3.6.57 \(\int e^{\coth ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\) [557]

Optimal. Leaf size=313 \[ \frac {5}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {5}{48} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {1}{24} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{56} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{14} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {5}{42} a^5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7+\frac {5 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{16 a} \]

[Out]

5/42*a^5*c^3*(1-1/a/x)^(3/2)*(1+1/a/x)^(9/2)*x^6-1/7*a^6*c^3*(1-1/a/x)^(5/2)*(1+1/a/x)^(9/2)*x^7+5/16*c^3*arct
anh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a+5/48*a*c^3*(1+1/a/x)^(3/2)*x^2*(1-1/a/x)^(1/2)+1/24*a^2*c^3*(1+1/a/x)^(
5/2)*x^3*(1-1/a/x)^(1/2)+1/56*a^3*c^3*(1+1/a/x)^(7/2)*x^4*(1-1/a/x)^(1/2)-1/14*a^4*c^3*(1+1/a/x)^(9/2)*x^5*(1-
1/a/x)^(1/2)+5/16*c^3*x*(1-1/a/x)^(1/2)*(1+1/a/x)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6326, 6330, 96, 94, 214} \begin {gather*} -\frac {1}{7} a^6 c^3 x^7 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {5}{42} a^5 c^3 x^6 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}-\frac {1}{14} a^4 c^3 x^5 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {1}{56} 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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]*(c - a^2*c*x^2)^3,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)/56
- (a^4*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)*x^5)/14 + (5*a^5*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)*
x^6)/42 - (a^6*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2)*x^7)/7 + (5*c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[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*} \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 ) \text {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}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{7} \left (5 a^5 c^3\right ) \text {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}{42} a^5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7+\frac {1}{14} \left (5 a^4 c^3\right ) \text {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}{14} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {5}{42} a^5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{14} \left (a^3 c^3\right ) \text {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}{56} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{14} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {5}{42} a^5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{8} \left (a^2 c^3\right ) \text {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}{56} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{14} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {5}{42} a^5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{24} \left (5 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 {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}{56} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{14} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {5}{42} a^5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{16} \left (5 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 {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}{56} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{14} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {5}{42} a^5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {\left (5 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 {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}{56} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{14} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {5}{42} a^5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7+\frac {\left (5 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 {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}{56} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{14} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {5}{42} a^5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} 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*}

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Mathematica [A]
time = 0.10, size = 95, normalized size = 0.30 \begin {gather*} \frac {c^3 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (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\right )+105 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{336 a} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2)^3,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 = 0.10, size = 231, normalized size = 0.74

method	result	size
risch	\(-\frac {\left (48 a^{6} x^{6}+56 a^{5} x^{5}-144 a^{4} x^{4}-182 a^{3} x^{3}+144 a^{2} x^{2}+231 a x -48\right ) \left (a x -1\right ) c^{3}}{336 a \sqrt {\frac {a x -1}{a x +1}}}+\frac {5 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{3} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{16 \sqrt {a^{2}}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(144\)
default	\(-\frac {\left (a x -1\right ) c^{3} \left (48 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{4} x^{4}+56 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-96 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-126 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -64 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+105 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +112 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-105 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{336 a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}}\)	\(231\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/336*(a*x-1)*c^3/a*(48*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*a^4*x^4+56*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*a^3*x^3-96*(a^
2*x^2-1)^(3/2)*(a^2)^(1/2)*a^2*x^2-126*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*a*x-64*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)+105*
(a^2)^(1/2)*(a^2*x^2-1)^(1/2)*a*x+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*x+1)*(a*x-1))^(1/2)/(a^2)^(1/2)

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Maxima [A]
time = 0.26, size = 337, normalized size = 1.08 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-a^2*c*x^2+c)^3,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*(10
5*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 + 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

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Fricas [A]
time = 0.34, size = 148, normalized size = 0.47 \begin {gather*} \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} + 104 \, a^{6} c^{3} x^{6} - 88 \, a^{5} c^{3} x^{5} - 326 \, a^{4} c^{3} x^{4} - 38 \, a^{3} c^{3} x^{3} + 375 \, a^{2} c^{3} x^{2} + 183 \, a c^{3} x - 48 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{336 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-a^2*c*x^2+c)^3,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 + 104*a^6*c^3*x^6 - 88*a^5*c^3*x^5 - 326*a^4*c^3*x^4 - 38*a^3*c^3*x^3 + 375*a^2*c^3*x^2 + 183*a*c^3*x - 48*
c^3)*sqrt((a*x - 1)/(a*x + 1)))/a

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - c^{3} \left (\int \frac {3 a^{2} x^{2}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {a^{6} x^{6}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {1}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.45, size = 178, normalized size = 0.57 \begin {gather*} -\frac {5 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{16 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} - \frac {1}{336} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (2 \, {\left (\frac {72 \, a c^{3}}{\mathrm {sgn}\left (a x + 1\right )} - {\left (\frac {91 \, a^{2} c^{3}}{\mathrm {sgn}\left (a x + 1\right )} + 4 \, {\left (\frac {18 \, a^{3} c^{3}}{\mathrm {sgn}\left (a x + 1\right )} - {\left (\frac {6 \, a^{5} c^{3} x}{\mathrm {sgn}\left (a x + 1\right )} + \frac {7 \, a^{4} c^{3}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x\right )} x\right )} x\right )} x + \frac {231 \, c^{3}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x - \frac {48 \, c^{3}}{a \mathrm {sgn}\left (a x + 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/16*c^3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(abs(a)*sgn(a*x + 1)) - 1/336*sqrt(a^2*x^2 - 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 + 231*c^3/sgn(a*x + 1))*x - 48*c^3/(a*sgn(a*x + 1)))

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Mupad [B]
time = 1.31, size = 289, normalized size = 0.92 \begin {gather*} \frac {5\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a}-\frac {\frac {5\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}-\frac {25\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{6}+\frac {283\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{24}+\frac {128\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{7}-\frac {283\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{24}+\frac {25\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{6}-\frac {5\,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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*c^3*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(8*a) - ((5*c^3*((a*x - 1)/(a*x + 1))^(1/2))/8 - (25*c^3*((a*x - 1)
/(a*x + 1))^(3/2))/6 + (283*c^3*((a*x - 1)/(a*x + 1))^(5/2))/24 + (128*c^3*((a*x - 1)/(a*x + 1))^(7/2))/7 - (2
83*c^3*((a*x - 1)/(a*x + 1))^(9/2))/24 + (25*c^3*((a*x - 1)/(a*x + 1))^(11/2))/6 - (5*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)

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