3.789 \(\int e^{3 \coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^4 \, dx\)

Optimal. Leaf size=343 \[ \frac {8 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{7 a}+c^4 x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}+\frac {15 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}}{14 a}-\frac {57 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}{70 a}-\frac {303 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}{280 a}-\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}{40 a}-\frac {37 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{16 a}-\frac {63 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{16 a}+\frac {15 c^4 \csc ^{-1}(a x)}{16 a}+\frac {3 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \]

[Out]

8/7*c^4*(1-1/a/x)^(3/2)*(1+1/a/x)^(11/2)/a+c^4*(1-1/a/x)^(5/2)*(1+1/a/x)^(11/2)*x+15/16*c^4*arccsc(a*x)/a+3*c^
4*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a-37/16*c^4*(1+1/a/x)^(3/2)*(1-1/a/x)^(1/2)/a-61/40*c^4*(1+1/a/x)^(
5/2)*(1-1/a/x)^(1/2)/a-303/280*c^4*(1+1/a/x)^(7/2)*(1-1/a/x)^(1/2)/a-57/70*c^4*(1+1/a/x)^(9/2)*(1-1/a/x)^(1/2)
/a+15/14*c^4*(1+1/a/x)^(11/2)*(1-1/a/x)^(1/2)/a-63/16*c^4*(1-1/a/x)^(1/2)*(1+1/a/x)^(1/2)/a

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Rubi [A]  time = 0.26, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6194, 97, 154, 157, 41, 216, 92, 208} \[ \frac {8 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{7 a}+c^4 x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}+\frac {15 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}}{14 a}-\frac {57 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}{70 a}-\frac {303 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}{280 a}-\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}{40 a}-\frac {37 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{16 a}-\frac {63 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{16 a}+\frac {15 c^4 \csc ^{-1}(a x)}{16 a}+\frac {3 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^4,x]

[Out]

(-63*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(16*a) - (37*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(16*a) -
(61*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(40*a) - (303*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))/(280*
a) - (57*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2))/(70*a) + (15*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(11/2))/(
14*a) + (8*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(11/2))/(7*a) + c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(11/2)*
x + (15*c^4*ArcCsc[a*x])/(16*a) + (3*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 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 97

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 154

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 157

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 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 216

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 6194

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^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx &=-\left (c^4 \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{11/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x-c^4 \operatorname {Subst}\left (\int \frac {\left (\frac {3}{a}-\frac {8 x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{9/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x-\frac {1}{7} \left (a c^4\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {21}{a^2}-\frac {45 x}{a^3}\right ) \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{9/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {15 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x-\frac {1}{42} \left (a^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {126}{a^3}-\frac {171 x}{a^4}\right ) \left (1+\frac {x}{a}\right )^{9/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {57 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {15 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x+\frac {1}{210} \left (a^3 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {630}{a^4}+\frac {909 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 {303 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}-\frac {57 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {15 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x-\frac {1}{840} \left (a^4 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {2520}{a^5}-\frac {3843 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 {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{40 a}-\frac {303 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}-\frac {57 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {15 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x+\frac {\left (a^5 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {7560}{a^6}+\frac {11655 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 {37 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}-\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{40 a}-\frac {303 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}-\frac {57 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {15 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x-\frac {\left (a^6 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {15120}{a^7}-\frac {19845 x}{a^8}\right ) \sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5040}\\ &=-\frac {63 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {37 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}-\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{40 a}-\frac {303 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}-\frac {57 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {15 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x+\frac {\left (a^7 c^4\right ) \operatorname {Subst}\left (\int \frac {-\frac {15120}{a^8}+\frac {4725 x}{a^9}}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5040}\\ &=-\frac {63 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {37 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}-\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{40 a}-\frac {303 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}-\frac {57 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {15 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x+\frac {\left (15 c^4\right ) \operatorname {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 {\left (3 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 )}{a}\\ &=-\frac {63 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {37 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}-\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{40 a}-\frac {303 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}-\frac {57 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {15 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x+\frac {\left (15 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{16 a^2}+\frac {\left (3 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 )}{a^2}\\ &=-\frac {63 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {37 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}-\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{40 a}-\frac {303 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}-\frac {57 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {15 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x+\frac {15 c^4 \csc ^{-1}(a x)}{16 a}+\frac {3 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 = 0.27, size = 126, normalized size = 0.37 \[ \frac {c^4 \left (525 a^6 x^6 \sin ^{-1}\left (\frac {1}{a x}\right )+1680 a^6 x^6 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+\sqrt {1-\frac {1}{a^2 x^2}} \left (560 a^7 x^7-2496 a^6 x^6-525 a^5 x^5+992 a^4 x^4+770 a^3 x^3-96 a^2 x^2-280 a x-80\right )\right )}{560 a^7 x^6} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^4,x]

[Out]

(c^4*(Sqrt[1 - 1/(a^2*x^2)]*(-80 - 280*a*x - 96*a^2*x^2 + 770*a^3*x^3 + 992*a^4*x^4 - 525*a^5*x^5 - 2496*a^6*x
^6 + 560*a^7*x^7) + 525*a^6*x^6*ArcSin[1/(a*x)] + 1680*a^6*x^6*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(560*a^7*x
^6)

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fricas [A]  time = 0.49, size = 201, normalized size = 0.59 \[ -\frac {1050 \, a^{7} c^{4} x^{7} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (560 \, a^{8} c^{4} x^{8} - 1936 \, a^{7} c^{4} x^{7} - 3021 \, a^{6} c^{4} x^{6} + 467 \, a^{5} c^{4} x^{5} + 1762 \, a^{4} c^{4} x^{4} + 674 \, a^{3} c^{4} x^{3} - 376 \, a^{2} c^{4} x^{2} - 360 \, a c^{4} x - 80 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{560 \, a^{8} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^4,x, algorithm="fricas")

[Out]

-1/560*(1050*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) - (560*a^8*c^4*x^8 - 1936*a^7*c^4*x^7 - 3021*a^6*c^4*
x^6 + 467*a^5*c^4*x^5 + 1762*a^4*c^4*x^4 + 674*a^3*c^4*x^3 - 376*a^2*c^4*x^2 - 360*a*c^4*x - 80*c^4)*sqrt((a*x
- 1)/(a*x + 1)))/(a^8*x^7)

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giac [A]  time = 0.19, size = 335, normalized size = 0.98 \[ -\frac {1}{280} \, a c^{4} {\left (\frac {525 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {840 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {840 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac {560 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}} + \frac {\frac {13300 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {45871 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + \frac {52672 \, {\left (a x - 1\right )}^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + \frac {33201 \, {\left (a x - 1\right )}^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} + \frac {11340 \, {\left (a x - 1\right )}^{5} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{5}} + \frac {1645 \, {\left (a x - 1\right )}^{6} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{6}} + 1715 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} + 1\right )}^{7}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^4,x, algorithm="giac")

[Out]

-1/280*a*c^4*(525*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 - 840*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 + 840*log
(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/a^2 + 560*sqrt((a*x - 1)/(a*x + 1))/(a^2*((a*x - 1)/(a*x + 1) - 1)) + (13
300*(a*x - 1)*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1) + 45871*(a*x - 1)^2*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^2 +
52672*(a*x - 1)^3*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^3 + 33201*(a*x - 1)^4*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1
)^4 + 11340*(a*x - 1)^5*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^5 + 1645*(a*x - 1)^6*sqrt((a*x - 1)/(a*x + 1))/(a*
x + 1)^6 + 1715*sqrt((a*x - 1)/(a*x + 1)))/(a^2*((a*x - 1)/(a*x + 1) + 1)^7))

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maple [A]  time = 0.07, size = 329, normalized size = 0.96 \[ \frac {\left (a x -1\right )^{2} c^{4} \left (-1680 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{8} a^{8}+1680 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{6} a^{6}+525 a^{7} x^{7} \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}+525 a^{7} x^{7} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+1680 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{7} a^{8}+35 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}-816 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}-490 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}+176 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}+280 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +80 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{560 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{8} x^{7} \sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^4,x)

[Out]

1/560*(a*x-1)^2*c^4*(-1680*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^8*a^8+1680*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^6*a^6+52
5*a^7*x^7*(a^2)^(1/2)*(a^2*x^2-1)^(1/2)+525*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))*x^7*a^8+35*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^5*a^5-816*(a^2*x^2-1)^(3/2
)*(a^2)^(1/2)*x^4*a^4-490*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^3*a^3+176*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^2*a^2+280*
(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x*a+80*(a^2*x^2-1)^(3/2)*(a^2)^(1/2))/((a*x-1)/(a*x+1))^(3/2)/(a*x+1)/((a*x-1)*(
a*x+1))^(1/2)/a^8/x^7/(a^2)^(1/2)

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maxima [A]  time = 0.43, size = 380, normalized size = 1.11 \[ -\frac {1}{280} \, {\left (\frac {525 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2205 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 13615 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} + 33621 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 39071 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 12799 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 20811 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 7665 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 1155 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {6 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {14 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {14 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {14 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {14 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} - \frac {6 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {{\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + a^{2}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^4,x, algorithm="maxima")

[Out]

-1/280*(525*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 - (2205*c^4*((a*x - 1)/(a*x + 1))^(15/2) + 13615*c^4*((a*x - 1)/(a*x
+ 1))^(13/2) + 33621*c^4*((a*x - 1)/(a*x + 1))^(11/2) + 39071*c^4*((a*x - 1)/(a*x + 1))^(9/2) + 12799*c^4*((a
*x - 1)/(a*x + 1))^(7/2) - 20811*c^4*((a*x - 1)/(a*x + 1))^(5/2) - 7665*c^4*((a*x - 1)/(a*x + 1))^(3/2) - 1155
*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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mupad [B]  time = 1.38, size = 332, normalized size = 0.97 \[ \frac {\frac {12799\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{280}-\frac {219\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{8}-\frac {2973\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{40}-\frac {33\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}+\frac {39071\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{280}+\frac {4803\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{40}+\frac {389\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{8}+\frac {63\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{8}}{a+\frac {6\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {14\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {14\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {14\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {14\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {6\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}-\frac {a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}}-\frac {15\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a}+\frac {6\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]

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))^(3/2),x)

[Out]

((12799*c^4*((a*x - 1)/(a*x + 1))^(7/2))/280 - (219*c^4*((a*x - 1)/(a*x + 1))^(3/2))/8 - (2973*c^4*((a*x - 1)/
(a*x + 1))^(5/2))/40 - (33*c^4*((a*x - 1)/(a*x + 1))^(1/2))/8 + (39071*c^4*((a*x - 1)/(a*x + 1))^(9/2))/280 +
(4803*c^4*((a*x - 1)/(a*x + 1))^(11/2))/40 + (389*c^4*((a*x - 1)/(a*x + 1))^(13/2))/8 + (63*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) - (15*c^4*atan(((a*x - 1)/(a*x + 1))^(1/2)))/(8*a) + (6*c^4*atanh(((a*x - 1)/(a*x + 1
))^(1/2)))/a

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {c^{4} \left (\int \left (- \frac {4 a^{2}}{\frac {a x^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {6 a^{4}}{\frac {a x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {4 a^{6}}{\frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{8}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \frac {1}{\frac {a x^{9} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx\right )}{a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a**2/x**2)**4,x)

[Out]

c**4*(Integral(-4*a**2/(a*x**7*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**6*sqrt(a*x/(a*x + 1) - 1/(a*x
+ 1))/(a*x + 1)), x) + Integral(6*a**4/(a*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**4*sqrt(a*x/(a*
x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(-4*a**6/(a*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) -
x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(a**8/(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(
a*x + 1) - sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(1/(a*x**9*sqrt(a*x/(a*x + 1) - 1/(a*x +
1))/(a*x + 1) - x**8*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x))/a**8

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