\(\int \frac {1}{x^3 (a+b \text {sech}^{-1}(c x))^3} \, dx\) [67]
Optimal result
Integrand size = 14, antiderivative size = 112 \[
\int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\frac {c^2 \cosh \left (2 \text {sech}^{-1}(c x)\right )}{2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b^3}+\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b^3}
\]
[Out]
1/2*c^2*cosh(2*arcsech(c*x))/b^2/(a+b*arcsech(c*x))-c^2*cosh(2*a/b)*Shi(2*a/b+2*arcsech(c*x))/b^3+c^2*Chi(2*a/
b+2*arcsech(c*x))*sinh(2*a/b)/b^3+1/4*c^2*sinh(2*arcsech(c*x))/b/(a+b*arcsech(c*x))^2
Rubi [A] (verified)
Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6420, 5556, 12, 3378, 3384,
3379, 3382} \[
\int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\frac {c^2 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b^3}-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b^3}+\frac {c^2 \cosh \left (2 \text {sech}^{-1}(c x)\right )}{2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2}
\]
[In]
Int[1/(x^3*(a + b*ArcSech[c*x])^3),x]
[Out]
(c^2*Cosh[2*ArcSech[c*x]])/(2*b^2*(a + b*ArcSech[c*x])) + (c^2*CoshIntegral[(2*a)/b + 2*ArcSech[c*x]]*Sinh[(2*
a)/b])/b^3 + (c^2*Sinh[2*ArcSech[c*x]])/(4*b*(a + b*ArcSech[c*x])^2) - (c^2*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b
+ 2*ArcSech[c*x]])/b^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 3378
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]
Rule 3379
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Rule 3382
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Rule 3384
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]
Rule 5556
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]
Rule 6420
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])
Rubi steps \begin{align*}
\text {integral}& = -\left (c^2 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{(a+b x)^3} \, dx,x,\text {sech}^{-1}(c x)\right )\right ) \\ & = -\left (c^2 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 (a+b x)^3} \, dx,x,\text {sech}^{-1}(c x)\right )\right ) \\ & = -\left (\frac {1}{2} c^2 \text {Subst}\left (\int \frac {\sinh (2 x)}{(a+b x)^3} \, dx,x,\text {sech}^{-1}(c x)\right )\right ) \\ & = \frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^2 \text {Subst}\left (\int \frac {\cosh (2 x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 b} \\ & = \frac {c^2 \cosh \left (2 \text {sech}^{-1}(c x)\right )}{2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^2 \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{b^2} \\ & = \frac {c^2 \cosh \left (2 \text {sech}^{-1}(c x)\right )}{2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {\left (c^2 \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{b^2}+\frac {\left (c^2 \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{b^2} \\ & = \frac {c^2 \cosh \left (2 \text {sech}^{-1}(c x)\right )}{2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b^3}+\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b^3} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.47 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.09
\[
\int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\frac {\frac {b^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {b \left (2-c^2 x^2\right )}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )}+2 c^2 \left (\text {Chi}\left (2 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )\right )}{2 b^3}
\]
[In]
Integrate[1/(x^3*(a + b*ArcSech[c*x])^3),x]
[Out]
((b^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(x^2*(a + b*ArcSech[c*x])^2) + (b*(2 - c^2*x^2))/(x^2*(a + b*ArcSec
h[c*x])) + 2*c^2*(CoshIntegral[2*(a/b + ArcSech[c*x])]*Sinh[(2*a)/b] - Cosh[(2*a)/b]*SinhIntegral[2*(a/b + Arc
Sech[c*x])]))/(2*b^3)
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(276\) vs. \(2(108)=216\).
Time = 0.79 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.47
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method | result | size |
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derivativedivides |
\(c^{2} \left (-\frac {\left (2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+c^{2} x^{2}-2\right ) \left (2 b \,\operatorname {arcsech}\left (c x \right )+2 a -b \right )}{8 c^{2} x^{2} b^{2} \left (b^{2} \operatorname {arcsech}\left (c x \right )^{2}+2 a b \,\operatorname {arcsech}\left (c x \right )+a^{2}\right )}-\frac {{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (\frac {2 a}{b}+2 \,\operatorname {arcsech}\left (c x \right )\right )}{2 b^{3}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{8 b \,c^{2} x^{2} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{2}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{4 b^{2} c^{2} x^{2} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsech}\left (c x \right )-\frac {2 a}{b}\right )}{2 b^{3}}\right )\) |
\(277\) |
default |
\(c^{2} \left (-\frac {\left (2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+c^{2} x^{2}-2\right ) \left (2 b \,\operatorname {arcsech}\left (c x \right )+2 a -b \right )}{8 c^{2} x^{2} b^{2} \left (b^{2} \operatorname {arcsech}\left (c x \right )^{2}+2 a b \,\operatorname {arcsech}\left (c x \right )+a^{2}\right )}-\frac {{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (\frac {2 a}{b}+2 \,\operatorname {arcsech}\left (c x \right )\right )}{2 b^{3}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{8 b \,c^{2} x^{2} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{2}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{4 b^{2} c^{2} x^{2} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsech}\left (c x \right )-\frac {2 a}{b}\right )}{2 b^{3}}\right )\) |
\(277\) |
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[In]
int(1/x^3/(a+b*arcsech(c*x))^3,x,method=_RETURNVERBOSE)
[Out]
c^2*(-1/8*(2*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)+c^2*x^2-2)*(2*b*arcsech(c*x)+2*a-b)/c^2/x^2/b^2/(b^2
*arcsech(c*x)^2+2*a*b*arcsech(c*x)+a^2)-1/2/b^3*exp(2*a/b)*Ei(1,2*a/b+2*arcsech(c*x))-1/8/b*(c^2*x^2-2-2*(-(c*
x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2))/c^2/x^2/(a+b*arcsech(c*x))^2-1/4/b^2*(c^2*x^2-2-2*(-(c*x-1)/c/x)^(1/2
)*c*x*((c*x+1)/c/x)^(1/2))/c^2/x^2/(a+b*arcsech(c*x))+1/2/b^3*exp(-2*a/b)*Ei(1,-2*arcsech(c*x)-2*a/b))
Fricas [F]
\[
\int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{3}} \,d x }
\]
[In]
integrate(1/x^3/(a+b*arcsech(c*x))^3,x, algorithm="fricas")
[Out]
integral(1/(b^3*x^3*arcsech(c*x)^3 + 3*a*b^2*x^3*arcsech(c*x)^2 + 3*a^2*b*x^3*arcsech(c*x) + a^3*x^3), x)
Sympy [F]
\[
\int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}\, dx
\]
[In]
integrate(1/x**3/(a+b*asech(c*x))**3,x)
[Out]
Integral(1/(x**3*(a + b*asech(c*x))**3), x)
Maxima [F]
\[
\int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{3}} \,d x }
\]
[In]
integrate(1/x^3/(a+b*arcsech(c*x))^3,x, algorithm="maxima")
[Out]
-1/2*((b*c^6*(2*log(c) - 1) - 2*a*c^6)*x^7 - 3*(b*c^4*(2*log(c) - 1) - 2*a*c^4)*x^5 + ((b*c^2*(2*log(c) - 1) -
2*a*c^2)*x^3 - (b*(2*log(c) - 1) - 2*a)*x + 2*(b*c^2*x^3 - b*x)*log(x))*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) + 3*
(b*c^2*(2*log(c) - 1) - 2*a*c^2)*x^3 - ((b*c^6*log(c) - a*c^6)*x^7 - (b*c^4*(5*log(c) - 2) - 5*a*c^4)*x^5 + 5*
(b*c^2*(2*log(c) - 1) - 2*a*c^2)*x^3 - 3*(b*(2*log(c) - 1) - 2*a)*x + (b*c^6*x^7 - 5*b*c^4*x^5 + 10*b*c^2*x^3
- 6*b*x)*log(x))*(c*x + 1)*(c*x - 1) + ((b*c^6*(3*log(c) - 1) - 3*a*c^6)*x^7 - (b*c^4*(11*log(c) - 5) - 11*a*c
^4)*x^5 + 7*(b*c^2*(2*log(c) - 1) - 2*a*c^2)*x^3 - 3*(b*(2*log(c) - 1) - 2*a)*x + (3*b*c^6*x^7 - 11*b*c^4*x^5
+ 14*b*c^2*x^3 - 6*b*x)*log(x))*sqrt(c*x + 1)*sqrt(-c*x + 1) - (b*(2*log(c) - 1) - 2*a)*x - (2*b*c^6*x^7 - 6*b
*c^4*x^5 + 6*b*c^2*x^3 + 2*(b*c^2*x^3 - b*x)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) - (b*c^6*x^7 - 5*b*c^4*x^5 + 10*
b*c^2*x^3 - 6*b*x)*(c*x + 1)*(c*x - 1) + (3*b*c^6*x^7 - 11*b*c^4*x^5 + 14*b*c^2*x^3 - 6*b*x)*sqrt(c*x + 1)*sqr
t(-c*x + 1) - 2*b*x)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) + 2*(b*c^6*x^7 - 3*b*c^4*x^5 + 3*b*c^2*x^3 - b*x)*l
og(x))/((b^4*c^6*x^6 - 3*b^4*c^4*x^4 + 3*b^4*c^2*x^2 - b^4)*x^3*log(x)^2 + 2*((b^4*c^6*log(c) - a*b^3*c^6)*x^6
- 3*(b^4*c^4*log(c) - a*b^3*c^4)*x^4 - b^4*log(c) + a*b^3 + 3*(b^4*c^2*log(c) - a*b^3*c^2)*x^2)*x^3*log(x) -
(b^4*x^3*log(x)^2 + 2*(b^4*log(c) - a*b^3)*x^3*log(x) + (b^4*log(c)^2 - 2*a*b^3*log(c) + a^2*b^2)*x^3)*(c*x +
1)^(3/2)*(-c*x + 1)^(3/2) + ((b^4*c^6*log(c)^2 - 2*a*b^3*c^6*log(c) + a^2*b^2*c^6)*x^6 - b^4*log(c)^2 - 3*(b^4
*c^4*log(c)^2 - 2*a*b^3*c^4*log(c) + a^2*b^2*c^4)*x^4 + 2*a*b^3*log(c) - a^2*b^2 + 3*(b^4*c^2*log(c)^2 - 2*a*b
^3*c^2*log(c) + a^2*b^2*c^2)*x^2)*x^3 - 3*((b^4*c^2*x^2 - b^4)*x^3*log(x)^2 - 2*(b^4*log(c) - a*b^3 - (b^4*c^2
*log(c) - a*b^3*c^2)*x^2)*x^3*log(x) - (b^4*log(c)^2 - 2*a*b^3*log(c) + a^2*b^2 - (b^4*c^2*log(c)^2 - 2*a*b^3*
c^2*log(c) + a^2*b^2*c^2)*x^2)*x^3)*(c*x + 1)*(c*x - 1) - ((c*x + 1)^(3/2)*(-c*x + 1)^(3/2)*b^4*x^3 + 3*(b^4*c
^2*x^2 - b^4)*(c*x + 1)*(c*x - 1)*x^3 + 3*(b^4*c^4*x^4 - 2*b^4*c^2*x^2 + b^4)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^3
- (b^4*c^6*x^6 - 3*b^4*c^4*x^4 + 3*b^4*c^2*x^2 - b^4)*x^3)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)^2 - 3*((b^4*
c^4*x^4 - 2*b^4*c^2*x^2 + b^4)*x^3*log(x)^2 + 2*((b^4*c^4*log(c) - a*b^3*c^4)*x^4 + b^4*log(c) - a*b^3 - 2*(b^
4*c^2*log(c) - a*b^3*c^2)*x^2)*x^3*log(x) + (b^4*log(c)^2 + (b^4*c^4*log(c)^2 - 2*a*b^3*c^4*log(c) + a^2*b^2*c
^4)*x^4 - 2*a*b^3*log(c) + a^2*b^2 - 2*(b^4*c^2*log(c)^2 - 2*a*b^3*c^2*log(c) + a^2*b^2*c^2)*x^2)*x^3)*sqrt(c*
x + 1)*sqrt(-c*x + 1) - 2*((b^4*c^6*x^6 - 3*b^4*c^4*x^4 + 3*b^4*c^2*x^2 - b^4)*x^3*log(x) - (b^4*x^3*log(x) +
(b^4*log(c) - a*b^3)*x^3)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) + ((b^4*c^6*log(c) - a*b^3*c^6)*x^6 - 3*(b^4*c^4*lo
g(c) - a*b^3*c^4)*x^4 - b^4*log(c) + a*b^3 + 3*(b^4*c^2*log(c) - a*b^3*c^2)*x^2)*x^3 - 3*((b^4*c^2*x^2 - b^4)*
x^3*log(x) - (b^4*log(c) - a*b^3 - (b^4*c^2*log(c) - a*b^3*c^2)*x^2)*x^3)*(c*x + 1)*(c*x - 1) - 3*((b^4*c^4*x^
4 - 2*b^4*c^2*x^2 + b^4)*x^3*log(x) + ((b^4*c^4*log(c) - a*b^3*c^4)*x^4 + b^4*log(c) - a*b^3 - 2*(b^4*c^2*log(
c) - a*b^3*c^2)*x^2)*x^3)*sqrt(c*x + 1)*sqrt(-c*x + 1))*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)) + integrate(-1/
2*(4*c^8*x^8 - 16*c^6*x^6 + 24*c^4*x^4 + 4*(c*x + 1)^2*(c*x - 1)^2 + (3*c^6*x^6 - 16*c^2*x^2 + 16)*(c*x + 1)^(
3/2)*(-c*x + 1)^(3/2) - 16*c^2*x^2 - 24*(c^4*x^4 - 2*c^2*x^2 + 1)*(c*x + 1)*(c*x - 1) + (3*c^8*x^8 - 19*c^6*x^
6 + 48*c^4*x^4 - 48*c^2*x^2 + 16)*sqrt(c*x + 1)*sqrt(-c*x + 1) + 4)/((b^3*x^3*log(x) + (b^3*log(c) - a*b^2)*x^
3)*(c*x + 1)^2*(c*x - 1)^2 + (b^3*c^8*x^8 - 4*b^3*c^6*x^6 + 6*b^3*c^4*x^4 - 4*b^3*c^2*x^2 + b^3)*x^3*log(x) -
4*((b^3*c^2*x^2 - b^3)*x^3*log(x) - (b^3*log(c) - a*b^2 - (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*x^3)*(c*x + 1)^(3/
2)*(-c*x + 1)^(3/2) + ((b^3*c^8*log(c) - a*b^2*c^8)*x^8 - 4*(b^3*c^6*log(c) - a*b^2*c^6)*x^6 + 6*(b^3*c^4*log(
c) - a*b^2*c^4)*x^4 + b^3*log(c) - a*b^2 - 4*(b^3*c^2*log(c) - a*b^2*c^2)*x^2)*x^3 - 6*((b^3*c^4*x^4 - 2*b^3*c
^2*x^2 + b^3)*x^3*log(x) + ((b^3*c^4*log(c) - a*b^2*c^4)*x^4 + b^3*log(c) - a*b^2 - 2*(b^3*c^2*log(c) - a*b^2*
c^2)*x^2)*x^3)*(c*x + 1)*(c*x - 1) - 4*((b^3*c^6*x^6 - 3*b^3*c^4*x^4 + 3*b^3*c^2*x^2 - b^3)*x^3*log(x) + ((b^3
*c^6*log(c) - a*b^2*c^6)*x^6 - 3*(b^3*c^4*log(c) - a*b^2*c^4)*x^4 - b^3*log(c) + a*b^2 + 3*(b^3*c^2*log(c) - a
*b^2*c^2)*x^2)*x^3)*sqrt(c*x + 1)*sqrt(-c*x + 1) - ((c*x + 1)^2*(c*x - 1)^2*b^3*x^3 - 4*(b^3*c^2*x^2 - b^3)*(c
*x + 1)^(3/2)*(-c*x + 1)^(3/2)*x^3 - 6*(b^3*c^4*x^4 - 2*b^3*c^2*x^2 + b^3)*(c*x + 1)*(c*x - 1)*x^3 - 4*(b^3*c^
6*x^6 - 3*b^3*c^4*x^4 + 3*b^3*c^2*x^2 - b^3)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^3 + (b^3*c^8*x^8 - 4*b^3*c^6*x^6 +
6*b^3*c^4*x^4 - 4*b^3*c^2*x^2 + b^3)*x^3)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)), x)
Giac [F]
\[
\int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{3}} \,d x }
\]
[In]
integrate(1/x^3/(a+b*arcsech(c*x))^3,x, algorithm="giac")
[Out]
integrate(1/((b*arcsech(c*x) + a)^3*x^3), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^3\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3} \,d x
\]
[In]
int(1/(x^3*(a + b*acosh(1/(c*x)))^3),x)
[Out]
int(1/(x^3*(a + b*acosh(1/(c*x)))^3), x)