 200*a^5*c^3*x^5 + 38*a^4*c^3*x^4 + 326*a^3*c^3*x^3 - 87*a^2*c^3*x^2 - 279*a*c^3*x - 48*c
^3)*sqrt((a*x - 1)/(a*x + 1)))/a

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**3*((a*x-1)/(a*x+1))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.17291, size = 217, normalized size = 0.69 \begin{align*} \frac{5 \, c^{3} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{16 \,{\left | a \right |}} + \frac{1}{336} \, \sqrt{a^{2} x^{2} - 1}{\left (\frac{48 \, c^{3} \mathrm{sgn}\left (a x + 1\right )}{a} +{\left (231 \, c^{3} \mathrm{sgn}\left (a x + 1\right ) - 2 \,{\left (72 \, a c^{3} \mathrm{sgn}\left (a x + 1\right ) +{\left (91 \, a^{2} c^{3} \mathrm{sgn}\left (a x + 1\right ) - 4 \,{\left (18 \, a^{3} c^{3} \mathrm{sgn}\left (a x + 1\right ) -{\left (6 \, a^{5} c^{3} x \mathrm{sgn}\left (a x + 1\right ) - 7 \, a^{4} c^{3} \mathrm{sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^3*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")

[Out]

5/16*c^3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + 1/336*sqrt(a^2*x^2 - 1)*(48*c^3*sgn(a*x
+ 1)/a + (231*c^3*sgn(a*x + 1) - 2*(72*a*c^3*sgn(a*x + 1) + (91*a^2*c^3*sgn(a*x + 1) - 4*(18*a^3*c^3*sgn(a*x
+ 1) - (6*a^5*c^3*x*sgn(a*x + 1) - 7*a^4*c^3*sgn(a*x + 1))*x)*x)*x)*x)*x)