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

Optimal. Leaf size=329 \[ \frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {\frac {1}{a x}+1}}+\frac {2609 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {1462}{105 a c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {1147}{315 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {208}{105 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {29}{21 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^4} \]

[Out]

-10/9/a/c^4/(1-1/a/x)^(9/2)/(1+1/a/x)^(3/2)-29/21/a/c^4/(1-1/a/x)^(7/2)/(1+1/a/x)^(3/2)-208/105/a/c^4/(1-1/a/x
)^(5/2)/(1+1/a/x)^(3/2)-1147/315/a/c^4/(1-1/a/x)^(3/2)/(1+1/a/x)^(3/2)+x/c^4/(1-1/a/x)^(9/2)/(1+1/a/x)^(3/2)+3
*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c^4-1462/105/a/c^4/(1+1/a/x)^(3/2)/(1-1/a/x)^(1/2)+2609/315*(1-1/a
/x)^(1/2)/a/c^4/(1+1/a/x)^(3/2)+1664/315*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(1/2)

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Rubi [A]  time = 0.24, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6194, 103, 152, 12, 92, 208} \[ \frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {\frac {1}{a x}+1}}+\frac {2609 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {1462}{105 a c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {1147}{315 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {208}{105 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {29}{21 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-10/(9*a*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(3/2)) - 29/(21*a*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2))
- 208/(105*a*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) - 1147/(315*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^
(3/2)) - 1462/(105*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) + (2609*Sqrt[1 - 1/(a*x)])/(315*a*c^4*(1 + 1/(
a*x))^(3/2)) + (1664*Sqrt[1 - 1/(a*x)])/(315*a*c^4*Sqrt[1 + 1/(a*x)]) + x/(c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x
))^(3/2)) + (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

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 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 \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^{11/2} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c^4}\\ &=\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{a}-\frac {7 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{11/2} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c^4}\\ &=-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a \operatorname {Subst}\left (\int \frac {\frac {27}{a^2}+\frac {60 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{9/2} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{9 c^4}\\ &=-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{21 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {a^2 \operatorname {Subst}\left (\int \frac {-\frac {189}{a^3}-\frac {435 x}{a^4}}{x \left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{63 c^4}\\ &=-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{21 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {208}{105 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a^3 \operatorname {Subst}\left (\int \frac {\frac {945}{a^4}+\frac {2496 x}{a^5}}{x \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{315 c^4}\\ &=-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{21 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {208}{105 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1147}{315 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {a^4 \operatorname {Subst}\left (\int \frac {-\frac {2835}{a^5}-\frac {10323 x}{a^6}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{945 c^4}\\ &=-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{21 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {208}{105 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1147}{315 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1462}{105 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a^5 \operatorname {Subst}\left (\int \frac {\frac {2835}{a^6}+\frac {26316 x}{a^7}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{945 c^4}\\ &=-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{21 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {208}{105 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1147}{315 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1462}{105 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {2609 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a^6 \operatorname {Subst}\left (\int \frac {\frac {8505}{a^7}+\frac {23481 x}{a^8}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2835 c^4}\\ &=-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{21 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {208}{105 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1147}{315 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1462}{105 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {2609 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a^7 \operatorname {Subst}\left (\int \frac {8505}{a^8 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2835 c^4}\\ &=-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{21 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {208}{105 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1147}{315 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1462}{105 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {2609 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c^4}\\ &=-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{21 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {208}{105 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1147}{315 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1462}{105 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {2609 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {3 \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 c^4}\\ &=-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{21 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {208}{105 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1147}{315 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1462}{105 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {2609 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^4}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 117, normalized size = 0.36 \[ \frac {3 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (315 a^7 x^7-2669 a^6 x^6+2967 a^5 x^5+4029 a^4 x^4-7399 a^3 x^3+339 a^2 x^2+4047 a x-1664\right )}{315 (a x-1)^5 (a x+1)^2}}{a c^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1664 + 4047*a*x + 339*a^2*x^2 - 7399*a^3*x^3 + 4029*a^4*x^4 + 2967*a^5*x^5 - 266
9*a^6*x^6 + 315*a^7*x^7))/(315*(-1 + a*x)^5*(1 + a*x)^2) + 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)

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fricas [A]  time = 0.54, size = 248, normalized size = 0.75 \[ \frac {945 \, {\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 945 \, {\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (315 \, a^{7} x^{7} - 2669 \, a^{6} x^{6} + 2967 \, a^{5} x^{5} + 4029 \, a^{4} x^{4} - 7399 \, a^{3} x^{3} + 339 \, a^{2} x^{2} + 4047 \, a x - 1664\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} \]

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="fricas")

[Out]

1/315*(945*(a^6*x^6 - 4*a^5*x^5 + 5*a^4*x^4 - 5*a^2*x^2 + 4*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 945*
(a^6*x^6 - 4*a^5*x^5 + 5*a^4*x^4 - 5*a^2*x^2 + 4*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (315*a^7*x^7 -
2669*a^6*x^6 + 2967*a^5*x^5 + 4029*a^4*x^4 - 7399*a^3*x^3 + 339*a^2*x^2 + 4047*a*x - 1664)*sqrt((a*x - 1)/(a*x
+ 1)))/(a^7*c^4*x^6 - 4*a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^3*c^4*x^2 + 4*a^2*c^4*x - a*c^4)

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giac [A]  time = 0.17, size = 264, normalized size = 0.80 \[ \frac {1}{20160} \, a {\left (\frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {60480 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{4}} - \frac {{\left (a x + 1\right )}^{4} {\left (\frac {450 \, {\left (a x - 1\right )}}{a x + 1} + \frac {2961 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {14700 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {95445 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 35\right )}}{{\left (a x - 1\right )}^{4} a^{2} c^{4} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {40320 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}} + \frac {105 \, {\left (\frac {{\left (a x - 1\right )} a^{4} c^{8} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + 30 \, a^{4} c^{8} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{6} c^{12}}\right )} \]

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]

1/20160*a*(60480*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 60480*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/
(a^2*c^4) - (a*x + 1)^4*(450*(a*x - 1)/(a*x + 1) + 2961*(a*x - 1)^2/(a*x + 1)^2 + 14700*(a*x - 1)^3/(a*x + 1)^
3 + 95445*(a*x - 1)^4/(a*x + 1)^4 + 35)/((a*x - 1)^4*a^2*c^4*sqrt((a*x - 1)/(a*x + 1))) - 40320*sqrt((a*x - 1)
/(a*x + 1))/(a^2*c^4*((a*x - 1)/(a*x + 1) - 1)) + 105*((a*x - 1)*a^4*c^8*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1) +
30*a^4*c^8*sqrt((a*x - 1)/(a*x + 1)))/(a^6*c^12))

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maple [B]  time = 0.08, size = 766, normalized size = 2.33 \[ -\frac {-138915 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{9} a^{9}-120960 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{9} a^{10}+98595 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{7} a^{7}+416745 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{8} a^{8}+362880 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{8} a^{9}-75113 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{6} a^{6}-240861 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}-1111320 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{6} a^{6}-967680 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{6} a^{7}+178863 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}+833490 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{5} a^{5}+725760 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{5} a^{6}+252497 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{3} a^{3}+833490 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}+725760 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{4} a^{5}-182307 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}-1111320 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}-967680 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-101271 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a +74077 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+416745 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +362880 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-138915 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-120960 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{40320 a \sqrt {a^{2}}\, \left (a x -1\right )^{4} c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right )^{4} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{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]

-1/40320*(-138915*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^9*a^9-120960*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(
1/2))/(a^2)^(1/2))*x^9*a^10+98595*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)*x^7*a^7+416745*((a*x-1)*(a*x+1))^(1/2)*(
a^2)^(1/2)*x^8*a^8+362880*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^8*a^9-75113*((a*x-1)*(
a*x+1))^(3/2)*(a^2)^(1/2)*x^6*a^6-240861*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)*x^5*a^5-1111320*((a*x-1)*(a*x+1))
^(1/2)*(a^2)^(1/2)*x^6*a^6-967680*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^6*a^7+178863*(
(a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)*x^4*a^4+833490*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^5*a^5+725760*ln((a^2*x
+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^5*a^6+252497*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3+
833490*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4+725760*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)
^(1/2))*x^4*a^5-182307*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2-1111320*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)
*x^3*a^3-967680*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4-101271*(a^2)^(1/2)*((a*x-1
)*(a*x+1))^(3/2)*x*a+74077*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)+416745*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x*a+
362880*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x*a^2-138915*((a*x-1)*(a*x+1))^(1/2)*(a^2)^
(1/2)-120960*a*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2)))/a/(a^2)^(1/2)/(a*x-1)^4/c^4/((a*x-
1)*(a*x+1))^(1/2)/(a*x+1)^4/((a*x-1)/(a*x+1))^(3/2)

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maxima [A]  time = 0.32, size = 226, normalized size = 0.69 \[ \frac {1}{20160} \, a {\left (\frac {\frac {415 \, {\left (a x - 1\right )}}{a x + 1} + \frac {2511 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {11739 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {80745 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - \frac {135765 \, {\left (a x - 1\right )}^{5}}{{\left (a x + 1\right )}^{5}} + 35}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}}} + \frac {105 \, {\left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 30 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} + \frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]

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="maxima")

[Out]

1/20160*a*((415*(a*x - 1)/(a*x + 1) + 2511*(a*x - 1)^2/(a*x + 1)^2 + 11739*(a*x - 1)^3/(a*x + 1)^3 + 80745*(a*
x - 1)^4/(a*x + 1)^4 - 135765*(a*x - 1)^5/(a*x + 1)^5 + 35)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(11/2) - a^2*c^4*((
a*x - 1)/(a*x + 1))^(9/2)) + 105*(((a*x - 1)/(a*x + 1))^(3/2) + 30*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^4) + 6048
0*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 60480*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^4))

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mupad [B]  time = 0.11, size = 203, normalized size = 0.62 \[ \frac {5\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{32\,a\,c^4}-\frac {\frac {279\,{\left (a\,x-1\right )}^2}{35\,{\left (a\,x+1\right )}^2}+\frac {559\,{\left (a\,x-1\right )}^3}{15\,{\left (a\,x+1\right )}^3}+\frac {769\,{\left (a\,x-1\right )}^4}{3\,{\left (a\,x+1\right )}^4}-\frac {431\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {83\,\left (a\,x-1\right )}{63\,\left (a\,x+1\right )}+\frac {1}{9}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}-64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{192\,a\,c^4}-\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*((a*x - 1)/(a*x + 1))^(1/2))/(32*a*c^4) - ((279*(a*x - 1)^2)/(35*(a*x + 1)^2) + (559*(a*x - 1)^3)/(15*(a*x
+ 1)^3) + (769*(a*x - 1)^4)/(3*(a*x + 1)^4) - (431*(a*x - 1)^5)/(a*x + 1)^5 + (83*(a*x - 1))/(63*(a*x + 1)) +
1/9)/(64*a*c^4*((a*x - 1)/(a*x + 1))^(9/2) - 64*a*c^4*((a*x - 1)/(a*x + 1))^(11/2)) + ((a*x - 1)/(a*x + 1))^(3
/2)/(192*a*c^4) - (atan(((a*x - 1)/(a*x + 1))^(1/2)*1i)*6i)/(a*c^4)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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