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

Optimal. Leaf size=328 \[ \frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {\frac {1}{a x}+1}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^4} \]

[Out]

-8/7/a/c^4/(1-1/a/x)^(7/2)/(1+1/a/x)^(5/2)-11/7/a/c^4/(1-1/a/x)^(5/2)/(1+1/a/x)^(5/2)-62/21/a/c^4/(1-1/a/x)^(3
/2)/(1+1/a/x)^(5/2)+x/c^4/(1-1/a/x)^(7/2)/(1+1/a/x)^(5/2)+arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c^4-269/2
1/a/c^4/(1+1/a/x)^(5/2)/(1-1/a/x)^(1/2)+262/35*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(5/2)+163/35*(1-1/a/x)^(1/2)/a/
c^4/(1+1/a/x)^(3/2)+128/35*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(1/2)

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Rubi [A]  time = 0.23, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6194, 103, 152, 12, 92, 208} \[ \frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {\frac {1}{a x}+1}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-8/(7*a*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)) - 11/(7*a*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)) -
62/(21*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)) - 269/(21*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)) +
(262*Sqrt[1 - 1/(a*x)])/(35*a*c^4*(1 + 1/(a*x))^(5/2)) + (163*Sqrt[1 - 1/(a*x)])/(35*a*c^4*(1 + 1/(a*x))^(3/2
)) + (128*Sqrt[1 - 1/(a*x)])/(35*a*c^4*Sqrt[1 + 1/(a*x)]) + x/(c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)) +
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^{\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 )^{9/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^4}\\ &=\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{a}-\frac {7 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{9/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^4}\\ &=-\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {a \operatorname {Subst}\left (\int \frac {\frac {7}{a^2}+\frac {48 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 c^4}\\ &=-\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^2 \operatorname {Subst}\left (\int \frac {-\frac {35}{a^3}-\frac {275 x}{a^4}}{x \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{35 c^4}\\ &=-\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {a^3 \operatorname {Subst}\left (\int \frac {\frac {105}{a^4}+\frac {1240 x}{a^5}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{105 c^4}\\ &=-\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^4 \operatorname {Subst}\left (\int \frac {-\frac {105}{a^5}-\frac {4035 x}{a^6}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{105 c^4}\\ &=-\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^5 \operatorname {Subst}\left (\int \frac {-\frac {525}{a^6}-\frac {7860 x}{a^7}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{525 c^4}\\ &=-\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^6 \operatorname {Subst}\left (\int \frac {-\frac {1575}{a^7}-\frac {7335 x}{a^8}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{1575 c^4}\\ &=-\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^7 \operatorname {Subst}\left (\int -\frac {1575}{a^8 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{1575 c^4}\\ &=-\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {\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 {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {\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 {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {\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 = 0.35, size = 115, normalized size = 0.35 \[ \frac {\log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (105 a^7 x^7-281 a^6 x^6-559 a^5 x^5+965 a^4 x^4+715 a^3 x^3-1065 a^2 x^2-279 a x+384\right )}{105 (a x-1)^4 (a x+1)^3}}{a c^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(384 - 279*a*x - 1065*a^2*x^2 + 715*a^3*x^3 + 965*a^4*x^4 - 559*a^5*x^5 - 281*a^6*
x^6 + 105*a^7*x^7))/(105*(-1 + a*x)^4*(1 + a*x)^3) + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)

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fricas [A]  time = 0.68, size = 274, normalized size = 0.84 \[ \frac {105 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (105 \, a^{7} x^{7} - 281 \, a^{6} x^{6} - 559 \, a^{5} x^{5} + 965 \, a^{4} x^{4} + 715 \, a^{3} x^{3} - 1065 \, a^{2} x^{2} - 279 \, a x + 384\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(105*(a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1
) - 105*(a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) +
(105*a^7*x^7 - 281*a^6*x^6 - 559*a^5*x^5 + 965*a^4*x^4 + 715*a^3*x^3 - 1065*a^2*x^2 - 279*a*x + 384)*sqrt((a*
x - 1)/(a*x + 1)))/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*
c^4)

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giac [A]  time = 0.17, size = 286, normalized size = 0.87 \[ \frac {1}{6720} \, a {\left (\frac {6720 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {6720 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{4}} - \frac {5 \, {\left (a x + 1\right )}^{3} {\left (\frac {42 \, {\left (a x - 1\right )}}{a x + 1} + \frac {329 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2940 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{4} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {13440 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}} + \frac {7 \, {\left (\frac {50 \, {\left (a x - 1\right )} a^{8} c^{16} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2} a^{8} c^{16} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 705 \, a^{8} c^{16} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{10} c^{20}}\right )} \]

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]

1/6720*a*(6720*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 6720*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a^
2*c^4) - 5*(a*x + 1)^3*(42*(a*x - 1)/(a*x + 1) + 329*(a*x - 1)^2/(a*x + 1)^2 + 2940*(a*x - 1)^3/(a*x + 1)^3 +
3)/((a*x - 1)^3*a^2*c^4*sqrt((a*x - 1)/(a*x + 1))) - 13440*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^4*((a*x - 1)/(a*x
+ 1) - 1)) + 7*(50*(a*x - 1)*a^8*c^16*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1) + 3*(a*x - 1)^2*a^8*c^16*sqrt((a*x -
1)/(a*x + 1))/(a*x + 1)^2 + 705*a^8*c^16*sqrt((a*x - 1)/(a*x + 1)))/(a^10*c^20))

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maple [B]  time = 0.08, size = 898, normalized size = 2.74 \[ \text {result too large to display} \]

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)

[Out]

-1/13440*(-132300*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^2*a^2+132300*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^7*a
^7-27673*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)*x^5*a^5+7705*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3-198450*(
a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^5*a^5+198450*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4-132300*((a*x-1)*
(a*x+1))^(1/2)*(a^2)^(1/2)*x^6*a^6-24295*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)*x^4*a^4-33075*((a*x-1)*(a*x+1))^(
1/2)*(a^2)^(1/2)*x*a+37095*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2+2637*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2
)*x*a+132300*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3-13440*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/
(a^2)^(1/2))*x*a^2-16077*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)+33075*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)+13440*a
*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))+53760*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/
2))/(a^2)^(1/2))*x^3*a^4-53760*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3-13440*ln((a
^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^9*a^10+13440*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^
(1/2))/(a^2)^(1/2))*x^8*a^9-33075*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^9*a^9+19635*((a*x-1)*(a*x+1))^(3/2)*(a
^2)^(1/2)*x^7*a^7+33075*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^8*a^8+2893*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)*x
^6*a^6+53760*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^7*a^8-53760*ln((a^2*x+((a*x-1)*(a*x
+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^6*a^7-80640*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))
*x^5*a^6+80640*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^4*a^5)/a/(a*x+1)^4/(a^2)^(1/2)/(a
*x-1)^4/c^4/((a*x-1)*(a*x+1))^(1/2)/((a*x-1)/(a*x+1))^(1/2)

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maxima [A]  time = 0.33, size = 230, normalized size = 0.70 \[ \frac {1}{6720} \, a {\left (\frac {5 \, {\left (\frac {39 \, {\left (a x - 1\right )}}{a x + 1} + \frac {287 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2611 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {5628 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 3\right )}}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}} + \frac {7 \, {\left (3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 50 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 705 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} + \frac {6720 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {6720 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6720*a*(5*(39*(a*x - 1)/(a*x + 1) + 287*(a*x - 1)^2/(a*x + 1)^2 + 2611*(a*x - 1)^3/(a*x + 1)^3 - 5628*(a*x -
1)^4/(a*x + 1)^4 + 3)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(9/2) - a^2*c^4*((a*x - 1)/(a*x + 1))^(7/2)) + 7*(3*((a*
x - 1)/(a*x + 1))^(5/2) + 50*((a*x - 1)/(a*x + 1))^(3/2) + 705*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^4) + 6720*log
(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 6720*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^4))

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mupad [B]  time = 0.07, size = 210, normalized size = 0.64 \[ \frac {47\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64\,a\,c^4}-\frac {\frac {41\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}+\frac {373\,{\left (a\,x-1\right )}^3}{3\,{\left (a\,x+1\right )}^3}-\frac {268\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {13\,\left (a\,x-1\right )}{7\,\left (a\,x+1\right )}+\frac {1}{7}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}-64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}+\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{96\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{320\,a\,c^4}-\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,c^4} \]

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))^(1/2)),x)

[Out]

(47*((a*x - 1)/(a*x + 1))^(1/2))/(64*a*c^4) - ((41*(a*x - 1)^2)/(3*(a*x + 1)^2) + (373*(a*x - 1)^3)/(3*(a*x +
1)^3) - (268*(a*x - 1)^4)/(a*x + 1)^4 + (13*(a*x - 1))/(7*(a*x + 1)) + 1/7)/(64*a*c^4*((a*x - 1)/(a*x + 1))^(7
/2) - 64*a*c^4*((a*x - 1)/(a*x + 1))^(9/2)) + (5*((a*x - 1)/(a*x + 1))^(3/2))/(96*a*c^4) + ((a*x - 1)/(a*x + 1
))^(5/2)/(320*a*c^4) - (atan(((a*x - 1)/(a*x + 1))^(1/2)*1i)*2i)/(a*c^4)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{8} \int \frac {x^{8}}{a^{8} x^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 6 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{4}} \]

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]

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

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