\(\int e^{-3 \coth ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\) [607]

Optimal result

Integrand size = 22, antiderivative size = 313 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {9}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{8} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7+\frac {9 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{16 a} \]

[Out]

3/8*a^3*c^3*(1-1/a/x)^(3/2)*(1+1/a/x)^(5/2)*x^4-3/10*a^4*c^3*(1-1/a/x)^(5/2)*(1+1/a/x)^(5/2)*x^5+3/14*a^5*c^3*
(1-1/a/x)^(7/2)*(1+1/a/x)^(5/2)*x^6-1/7*a^6*c^3*(1-1/a/x)^(9/2)*(1+1/a/x)^(5/2)*x^7+9/16*c^3*arctanh((1-1/a/x)
^(1/2)*(1+1/a/x)^(1/2))/a+3/16*a*c^3*(1+1/a/x)^(3/2)*x^2*(1-1/a/x)^(1/2)-3/8*a^2*c^3*(1+1/a/x)^(5/2)*x^3*(1-1/
a/x)^(1/2)+9/16*c^3*x*(1-1/a/x)^(1/2)*(1+1/a/x)^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6326, 6330, 96, 94, 214} \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {1}{7} a^6 c^3 x^7 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {3}{14} a^5 c^3 x^6 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {3}{10} a^4 c^3 x^5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {3}{8} a^3 c^3 x^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {3}{8} a^2 c^3 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {9 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{16 a}+\frac {3}{16} a c^3 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {9}{16} c^3 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \]

[In]

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

[Out]

(9*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/16 + (3*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2)/16 - (3
*a^2*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3)/8 + (3*a^3*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)*x^4
)/8 - (3*a^4*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)*x^5)/10 + (3*a^5*c^3*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x)
)^(5/2)*x^6)/14 - (a^6*c^3*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(5/2)*x^7)/7 + (9*c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*S
qrt[1 + 1/(a*x)]])/(16*a)

Rule 94

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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 214

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

Rule 6326

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 6330

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*} \text {integral}& = -\left (\left (a^6 c^3\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^3 x^6 \, dx\right ) \\ & = \left (a^6 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{9/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^8} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7-\frac {1}{7} \left (9 a^5 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^7} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7+\frac {1}{2} \left (3 a^4 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^6} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7-\frac {1}{2} \left (3 a^3 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^5} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7+\frac {1}{8} \left (9 a^2 c^3\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}}{x^4} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {3}{8} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7-\frac {1}{8} \left (3 a c^3\right ) \text {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 {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{8} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7-\frac {1}{16} \left (9 c^3\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^2 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {9}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{8} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7-\frac {\left (9 c^3\right ) \text {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 {9}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{8} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7+\frac {\left (9 c^3\right ) \text {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 {9}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{8} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7+\frac {9 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{16 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.30 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {c^3 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (368+245 a x-656 a^2 x^2+350 a^3 x^3+208 a^4 x^4-280 a^5 x^5+80 a^6 x^6\right )-315 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{560 a} \]

[In]

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

[Out]

-1/560*(c^3*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(368 + 245*a*x - 656*a^2*x^2 + 350*a^3*x^3 + 208*a^4*x^4 - 280*a^5*x^5
+ 80*a^6*x^6) - 315*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/a

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.46

method	result	size
risch	\(-\frac {\left (80 a^{6} x^{6}-280 a^{5} x^{5}+208 a^{4} x^{4}+350 a^{3} x^{3}-656 a^{2} x^{2}+245 a x +368\right ) \left (a x +1\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{560 a}+\frac {9 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{16 \sqrt {a^{2}}\, \left (a x -1\right )}\)	\(144\)
default	\(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )^{2} c^{3} \left (80 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-280 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+288 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+70 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +192 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+315 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -560 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-315 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{560 a \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\)	\(240\)

[In]

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

[Out]

-1/560*(80*a^6*x^6-280*a^5*x^5+208*a^4*x^4+350*a^3*x^3-656*a^2*x^2+245*a*x+368)*(a*x+1)/a*c^3*((a*x-1)/(a*x+1)
)^(1/2)+9/16*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)*c^3*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))
^(1/2)/(a*x-1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.47 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (80 \, a^{7} c^{3} x^{7} - 200 \, a^{6} c^{3} x^{6} - 72 \, a^{5} c^{3} x^{5} + 558 \, a^{4} c^{3} x^{4} - 306 \, a^{3} c^{3} x^{3} - 411 \, a^{2} c^{3} x^{2} + 613 \, a c^{3} x + 368 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{560 \, a} \]

[In]

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

[Out]

1/560*(315*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 315*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (80*a^7*c^3*x
^7 - 200*a^6*c^3*x^6 - 72*a^5*c^3*x^5 + 558*a^4*c^3*x^4 - 306*a^3*c^3*x^3 - 411*a^2*c^3*x^2 + 613*a*c^3*x + 36
8*c^3)*sqrt((a*x - 1)/(a*x + 1)))/a

Sympy [F]

\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=- c^{3} \left (\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \left (- \frac {3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {3 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \frac {3 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {3 a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \left (- \frac {a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{7} x^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right ) \]

[In]

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

[Out]

-c**3*(Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(-a*x*sqrt(a*x/(a*x + 1) - 1/(a*x +
1))/(a*x + 1), x) + Integral(-3*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(3*a**3*x*
*3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(3*a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a
*x + 1), x) + Integral(-3*a**5*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(-a**6*x**6*sqrt
(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(a**7*x**7*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1),
x))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.08 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {1}{560} \, {\left (\frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (315 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} - 2100 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} - 8393 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 9216 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 5943 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2100 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 315 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {7 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {21 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {35 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {35 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {21 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {7 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac {{\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - a^{2}}\right )} a \]

[In]

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

[Out]

1/560*(315*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 315*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(31
5*c^3*((a*x - 1)/(a*x + 1))^(13/2) - 2100*c^3*((a*x - 1)/(a*x + 1))^(11/2) - 8393*c^3*((a*x - 1)/(a*x + 1))^(9
/2) + 9216*c^3*((a*x - 1)/(a*x + 1))^(7/2) - 5943*c^3*((a*x - 1)/(a*x + 1))^(5/2) + 2100*c^3*((a*x - 1)/(a*x +
1))^(3/2) - 315*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 +
35*(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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.52 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {9 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{16 \, {\left | a \right |}} - \frac {1}{560} \, \sqrt {a^{2} x^{2} - 1} {\left (\frac {368 \, c^{3} \mathrm {sgn}\left (a x + 1\right )}{a} + {\left (245 \, c^{3} \mathrm {sgn}\left (a x + 1\right ) - 2 \, {\left (328 \, a c^{3} \mathrm {sgn}\left (a x + 1\right ) - {\left (175 \, a^{2} c^{3} \mathrm {sgn}\left (a x + 1\right ) + 4 \, {\left (26 \, a^{3} c^{3} \mathrm {sgn}\left (a x + 1\right ) + 5 \, {\left (2 \, a^{5} c^{3} x \mathrm {sgn}\left (a x + 1\right ) - 7 \, a^{4} c^{3} \mathrm {sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

[In]

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

[Out]

-9/16*c^3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) - 1/560*sqrt(a^2*x^2 - 1)*(368*c^3*sgn(a
*x + 1)/a + (245*c^3*sgn(a*x + 1) - 2*(328*a*c^3*sgn(a*x + 1) - (175*a^2*c^3*sgn(a*x + 1) + 4*(26*a^3*c^3*sgn(
a*x + 1) + 5*(2*a^5*c^3*x*sgn(a*x + 1) - 7*a^4*c^3*sgn(a*x + 1))*x)*x)*x)*x)*x)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.92 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {9\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a}-\frac {\frac {9\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}-\frac {15\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{2}+\frac {849\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{40}-\frac {1152\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{35}+\frac {1199\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{40}+\frac {15\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{2}-\frac {9\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{8}}{a-\frac {7\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {21\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {35\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {35\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {21\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {7\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}} \]

[In]

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

[Out]

(9*c^3*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(8*a) - ((9*c^3*((a*x - 1)/(a*x + 1))^(1/2))/8 - (15*c^3*((a*x - 1)
/(a*x + 1))^(3/2))/2 + (849*c^3*((a*x - 1)/(a*x + 1))^(5/2))/40 - (1152*c^3*((a*x - 1)/(a*x + 1))^(7/2))/35 +
(1199*c^3*((a*x - 1)/(a*x + 1))^(9/2))/40 + (15*c^3*((a*x - 1)/(a*x + 1))^(11/2))/2 - (9*c^3*((a*x - 1)/(a*x +
1))^(13/2))/8)/(a - (7*a*(a*x - 1))/(a*x + 1) + (21*a*(a*x - 1)^2)/(a*x + 1)^2 - (35*a*(a*x - 1)^3)/(a*x + 1)
^3 + (35*a*(a*x - 1)^4)/(a*x + 1)^4 - (21*a*(a*x - 1)^5)/(a*x + 1)^5 + (7*a*(a*x - 1)^6)/(a*x + 1)^6 - (a*(a*x
- 1)^7)/(a*x + 1)^7)