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

Optimal. Leaf size=254 \[ \frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {\frac {1}{a x}+1}}+\frac {21 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {34}{5 a c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {29}{15 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {6}{5 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^3} \]

[Out]

-6/5/a/c^3/(1-1/a/x)^(5/2)/(1+1/a/x)^(3/2)-29/15/a/c^3/(1-1/a/x)^(3/2)/(1+1/a/x)^(3/2)+x/c^3/(1-1/a/x)^(5/2)/(
1+1/a/x)^(3/2)+arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c^3-34/5/a/c^3/(1+1/a/x)^(3/2)/(1-1/a/x)^(1/2)+21/5*
(1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(3/2)+16/5*(1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(1/2)

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Rubi [A]  time = 0.17, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {\frac {1}{a x}+1}}+\frac {21 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {34}{5 a c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {29}{15 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {6}{5 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-6/(5*a*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) - 29/(15*a*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)) -
34/(5*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) + (21*Sqrt[1 - 1/(a*x)])/(5*a*c^3*(1 + 1/(a*x))^(3/2)) + (
16*Sqrt[1 - 1/(a*x)])/(5*a*c^3*Sqrt[1 + 1/(a*x)]) + x/(c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) + ArcTanh[
Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^3)

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 )^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{a}-\frac {5 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {6}{5 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a \operatorname {Subst}\left (\int \frac {\frac {5}{a^2}+\frac {24 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 c^3}\\ &=-\frac {6}{5 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{15 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {a^2 \operatorname {Subst}\left (\int \frac {-\frac {15}{a^3}-\frac {87 x}{a^4}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{15 c^3}\\ &=-\frac {6}{5 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{15 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {34}{5 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a^3 \operatorname {Subst}\left (\int \frac {\frac {15}{a^4}+\frac {204 x}{a^5}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{15 c^3}\\ &=-\frac {6}{5 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{15 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {34}{5 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {21 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a^4 \operatorname {Subst}\left (\int \frac {\frac {45}{a^5}+\frac {189 x}{a^6}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{45 c^3}\\ &=-\frac {6}{5 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{15 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {34}{5 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {21 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a^5 \operatorname {Subst}\left (\int \frac {45}{a^6 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{45 c^3}\\ &=-\frac {6}{5 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{15 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {34}{5 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {21 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\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^3}\\ &=-\frac {6}{5 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{15 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {34}{5 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {21 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\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^3}\\ &=-\frac {6}{5 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{15 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {34}{5 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {21 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^3}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 99, normalized size = 0.39 \[ \frac {\log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (15 a^5 x^5-38 a^4 x^4-52 a^3 x^3+87 a^2 x^2+33 a x-48\right )}{15 (a x-1)^3 (a x+1)^2}}{a c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-48 + 33*a*x + 87*a^2*x^2 - 52*a^3*x^3 - 38*a^4*x^4 + 15*a^5*x^5))/(15*(-1 + a*x)
^3*(1 + a*x)^2) + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^3)

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fricas [A]  time = 0.58, size = 178, normalized size = 0.70 \[ \frac {15 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (15 \, a^{5} x^{5} - 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} + 87 \, a^{2} x^{2} + 33 \, a x - 48\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

1/15*(15*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*(a^4*x^4 - 2*a^3*x^3 + 2*a*
x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (15*a^5*x^5 - 38*a^4*x^4 - 52*a^3*x^3 + 87*a^2*x^2 + 33*a*x - 48)*
sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^2*c^3*x - a*c^3)

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giac [A]  time = 0.18, size = 232, normalized size = 0.91 \[ \frac {1}{240} \, a {\left (\frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {240 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{3}} - \frac {{\left (a x + 1\right )}^{2} {\left (\frac {40 \, {\left (a x - 1\right )}}{a x + 1} + \frac {450 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 3\right )}}{{\left (a x - 1\right )}^{2} a^{2} c^{3} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {480 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3} {\left (\frac {a x - 1}{a x + 1} - 1\right )}} + \frac {5 \, {\left (\frac {{\left (a x - 1\right )} a^{4} c^{6} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + 24 \, a^{4} c^{6} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{6} c^{9}}\right )} \]

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)^3,x, algorithm="giac")

[Out]

1/240*a*(240*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 240*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a^2*c
^3) - (a*x + 1)^2*(40*(a*x - 1)/(a*x + 1) + 450*(a*x - 1)^2/(a*x + 1)^2 + 3)/((a*x - 1)^2*a^2*c^3*sqrt((a*x -
1)/(a*x + 1))) - 480*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^3*((a*x - 1)/(a*x + 1) - 1)) + 5*((a*x - 1)*a^4*c^6*sqrt
((a*x - 1)/(a*x + 1))/(a*x + 1) + 24*a^4*c^6*sqrt((a*x - 1)/(a*x + 1)))/(a^6*c^9))

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maple [B]  time = 0.07, size = 714, normalized size = 2.81 \[ -\frac {-525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{7} a^{7}-240 \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^{7} a^{8}+285 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}+525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{6} a^{6}+240 \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}+83 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}+1575 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{5} a^{5}+720 \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}-218 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{3} a^{3}-1575 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}-720 \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}-342 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}-1575 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}-720 \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}-3 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a +1575 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+720 \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^{2} a^{3}+243 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +240 \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}-525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-240 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 )}{240 a \left (a x +1\right )^{3} \sqrt {a^{2}}\, \left (a x -1\right )^{3} c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}} \]

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)^3,x)

[Out]

-1/240*(-525*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^7*a^7-240*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a
^2)^(1/2))*x^7*a^8+285*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)*x^5*a^5+525*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^6
*a^6+240*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^6*a^7+83*((a*x-1)*(a*x+1))^(3/2)*(a^2)^
(1/2)*x^4*a^4+1575*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^5*a^5+720*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/
2))/(a^2)^(1/2))*x^5*a^6-218*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3-1575*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1
/2)*x^4*a^4-720*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^4*a^5-342*(a^2)^(1/2)*((a*x-1)*(
a*x+1))^(3/2)*x^2*a^2-1575*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3-720*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(
a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4-3*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a+1575*((a*x-1)*(a*x+1))^(1/2)*(a^2)^
(1/2)*x^2*a^2+720*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3+243*((a*x-1)*(a*x+1))^(3
/2)*(a^2)^(1/2)+525*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x*a+240*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))
/(a^2)^(1/2))*x*a^2-525*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)-240*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)^3/(a^2)^(1/2)/(a*x-1)^3/c^3/((a*x-1)*(a*x+1))^(1/2)/((a*x-1)/(a*x+1))^(1/2)

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maxima [A]  time = 0.32, size = 194, normalized size = 0.76 \[ \frac {1}{240} \, a {\left (\frac {\frac {37 \, {\left (a x - 1\right )}}{a x + 1} + \frac {410 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {930 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {5 \, {\left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 24 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} c^{3}} + \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \]

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)^3,x, algorithm="maxima")

[Out]

1/240*a*((37*(a*x - 1)/(a*x + 1) + 410*(a*x - 1)^2/(a*x + 1)^2 - 930*(a*x - 1)^3/(a*x + 1)^3 + 3)/(a^2*c^3*((a
*x - 1)/(a*x + 1))^(7/2) - a^2*c^3*((a*x - 1)/(a*x + 1))^(5/2)) + 5*(((a*x - 1)/(a*x + 1))^(3/2) + 24*sqrt((a*
x - 1)/(a*x + 1)))/(a^2*c^3) + 240*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 240*log(sqrt((a*x - 1)/(a*x
+ 1)) - 1)/(a^2*c^3))

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mupad [B]  time = 0.10, size = 171, normalized size = 0.67 \[ \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a\,c^3}-\frac {\frac {82\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {62\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {37\,\left (a\,x-1\right )}{15\,\left (a\,x+1\right )}+\frac {1}{5}}{16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{48\,a\,c^3}-\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,c^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*x - 1)/(a*x + 1))^(1/2)/(2*a*c^3) - ((82*(a*x - 1)^2)/(3*(a*x + 1)^2) - (62*(a*x - 1)^3)/(a*x + 1)^3 + (37
*(a*x - 1))/(15*(a*x + 1)) + 1/5)/(16*a*c^3*((a*x - 1)/(a*x + 1))^(5/2) - 16*a*c^3*((a*x - 1)/(a*x + 1))^(7/2)
) + ((a*x - 1)/(a*x + 1))^(3/2)/(48*a*c^3) - (atan(((a*x - 1)/(a*x + 1))^(1/2)*1i)*2i)/(a*c^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{6} \int \frac {x^{6}}{a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 3 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{3}} \]

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)**3,x)

[Out]

a**6*Integral(x**6/(a**6*x**6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 3*a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)
) + 3*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**3

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