3.8.72 \(\int e^{\coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^4 \, dx\) [772]

Optimal. Leaf size=342 \[ -\frac {51 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{48 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {35 c^4 \csc ^{-1}(a x)}{16 a}+\frac {c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \]

[Out]

47/42*c^4*(1-1/a/x)^(3/2)*(1+1/a/x)^(9/2)/a+8/7*c^4*(1-1/a/x)^(5/2)*(1+1/a/x)^(9/2)/a+c^4*(1-1/a/x)^(7/2)*(1+1
/a/x)^(9/2)*x+35/16*c^4*arccsc(a*x)/a+c^4*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a-67/48*c^4*(1+1/a/x)^(3/2)
*(1-1/a/x)^(1/2)/a-91/120*c^4*(1+1/a/x)^(5/2)*(1-1/a/x)^(1/2)/a-131/280*c^4*(1+1/a/x)^(7/2)*(1-1/a/x)^(1/2)/a+
61/70*c^4*(1+1/a/x)^(9/2)*(1-1/a/x)^(1/2)/a-51/16*c^4*(1-1/a/x)^(1/2)*(1+1/a/x)^(1/2)/a

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Rubi [A]
time = 0.17, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6329, 99, 159, 163, 41, 222, 94, 214} \begin {gather*} \frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{9/2}}{7 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}{42 a}+c^4 x \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}{70 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}{280 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}{120 a}-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{48 a}-\frac {51 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{16 a}+\frac {35 c^4 \csc ^{-1}(a x)}{16 a}+\frac {c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-51*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(16*a) - (67*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(48*a) -
(91*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(120*a) - (131*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))/(280
*a) + (61*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2))/(70*a) + (47*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2))
/(42*a) + (8*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2))/(7*a) + c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(9/2)*
x + (35*c^4*ArcCsc[a*x])/(16*a) + (c^4*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

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 99

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/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 6329

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

Rubi steps

\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx &=-\left (c^4 \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{9/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x-c^4 \text {Subst}\left (\int \frac {\left (\frac {1}{a}-\frac {8 x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{7/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {1}{7} \left (a c^4\right ) \text {Subst}\left (\int \frac {\left (\frac {7}{a^2}-\frac {47 x}{a^3}\right ) \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{7/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {1}{42} \left (a^2 c^4\right ) \text {Subst}\left (\int \frac {\left (\frac {42}{a^3}-\frac {183 x}{a^4}\right ) \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {1}{210} \left (a^3 c^4\right ) \text {Subst}\left (\int \frac {\left (\frac {210}{a^4}-\frac {393 x}{a^5}\right ) \left (1+\frac {x}{a}\right )^{7/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {1}{840} \left (a^4 c^4\right ) \text {Subst}\left (\int \frac {\left (-\frac {840}{a^5}+\frac {1911 x}{a^6}\right ) \left (1+\frac {x}{a}\right )^{5/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {\left (a^5 c^4\right ) \text {Subst}\left (\int \frac {\left (\frac {2520}{a^6}-\frac {7035 x}{a^7}\right ) \left (1+\frac {x}{a}\right )^{3/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2520}\\ &=-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{48 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {\left (a^6 c^4\right ) \text {Subst}\left (\int \frac {\left (-\frac {5040}{a^7}+\frac {16065 x}{a^8}\right ) \sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5040}\\ &=-\frac {51 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{48 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {\left (a^7 c^4\right ) \text {Subst}\left (\int \frac {\frac {5040}{a^8}-\frac {11025 x}{a^9}}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5040}\\ &=-\frac {51 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{48 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {\left (35 c^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{16 a^2}-\frac {c^4 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {51 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{48 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {c^4 \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 )}{a^2}+\frac {\left (35 c^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{16 a^2}\\ &=-\frac {51 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{48 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {35 c^4 \csc ^{-1}(a x)}{16 a}+\frac {c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 120, normalized size = 0.35 \begin {gather*} \frac {c^4 \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (240+280 a x-1056 a^2 x^2-1330 a^3 x^3+1952 a^4 x^4+3045 a^5 x^5-2816 a^6 x^6+1680 a^7 x^7\right )}{x^6}+3675 a^6 \text {ArcSin}\left (\frac {1}{a x}\right )+1680 a^6 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{1680 a^7} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^4*((Sqrt[1 - 1/(a^2*x^2)]*(240 + 280*a*x - 1056*a^2*x^2 - 1330*a^3*x^3 + 1952*a^4*x^4 + 3045*a^5*x^5 - 2816
*a^6*x^6 + 1680*a^7*x^7))/x^6 + 3675*a^6*ArcSin[1/(a*x)] + 1680*a^6*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(1680
*a^7)

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Maple [A]
time = 0.09, size = 320, normalized size = 0.94

method	result	size
risch	\(-\frac {\left (a x -1\right ) \left (2816 a^{6} x^{6}-3045 a^{5} x^{5}-1952 a^{4} x^{4}+1330 a^{3} x^{3}+1056 a^{2} x^{2}-280 a x -240\right ) c^{4}}{1680 x^{7} a^{8} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (a^{7} \sqrt {\left (a x +1\right ) \left (a x -1\right )}+\frac {a^{8} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {35 a^{7} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{16}\right ) c^{4} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a^{8} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(188\)
default	\(\frac {\left (a x -1\right ) c^{4} \left (-1680 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{8} x^{8}+1680 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{6} x^{6}+3675 a^{7} x^{7} \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}+3675 a^{7} x^{7} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+1680 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{8} x^{7}-1995 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}-1136 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{4} x^{4}+1050 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+816 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-280 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -240 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{1680 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{8} x^{7} \sqrt {a^{2}}}\)	\(320\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(a*x-1)*c^4*(-1680*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*a^8*x^8+1680*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*a^6*x^6+367
5*a^7*x^7*(a^2)^(1/2)*(a^2*x^2-1)^(1/2)+3675*a^7*x^7*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))+1680*ln((a^2*x+(a
^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^8*x^7-1995*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*a^5*x^5-1136*(a^2)^(1/2)*
(a^2*x^2-1)^(3/2)*a^4*x^4+1050*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*a^3*x^3+816*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*a^2*x^2
-280*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*a*x-240*(a^2*x^2-1)^(3/2)*(a^2)^(1/2))/((a*x-1)/(a*x+1))^(1/2)/((a*x+1)*(a*
x-1))^(1/2)/a^8/x^7/(a^2)^(1/2)

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Maxima [A]
time = 0.47, size = 380, normalized size = 1.11 \begin {gather*} -\frac {1}{840} \, {\left (\frac {3675 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {5355 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 31465 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} + 72051 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 71801 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 4569 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 17619 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 10185 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 1995 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {6 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {14 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {14 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {14 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {14 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} - \frac {6 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {{\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + 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)*(c-c/a^2/x^2)^4,x, algorithm="maxima")

[Out]

-1/840*(3675*c^4*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 - 840*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 + 840*
c^4*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - (5355*c^4*((a*x - 1)/(a*x + 1))^(15/2) + 31465*c^4*((a*x - 1)/(a*
x + 1))^(13/2) + 72051*c^4*((a*x - 1)/(a*x + 1))^(11/2) + 71801*c^4*((a*x - 1)/(a*x + 1))^(9/2) + 4569*c^4*((a
*x - 1)/(a*x + 1))^(7/2) + 17619*c^4*((a*x - 1)/(a*x + 1))^(5/2) + 10185*c^4*((a*x - 1)/(a*x + 1))^(3/2) + 199
5*c^4*sqrt((a*x - 1)/(a*x + 1)))/(6*(a*x - 1)*a^2/(a*x + 1) + 14*(a*x - 1)^2*a^2/(a*x + 1)^2 + 14*(a*x - 1)^3*
a^2/(a*x + 1)^3 - 14*(a*x - 1)^5*a^2/(a*x + 1)^5 - 14*(a*x - 1)^6*a^2/(a*x + 1)^6 - 6*(a*x - 1)^7*a^2/(a*x + 1
)^7 - (a*x - 1)^8*a^2/(a*x + 1)^8 + a^2))*a

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Fricas [A]
time = 0.37, size = 201, normalized size = 0.59 \begin {gather*} -\frac {7350 \, a^{7} c^{4} x^{7} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (1680 \, a^{8} c^{4} x^{8} - 1136 \, a^{7} c^{4} x^{7} + 229 \, a^{6} c^{4} x^{6} + 4997 \, a^{5} c^{4} x^{5} + 622 \, a^{4} c^{4} x^{4} - 2386 \, a^{3} c^{4} x^{3} - 776 \, a^{2} c^{4} x^{2} + 520 \, a c^{4} x + 240 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{1680 \, a^{8} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1680*(7350*a^7*c^4*x^7*arctan(sqrt((a*x - 1)/(a*x + 1))) - 1680*a^7*c^4*x^7*log(sqrt((a*x - 1)/(a*x + 1)) +
1) + 1680*a^7*c^4*x^7*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (1680*a^8*c^4*x^8 - 1136*a^7*c^4*x^7 + 229*a^6*c^4
*x^6 + 4997*a^5*c^4*x^5 + 622*a^4*c^4*x^4 - 2386*a^3*c^4*x^3 - 776*a^2*c^4*x^2 + 520*a*c^4*x + 240*c^4)*sqrt((
a*x - 1)/(a*x + 1)))/(a^8*x^7)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.44, size = 461, normalized size = 1.35 \begin {gather*} -\frac {35 \, c^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{8 \, a \mathrm {sgn}\left (a x + 1\right )} - \frac {c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{4}}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {3045 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{13} c^{4} {\left | a \right |} + 6720 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{12} a c^{4} + 6860 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{11} c^{4} {\left | a \right |} + 20160 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{10} a c^{4} + 9065 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{9} c^{4} {\left | a \right |} + 49280 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{8} a c^{4} + 49280 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{6} a c^{4} - 9065 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} c^{4} {\left | a \right |} + 38976 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{4} - 6860 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} c^{4} {\left | a \right |} + 12992 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{4} - 3045 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{4} {\left | a \right |} + 2816 \, a c^{4}}{840 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{7} a {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35/8*c^4*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))/(a*sgn(a*x + 1)) - c^4*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))
/(abs(a)*sgn(a*x + 1)) + sqrt(a^2*x^2 - 1)*c^4/(a*sgn(a*x + 1)) - 1/840*(3045*(x*abs(a) - sqrt(a^2*x^2 - 1))^1
3*c^4*abs(a) + 6720*(x*abs(a) - sqrt(a^2*x^2 - 1))^12*a*c^4 + 6860*(x*abs(a) - sqrt(a^2*x^2 - 1))^11*c^4*abs(a
) + 20160*(x*abs(a) - sqrt(a^2*x^2 - 1))^10*a*c^4 + 9065*(x*abs(a) - sqrt(a^2*x^2 - 1))^9*c^4*abs(a) + 49280*(
x*abs(a) - sqrt(a^2*x^2 - 1))^8*a*c^4 + 49280*(x*abs(a) - sqrt(a^2*x^2 - 1))^6*a*c^4 - 9065*(x*abs(a) - sqrt(a
^2*x^2 - 1))^5*c^4*abs(a) + 38976*(x*abs(a) - sqrt(a^2*x^2 - 1))^4*a*c^4 - 6860*(x*abs(a) - sqrt(a^2*x^2 - 1))
^3*c^4*abs(a) + 12992*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a*c^4 - 3045*(x*abs(a) - sqrt(a^2*x^2 - 1))*c^4*abs(a)
+ 2816*a*c^4)/(((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^7*a*abs(a)*sgn(a*x + 1))

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Mupad [B]
time = 1.42, size = 332, normalized size = 0.97 \begin {gather*} \frac {\frac {19\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}+\frac {97\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{8}+\frac {839\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{40}+\frac {1523\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{280}+\frac {71801\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{840}+\frac {3431\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{40}+\frac {899\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{24}+\frac {51\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{8}}{a+\frac {6\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {14\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {14\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {14\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {14\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {6\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}-\frac {a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}}-\frac {35\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a}+\frac {2\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((19*c^4*((a*x - 1)/(a*x + 1))^(1/2))/8 + (97*c^4*((a*x - 1)/(a*x + 1))^(3/2))/8 + (839*c^4*((a*x - 1)/(a*x +
1))^(5/2))/40 + (1523*c^4*((a*x - 1)/(a*x + 1))^(7/2))/280 + (71801*c^4*((a*x - 1)/(a*x + 1))^(9/2))/840 + (34
31*c^4*((a*x - 1)/(a*x + 1))^(11/2))/40 + (899*c^4*((a*x - 1)/(a*x + 1))^(13/2))/24 + (51*c^4*((a*x - 1)/(a*x
+ 1))^(15/2))/8)/(a + (6*a*(a*x - 1))/(a*x + 1) + (14*a*(a*x - 1)^2)/(a*x + 1)^2 + (14*a*(a*x - 1)^3)/(a*x + 1
)^3 - (14*a*(a*x - 1)^5)/(a*x + 1)^5 - (14*a*(a*x - 1)^6)/(a*x + 1)^6 - (6*a*(a*x - 1)^7)/(a*x + 1)^7 - (a*(a*
x - 1)^8)/(a*x + 1)^8) - (35*c^4*atan(((a*x - 1)/(a*x + 1))^(1/2)))/(8*a) + (2*c^4*atanh(((a*x - 1)/(a*x + 1))
^(1/2)))/a

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