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

Optimal. Leaf size=255 \[ \frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {\frac {1}{a x}+1}}-\frac {281}{35 a c^3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {88}{35 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}-\frac {53}{35 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}-\frac {8}{7 a c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^3} \]

[Out]

3*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c^3-8/7/a/c^3/(1-1/a/x)^(7/2)/(1+1/a/x)^(1/2)-53/35/a/c^3/(1-1/a/
x)^(5/2)/(1+1/a/x)^(1/2)-88/35/a/c^3/(1-1/a/x)^(3/2)/(1+1/a/x)^(1/2)+x/c^3/(1-1/a/x)^(7/2)/(1+1/a/x)^(1/2)-281
/35/a/c^3/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)+176/35*(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 = 255, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {\frac {1}{a x}+1}}-\frac {281}{35 a c^3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {88}{35 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}-\frac {53}{35 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}-\frac {8}{7 a c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-8/(7*a*c^3*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]) - 53/(35*a*c^3*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]) - 88/
(35*a*c^3*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) - 281/(35*a*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (176*S
qrt[1 - 1/(a*x)])/(35*a*c^3*Sqrt[1 + 1/(a*x)]) + x/(c^3*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]) + (3*ArcTanh[Sq
rt[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^{3 \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 )^{9/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{a}-\frac {5 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{9/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {8}{7 a c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}-\frac {a \operatorname {Subst}\left (\int \frac {\frac {21}{a^2}+\frac {32 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{7 c^3}\\ &=-\frac {8}{7 a c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}-\frac {53}{35 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {a^2 \operatorname {Subst}\left (\int \frac {-\frac {105}{a^3}-\frac {159 x}{a^4}}{x \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{35 c^3}\\ &=-\frac {8}{7 a c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}-\frac {53}{35 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {88}{35 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}-\frac {a^3 \operatorname {Subst}\left (\int \frac {\frac {315}{a^4}+\frac {528 x}{a^5}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{105 c^3}\\ &=-\frac {8}{7 a c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}-\frac {53}{35 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {88}{35 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {281}{35 a c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {a^4 \operatorname {Subst}\left (\int \frac {-\frac {315}{a^5}-\frac {843 x}{a^6}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{105 c^3}\\ &=-\frac {8}{7 a c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}-\frac {53}{35 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {88}{35 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {281}{35 a c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {a^5 \operatorname {Subst}\left (\int -\frac {315}{a^6 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{105 c^3}\\ &=-\frac {8}{7 a c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}-\frac {53}{35 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {88}{35 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {281}{35 a c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}-\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^3}\\ &=-\frac {8}{7 a c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}-\frac {53}{35 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {88}{35 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {281}{35 a c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\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^3}\\ &=-\frac {8}{7 a c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}-\frac {53}{35 a c^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {88}{35 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {281}{35 a c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {3 \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 = 0.24, size = 101, normalized size = 0.40 \[ \frac {3 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (35 a^5 x^5-286 a^4 x^4+368 a^3 x^3+125 a^2 x^2-423 a x+176\right )}{35 (a x-1)^4 (a x+1)}}{a c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(176 - 423*a*x + 125*a^2*x^2 + 368*a^3*x^3 - 286*a^4*x^4 + 35*a^5*x^5))/(35*(-1 +
a*x)^4*(1 + a*x)) + 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^3)

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fricas [A]  time = 0.55, size = 204, normalized size = 0.80 \[ \frac {105 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (35 \, a^{5} x^{5} - 286 \, a^{4} x^{4} + 368 \, a^{3} x^{3} + 125 \, a^{2} x^{2} - 423 \, a x + 176\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(105*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 105*(a^4*x^4 - 4*
a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (35*a^5*x^5 - 286*a^4*x^4 + 368*a^3*x^3
+ 125*a^2*x^2 - 423*a*x + 176)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2
*c^3*x + a*c^3)

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giac [A]  time = 0.17, size = 205, normalized size = 0.80 \[ \frac {1}{560} \, a {\left (\frac {1680 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {1680 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{3}} - \frac {{\left (a x + 1\right )}^{3} {\left (\frac {56 \, {\left (a x - 1\right )}}{a x + 1} + \frac {350 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2520 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 5\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{3} \sqrt {\frac {a x - 1}{a x + 1}}} + \frac {35 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} - \frac {1120 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]

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)^3,x, algorithm="giac")

[Out]

1/560*a*(1680*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 1680*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a^2
*c^3) - (a*x + 1)^3*(56*(a*x - 1)/(a*x + 1) + 350*(a*x - 1)^2/(a*x + 1)^2 + 2520*(a*x - 1)^3/(a*x + 1)^3 + 5)/
((a*x - 1)^3*a^2*c^3*sqrt((a*x - 1)/(a*x + 1))) + 35*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^3) - 1120*sqrt((a*x - 1)
/(a*x + 1))/(a^2*c^3*((a*x - 1)/(a*x + 1) - 1)))

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maple [B]  time = 0.07, size = 714, normalized size = 2.80 \[ -\frac {-3675 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{7} a^{7}-3360 \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}+2555 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}+11025 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{6} a^{6}+10080 \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}-1873 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}-3675 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{5} a^{5}-3360 \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}-4426 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{3} a^{3}-18375 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}-16800 \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}+3350 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}+18375 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}+16800 \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}+2511 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a +3675 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+3360 \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}-1957 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-11025 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -10080 \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}+3675 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+3360 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 )}{1120 a \sqrt {a^{2}}\, \left (a x -1\right )^{3} c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right )^{3} \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)^3,x)

[Out]

-1/1120*(-3675*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^7*a^7-3360*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))
/(a^2)^(1/2))*x^7*a^8+2555*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)*x^5*a^5+11025*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/
2)*x^6*a^6+10080*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^6*a^7-1873*((a*x-1)*(a*x+1))^(3
/2)*(a^2)^(1/2)*x^4*a^4-3675*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^5*a^5-3360*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2
)*(a^2)^(1/2))/(a^2)^(1/2))*x^5*a^6-4426*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3-18375*(a^2)^(1/2)*((a*x-1
)*(a*x+1))^(1/2)*x^4*a^4-16800*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^4*a^5+3350*(a^2)^
(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2+18375*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3+16800*ln((a^2*x+((a*x-
1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4+2511*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a+3675*((a*x-1)
*(a*x+1))^(1/2)*(a^2)^(1/2)*x^2*a^2+3360*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3-1
957*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-11025*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x*a-10080*ln((a^2*x+((a*x-1)
*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x*a^2+3675*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)+3360*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)^3/c^3/((a*x-1)*(a*x+1))^(1/2)/(a*x+1)^3/(
(a*x-1)/(a*x+1))^(3/2)

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maxima [A]  time = 0.33, size = 192, normalized size = 0.75 \[ \frac {1}{560} \, a {\left (\frac {\frac {51 \, {\left (a x - 1\right )}}{a x + 1} + \frac {294 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2170 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {3640 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 5}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}} + \frac {1680 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {1680 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}} + \frac {35 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}}\right )} \]

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)^3,x, algorithm="maxima")

[Out]

1/560*a*((51*(a*x - 1)/(a*x + 1) + 294*(a*x - 1)^2/(a*x + 1)^2 + 2170*(a*x - 1)^3/(a*x + 1)^3 - 3640*(a*x - 1)
^4/(a*x + 1)^4 + 5)/(a^2*c^3*((a*x - 1)/(a*x + 1))^(9/2) - a^2*c^3*((a*x - 1)/(a*x + 1))^(7/2)) + 1680*log(sqr
t((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 1680*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^3) + 35*sqrt((a*x - 1)/
(a*x + 1))/(a^2*c^3))

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mupad [B]  time = 1.41, size = 160, normalized size = 0.63 \[ \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{16\,a\,c^3}-\frac {\frac {42\,{\left (a\,x-1\right )}^2}{5\,{\left (a\,x+1\right )}^2}+\frac {62\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {104\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {51\,\left (a\,x-1\right )}{35\,\left (a\,x+1\right )}+\frac {1}{7}}{16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}-16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}+\frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^3} \]

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))^(3/2)),x)

[Out]

((a*x - 1)/(a*x + 1))^(1/2)/(16*a*c^3) - ((42*(a*x - 1)^2)/(5*(a*x + 1)^2) + (62*(a*x - 1)^3)/(a*x + 1)^3 - (1
04*(a*x - 1)^4)/(a*x + 1)^4 + (51*(a*x - 1))/(35*(a*x + 1)) + 1/7)/(16*a*c^3*((a*x - 1)/(a*x + 1))^(7/2) - 16*
a*c^3*((a*x - 1)/(a*x + 1))^(9/2)) + (6*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{6} \int \frac {x^{6}}{\frac {a^{7} x^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {3 a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {3 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {3 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{3}} \]

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)**3,x)

[Out]

a**6*Integral(x**6/(a**7*x**7*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - a**6*x**6*sqrt(a*x/(a*x + 1) - 1/(
a*x + 1))/(a*x + 1) - 3*a**5*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) + 3*a**4*x**4*sqrt(a*x/(a*x + 1)
- 1/(a*x + 1))/(a*x + 1) + 3*a**3*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 3*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**3

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