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

Optimal. Leaf size=181 \[ \frac {x \sqrt {\frac {1}{a x}+1}}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {24 \sqrt {\frac {1}{a x}+1}}{5 a c^2 \sqrt {1-\frac {1}{a x}}}-\frac {9 \sqrt {\frac {1}{a x}+1}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {6 \sqrt {\frac {1}{a x}+1}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^2} \]

[Out]

3*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c^2-6/5*(1+1/a/x)^(1/2)/a/c^2/(1-1/a/x)^(5/2)-9/5*(1+1/a/x)^(1/2)
/a/c^2/(1-1/a/x)^(3/2)+x*(1+1/a/x)^(1/2)/c^2/(1-1/a/x)^(5/2)-24/5*(1+1/a/x)^(1/2)/a/c^2/(1-1/a/x)^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6194, 103, 21, 99, 152, 12, 92, 208} \[ \frac {x \sqrt {\frac {1}{a x}+1}}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {24 \sqrt {\frac {1}{a x}+1}}{5 a c^2 \sqrt {1-\frac {1}{a x}}}-\frac {9 \sqrt {\frac {1}{a x}+1}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {6 \sqrt {\frac {1}{a x}+1}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[1 + 1/(a*x)])/(5*a*c^2*(1 - 1/(a*x))^(5/2)) - (9*Sqrt[1 + 1/(a*x)])/(5*a*c^2*(1 - 1/(a*x))^(3/2)) - (
24*Sqrt[1 + 1/(a*x)])/(5*a*c^2*Sqrt[1 - 1/(a*x)]) + (Sqrt[1 + 1/(a*x)]*x)/(c^2*(1 - 1/(a*x))^(5/2)) + (3*ArcTa
nh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^2)

Rule 12

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + 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 99

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

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 )^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^{7/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{a}-\frac {3 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{7/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x \left (1-\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{a c^2}\\ &=-\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {6 \operatorname {Subst}\left (\int \frac {-\frac {5}{2}-\frac {2 x}{a}}{x \left (1-\frac {x}{a}\right )^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 a c^2}\\ &=-\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {9 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {2 \operatorname {Subst}\left (\int \frac {\frac {15}{2 a}+\frac {9 x}{2 a^2}}{x \left (1-\frac {x}{a}\right )^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 c^2}\\ &=-\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {9 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {24 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {(2 a) \operatorname {Subst}\left (\int -\frac {15}{2 a^2 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 c^2}\\ &=-\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {9 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {24 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\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^2}\\ &=-\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {9 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {24 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\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^2}\\ &=-\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {9 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {24 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^2}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 78, normalized size = 0.43 \[ \frac {3 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (5 a^3 x^3-39 a^2 x^2+57 a x-24\right )}{5 (a x-1)^3}}{a c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-24 + 57*a*x - 39*a^2*x^2 + 5*a^3*x^3))/(5*(-1 + a*x)^3) + 3*Log[(1 + Sqrt[1 - 1/
(a^2*x^2)])*x])/(a*c^2)

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fricas [A]  time = 0.71, size = 170, normalized size = 0.94 \[ \frac {15 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (5 \, a^{4} x^{4} - 34 \, a^{3} x^{3} + 18 \, a^{2} x^{2} + 33 \, a x - 24\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/5*(15*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*(a^3*x^3 - 3*a^2*x^2 + 3*a*x
- 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (5*a^4*x^4 - 34*a^3*x^3 + 18*a^2*x^2 + 33*a*x - 24)*sqrt((a*x - 1)/
(a*x + 1)))/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)

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giac [A]  time = 0.16, size = 166, normalized size = 0.92 \[ \frac {1}{20} \, a {\left (\frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {60 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{2}} - \frac {{\left (a x + 1\right )}^{2} {\left (\frac {10 \, {\left (a x - 1\right )}}{a x + 1} + \frac {85 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1\right )}}{{\left (a x - 1\right )}^{2} a^{2} c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {40 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]

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)^2,x, algorithm="giac")

[Out]

1/20*a*(60*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^2) - 60*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a^2*c^2)
- (a*x + 1)^2*(10*(a*x - 1)/(a*x + 1) + 85*(a*x - 1)^2/(a*x + 1)^2 + 1)/((a*x - 1)^2*a^2*c^2*sqrt((a*x - 1)/(
a*x + 1))) - 40*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^2*((a*x - 1)/(a*x + 1) - 1)))

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maple [B]  time = 0.06, size = 438, normalized size = 2.42 \[ -\frac {-125 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}-120 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{4} a^{5}+85 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}+500 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}+480 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-148 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -750 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}-720 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+67 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+500 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +480 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-125 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-120 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{40 a \sqrt {a^{2}}\, \left (a x -1\right )^{2} c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}} \]

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)^2,x)

[Out]

-1/40*(-125*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4-120*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^
2)^(1/2))*x^4*a^5+85*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2+500*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a
^3+480*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4-148*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(
3/2)*x*a-750*((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+67*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)+500*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x*a+480*ln((
a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x*a^2-125*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)-120*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)^2/c^2/((a*x-1)*(a*x+1))^(1/2)
/(a*x+1)/((a*x-1)/(a*x+1))^(3/2)

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maxima [A]  time = 0.32, size = 153, normalized size = 0.85 \[ \frac {1}{20} \, a {\left (\frac {\frac {9 \, {\left (a x - 1\right )}}{a x + 1} + \frac {75 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {125 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \]

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)^2,x, algorithm="maxima")

[Out]

1/20*a*((9*(a*x - 1)/(a*x + 1) + 75*(a*x - 1)^2/(a*x + 1)^2 - 125*(a*x - 1)^3/(a*x + 1)^3 + 1)/(a^2*c^2*((a*x
- 1)/(a*x + 1))^(7/2) - a^2*c^2*((a*x - 1)/(a*x + 1))^(5/2)) + 60*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^2)
- 60*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^2))

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mupad [B]  time = 0.09, size = 121, normalized size = 0.67 \[ \frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^2}-\frac {\frac {15\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {25\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {9\,\left (a\,x-1\right )}{5\,\left (a\,x+1\right )}+\frac {1}{5}}{4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^2) - ((15*(a*x - 1)^2)/(a*x + 1)^2 - (25*(a*x - 1)^3)/(a*x + 1)^3
+ (9*(a*x - 1))/(5*(a*x + 1)) + 1/5)/(4*a*c^2*((a*x - 1)/(a*x + 1))^(5/2) - 4*a*c^2*((a*x - 1)/(a*x + 1))^(7/2
))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{4} \int \frac {x^{4}}{\frac {a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {2 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {2 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{2}} \]

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)**2,x)

[Out]

a**4*Integral(x**4/(a**5*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(
a*x + 1))/(a*x + 1) - 2*a**3*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) + 2*a**2*x**2*sqrt(a*x/(a*x + 1)
- 1/(a*x + 1))/(a*x + 1) + 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)/c**2

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