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

Optimal. Leaf size=327 \[ \frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {\frac {1}{a x}+1}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (\frac {1}{a x}+1\right )^{7/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^4} \]

[Out]

-4/3/a/c^4/(1-1/a/x)^(3/2)/(1+1/a/x)^(9/2)+x/c^4/(1-1/a/x)^(3/2)/(1+1/a/x)^(9/2)-3*arctanh((1-1/a/x)^(1/2)*(1+
1/a/x)^(1/2))/a/c^4-5/a/c^4/(1+1/a/x)^(9/2)/(1-1/a/x)^(1/2)+28/9*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(9/2)+139/63*
(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(7/2)+202/105*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(5/2)+719/315*(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 = 0.22, antiderivative size = 327, 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} \[ \frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {\frac {1}{a x}+1}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (\frac {1}{a x}+1\right )^{7/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^4),x]

[Out]

-4/(3*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)) - 5/(a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)) + (28*S
qrt[1 - 1/(a*x)])/(9*a*c^4*(1 + 1/(a*x))^(9/2)) + (139*Sqrt[1 - 1/(a*x)])/(63*a*c^4*(1 + 1/(a*x))^(7/2)) + (20
2*Sqrt[1 - 1/(a*x)])/(105*a*c^4*(1 + 1/(a*x))^(5/2)) + (719*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))^(3/2)*(1 + 1/(a*x))^(9/2)) -
(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 )^{5/2} \left (1+\frac {x}{a}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{c^4}\\ &=\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {3}{a}-\frac {7 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {a \operatorname {Subst}\left (\int \frac {-\frac {9}{a^2}+\frac {24 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{3 c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^2 \operatorname {Subst}\left (\int \frac {\frac {9}{a^3}-\frac {75 x}{a^4}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{3 c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^3 \operatorname {Subst}\left (\int \frac {\frac {81}{a^4}-\frac {336 x}{a^5}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{27 c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^4 \operatorname {Subst}\left (\int \frac {\frac {567}{a^5}-\frac {1251 x}{a^6}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{189 c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^5 \operatorname {Subst}\left (\int \frac {\frac {2835}{a^6}-\frac {3636 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 {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \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 )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^6 \operatorname {Subst}\left (\int \frac {\frac {8505}{a^7}-\frac {6471 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 {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \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 )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\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 {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \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 )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\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 {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \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 )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\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 {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \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 )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\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 = 0.29, size = 117, normalized size = 0.36 \[ \frac {\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (315 a^7 x^7+2669 a^6 x^6+2967 a^5 x^5-4029 a^4 x^4-7399 a^3 x^3-339 a^2 x^2+4047 a x+1664\right )}{315 (a x-1)^2 (a x+1)^5}-3 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{a c^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(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 + 2669
*a^6*x^6 + 315*a^7*x^7))/(315*(-1 + a*x)^2*(1 + a*x)^5) - 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)

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fricas [A]  time = 0.61, size = 275, normalized size = 0.84 \[ -\frac {945 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 945 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (315 \, a^{7} x^{7} + 2669 \, a^{6} x^{6} + 2967 \, a^{5} x^{5} - 4029 \, a^{4} x^{4} - 7399 \, a^{3} x^{3} - 339 \, a^{2} x^{2} + 4047 \, a x + 1664\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((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 + 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) - 945*(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)
- (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)*sq
rt((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 [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.08, size = 766, normalized size = 2.34 \[ -\frac {\left (-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 )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{40320 a \left (a x +1\right )^{4} \sqrt {a^{2}}\, c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.33, size = 231, normalized size = 0.71 \[ \frac {1}{20160} \, a {\left (\frac {105 \, {\left (\frac {29 \, {\left (a x - 1\right )}}{a x + 1} - \frac {414 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1\right )}}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {35 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 450 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 2961 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 14700 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 95445 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4}} - \frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} + \frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20160*a*(105*(29*(a*x - 1)/(a*x + 1) - 414*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(5/2)
- a^2*c^4*((a*x - 1)/(a*x + 1))^(3/2)) + (35*((a*x - 1)/(a*x + 1))^(9/2) + 450*((a*x - 1)/(a*x + 1))^(7/2) +
2961*((a*x - 1)/(a*x + 1))^(5/2) + 14700*((a*x - 1)/(a*x + 1))^(3/2) + 95445*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c
^4) - 60480*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 = 0.06, size = 224, normalized size = 0.69 \[ \frac {303\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64\,a\,c^4}-\frac {\frac {29\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {138\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}+\frac {35\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{48\,a\,c^4}+\frac {47\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{320\,a\,c^4}+\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{224\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{576\,a\,c^4}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(303*((a*x - 1)/(a*x + 1))^(1/2))/(64*a*c^4) - ((29*(a*x - 1))/(3*(a*x + 1)) - (138*(a*x - 1)^2)/(a*x + 1)^2 +
1/3)/(64*a*c^4*((a*x - 1)/(a*x + 1))^(3/2) - 64*a*c^4*((a*x - 1)/(a*x + 1))^(5/2)) + (35*((a*x - 1)/(a*x + 1)
)^(3/2))/(48*a*c^4) + (47*((a*x - 1)/(a*x + 1))^(5/2))/(320*a*c^4) + (5*((a*x - 1)/(a*x + 1))^(7/2))/(224*a*c^
4) + ((a*x - 1)/(a*x + 1))^(9/2)/(576*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(((a*x-1)/(a*x+1))**(3/2)/(c-c/a**2/x**2)**4,x)

[Out]

Timed out

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