3.50 \(\int \frac{(a+b \text{sech}^{-1}(c x))^3}{x^5} \, dx\)

Optimal. Leaf size=242 \[ -\frac{9 b^2 c^2 \left (a+b \text{sech}^{-1}(c x)\right )}{32 x^2}-\frac{3 b^2 \left (a+b \text{sech}^{-1}(c x)\right )}{32 x^4}+\frac{9 b c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )^2}{32 x^2}+\frac{3}{32} c^4 \left (a+b \text{sech}^{-1}(c x)\right )^3-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{4 x^4}+\frac{3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )^2}{16 x^4}+\frac{45 b^3 c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{256 x^2}+\frac{45}{256} b^3 c^4 \text{sech}^{-1}(c x)+\frac{3 b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{128 x^4} \]

[Out]

(3*b^3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(128*x^4) + (45*b^3*c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(256*
x^2) + (45*b^3*c^4*ArcSech[c*x])/256 - (3*b^2*(a + b*ArcSech[c*x]))/(32*x^4) - (9*b^2*c^2*(a + b*ArcSech[c*x])
)/(32*x^2) + (3*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(16*x^4) + (9*b*c^2*Sqrt[(1 - c*
x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(32*x^2) + (3*c^4*(a + b*ArcSech[c*x])^3)/32 - (a + b*ArcSech[
c*x])^3/(4*x^4)

________________________________________________________________________________________

Rubi [A]  time = 0.196284, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6285, 5447, 3311, 32, 2635, 8} \[ -\frac{9 b^2 c^2 \left (a+b \text{sech}^{-1}(c x)\right )}{32 x^2}-\frac{3 b^2 \left (a+b \text{sech}^{-1}(c x)\right )}{32 x^4}+\frac{9 b c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )^2}{32 x^2}+\frac{3}{32} c^4 \left (a+b \text{sech}^{-1}(c x)\right )^3-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{4 x^4}+\frac{3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )^2}{16 x^4}+\frac{45 b^3 c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{256 x^2}+\frac{45}{256} b^3 c^4 \text{sech}^{-1}(c x)+\frac{3 b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{128 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSech[c*x])^3/x^5,x]

[Out]

(3*b^3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(128*x^4) + (45*b^3*c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(256*
x^2) + (45*b^3*c^4*ArcSech[c*x])/256 - (3*b^2*(a + b*ArcSech[c*x]))/(32*x^4) - (9*b^2*c^2*(a + b*ArcSech[c*x])
)/(32*x^2) + (3*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(16*x^4) + (9*b*c^2*Sqrt[(1 - c*
x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(32*x^2) + (3*c^4*(a + b*ArcSech[c*x])^3)/32 - (a + b*ArcSech[
c*x])^3/(4*x^4)

Rule 6285

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])

Rule 5447

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((c
+ d*x)^m*Cosh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{x^5} \, dx &=-\left (c^4 \operatorname{Subst}\left (\int (a+b x)^3 \cosh ^3(x) \sinh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\right )\\ &=-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{4 x^4}+\frac{1}{4} \left (3 b c^4\right ) \operatorname{Subst}\left (\int (a+b x)^2 \cosh ^4(x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=-\frac{3 b^2 \left (a+b \text{sech}^{-1}(c x)\right )}{32 x^4}+\frac{3 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{16 x^4}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{4 x^4}+\frac{1}{16} \left (9 b c^4\right ) \operatorname{Subst}\left (\int (a+b x)^2 \cosh ^2(x) \, dx,x,\text{sech}^{-1}(c x)\right )+\frac{1}{32} \left (3 b^3 c^4\right ) \operatorname{Subst}\left (\int \cosh ^4(x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=\frac{3 b^3 \sqrt{\frac{1-c x}{1+c x}} (1+c x)}{128 x^4}-\frac{3 b^2 \left (a+b \text{sech}^{-1}(c x)\right )}{32 x^4}-\frac{9 b^2 c^2 \left (a+b \text{sech}^{-1}(c x)\right )}{32 x^2}+\frac{3 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{16 x^4}+\frac{9 b c^2 \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{32 x^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{4 x^4}+\frac{1}{32} \left (9 b c^4\right ) \operatorname{Subst}\left (\int (a+b x)^2 \, dx,x,\text{sech}^{-1}(c x)\right )+\frac{1}{128} \left (9 b^3 c^4\right ) \operatorname{Subst}\left (\int \cosh ^2(x) \, dx,x,\text{sech}^{-1}(c x)\right )+\frac{1}{32} \left (9 b^3 c^4\right ) \operatorname{Subst}\left (\int \cosh ^2(x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=\frac{3 b^3 \sqrt{\frac{1-c x}{1+c x}} (1+c x)}{128 x^4}+\frac{45 b^3 c^2 \sqrt{\frac{1-c x}{1+c x}} (1+c x)}{256 x^2}-\frac{3 b^2 \left (a+b \text{sech}^{-1}(c x)\right )}{32 x^4}-\frac{9 b^2 c^2 \left (a+b \text{sech}^{-1}(c x)\right )}{32 x^2}+\frac{3 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{16 x^4}+\frac{9 b c^2 \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{32 x^2}+\frac{3}{32} c^4 \left (a+b \text{sech}^{-1}(c x)\right )^3-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{4 x^4}+\frac{1}{256} \left (9 b^3 c^4\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\text{sech}^{-1}(c x)\right )+\frac{1}{64} \left (9 b^3 c^4\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=\frac{3 b^3 \sqrt{\frac{1-c x}{1+c x}} (1+c x)}{128 x^4}+\frac{45 b^3 c^2 \sqrt{\frac{1-c x}{1+c x}} (1+c x)}{256 x^2}+\frac{45}{256} b^3 c^4 \text{sech}^{-1}(c x)-\frac{3 b^2 \left (a+b \text{sech}^{-1}(c x)\right )}{32 x^4}-\frac{9 b^2 c^2 \left (a+b \text{sech}^{-1}(c x)\right )}{32 x^2}+\frac{3 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{16 x^4}+\frac{9 b c^2 \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{32 x^2}+\frac{3}{32} c^4 \left (a+b \text{sech}^{-1}(c x)\right )^3-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.713887, size = 332, normalized size = 1.37 \[ \frac{3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (8 a^2 \left (3 c^2 x^2+2\right )+b^2 \left (15 c^2 x^2+2\right )\right )-9 b c^4 x^4 \left (8 a^2+5 b^2\right ) \log (x)+9 b c^4 x^4 \left (8 a^2+5 b^2\right ) \log \left (c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right )-24 b \text{sech}^{-1}(c x) \left (8 a^2-2 a b \sqrt{\frac{1-c x}{c x+1}} \left (3 c^3 x^3+3 c^2 x^2+2 c x+2\right )+b^2 \left (3 c^2 x^2+1\right )\right )-8 a \left (8 a^2+3 b^2\right )-72 a b^2 c^2 x^2+24 b^2 \text{sech}^{-1}(c x)^2 \left (a \left (3 c^4 x^4-8\right )+b \sqrt{\frac{1-c x}{c x+1}} \left (3 c^3 x^3+3 c^2 x^2+2 c x+2\right )\right )+8 b^3 \left (3 c^4 x^4-8\right ) \text{sech}^{-1}(c x)^3}{256 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSech[c*x])^3/x^5,x]

[Out]

(-8*a*(8*a^2 + 3*b^2) - 72*a*b^2*c^2*x^2 + 3*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(8*a^2*(2 + 3*c^2*x^2) + b^
2*(2 + 15*c^2*x^2)) - 24*b*(8*a^2 + b^2*(1 + 3*c^2*x^2) - 2*a*b*Sqrt[(1 - c*x)/(1 + c*x)]*(2 + 2*c*x + 3*c^2*x
^2 + 3*c^3*x^3))*ArcSech[c*x] + 24*b^2*(b*Sqrt[(1 - c*x)/(1 + c*x)]*(2 + 2*c*x + 3*c^2*x^2 + 3*c^3*x^3) + a*(-
8 + 3*c^4*x^4))*ArcSech[c*x]^2 + 8*b^3*(-8 + 3*c^4*x^4)*ArcSech[c*x]^3 - 9*b*(8*a^2 + 5*b^2)*c^4*x^4*Log[x] +
9*b*(8*a^2 + 5*b^2)*c^4*x^4*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/(256*x^4)

________________________________________________________________________________________

Maple [B]  time = 0.277, size = 553, normalized size = 2.3 \begin{align*}{c}^{4} \left ( -{\frac{{a}^{3}}{4\,{c}^{4}{x}^{4}}}+{b}^{3} \left ({\frac{ \left ({\rm arcsech} \left (cx\right ) \right ) ^{3} \left ( cx-1 \right ) \left ( cx+1 \right ) }{4\,{c}^{4}{x}^{4}}}-{\frac{ \left ({\rm arcsech} \left (cx\right ) \right ) ^{3}}{4\,{c}^{2}{x}^{2}}}+{\frac{3\, \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}}{16\,{c}^{3}{x}^{3}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{9\, \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}}{32\,cx}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{3\, \left ({\rm arcsech} \left (cx\right ) \right ) ^{3}}{32}}+{\frac{3\,{\rm arcsech} \left (cx\right ) \left ( cx-1 \right ) \left ( cx+1 \right ) }{32\,{c}^{4}{x}^{4}}}+{\frac{3}{128\,{c}^{3}{x}^{3}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{45}{256\,cx}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{45\,{\rm arcsech} \left (cx\right )}{256}}-{\frac{3\,{\rm arcsech} \left (cx\right )}{8\,{c}^{2}{x}^{2}}} \right ) +3\,a{b}^{2} \left ( 1/4\,{\frac{ \left ({\rm arcsech} \left (cx\right ) \right ) ^{2} \left ( cx-1 \right ) \left ( cx+1 \right ) }{{c}^{4}{x}^{4}}}-1/4\,{\frac{ \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}}{{c}^{2}{x}^{2}}}+1/8\,{\frac{{\rm arcsech} \left (cx\right )}{{c}^{3}{x}^{3}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+3/16\,{\frac{{\rm arcsech} \left (cx\right )}{cx}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{3\, \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}}{32}}+1/32\,{\frac{ \left ( cx+1 \right ) \left ( cx-1 \right ) }{{c}^{4}{x}^{4}}}-1/8\,{\frac{1}{{c}^{2}{x}^{2}}} \right ) +3\,{a}^{2}b \left ( -1/4\,{\frac{{\rm arcsech} \left (cx\right )}{{c}^{4}{x}^{4}}}+1/32\,{\frac{3\,{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ){c}^{4}{x}^{4}+3\,\sqrt{-{c}^{2}{x}^{2}+1}{c}^{2}{x}^{2}+2\,\sqrt{-{c}^{2}{x}^{2}+1}}{{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsech(c*x))^3/x^5,x)

[Out]

c^4*(-1/4*a^3/c^4/x^4+b^3*(1/4*arcsech(c*x)^3/c^4/x^4*(c*x-1)*(c*x+1)-1/4/c^2/x^2*arcsech(c*x)^3+3/16*arcsech(
c*x)^2/c^3/x^3*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)+9/32*arcsech(c*x)^2/c/x*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/
c/x)^(1/2)+3/32*arcsech(c*x)^3+3/32*arcsech(c*x)*(c*x-1)/c^4/x^4*(c*x+1)+3/128/c^3/x^3*(-(c*x-1)/c/x)^(1/2)*((
c*x+1)/c/x)^(1/2)+45/256/c/x*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)+45/256*arcsech(c*x)-3/8/c^2/x^2*arcsech(
c*x))+3*a*b^2*(1/4*arcsech(c*x)^2/c^4/x^4*(c*x-1)*(c*x+1)-1/4/c^2/x^2*arcsech(c*x)^2+1/8*arcsech(c*x)/c^3/x^3*
(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)+3/16*arcsech(c*x)/c/x*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)+3/32*a
rcsech(c*x)^2+1/32*(c*x-1)/c^4/x^4*(c*x+1)-1/8/c^2/x^2)+3*a^2*b*(-1/4/c^4/x^4*arcsech(c*x)+1/32*(-(c*x-1)/c/x)
^(1/2)/c^3/x^3*((c*x+1)/c/x)^(1/2)*(3*arctanh(1/(-c^2*x^2+1)^(1/2))*c^4*x^4+3*(-c^2*x^2+1)^(1/2)*c^2*x^2+2*(-c
^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3}{64} \, a^{2} b{\left (\frac{3 \, c^{5} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} - 1} + 1\right ) - 3 \, c^{5} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} - 1} - 1\right ) - \frac{2 \,{\left (3 \, c^{8} x^{3}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} - 5 \, c^{6} x \sqrt{\frac{1}{c^{2} x^{2}} - 1}\right )}}{c^{4} x^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{2} - 2 \, c^{2} x^{2}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + 1}}{c} - \frac{16 \, \operatorname{arsech}\left (c x\right )}{x^{4}}\right )} - \frac{a^{3}}{4 \, x^{4}} + \int \frac{b^{3} \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )^{3}}{x^{5}} + \frac{3 \, a b^{2} \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )^{2}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))^3/x^5,x, algorithm="maxima")

[Out]

3/64*a^2*b*((3*c^5*log(c*x*sqrt(1/(c^2*x^2) - 1) + 1) - 3*c^5*log(c*x*sqrt(1/(c^2*x^2) - 1) - 1) - 2*(3*c^8*x^
3*(1/(c^2*x^2) - 1)^(3/2) - 5*c^6*x*sqrt(1/(c^2*x^2) - 1))/(c^4*x^4*(1/(c^2*x^2) - 1)^2 - 2*c^2*x^2*(1/(c^2*x^
2) - 1) + 1))/c - 16*arcsech(c*x)/x^4) - 1/4*a^3/x^4 + integrate(b^3*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) +
1/(c*x))^3/x^5 + 3*a*b^2*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))^2/x^5, x)

________________________________________________________________________________________

Fricas [A]  time = 1.63898, size = 756, normalized size = 3.12 \begin{align*} -\frac{72 \, a b^{2} c^{2} x^{2} - 8 \,{\left (3 \, b^{3} c^{4} x^{4} - 8 \, b^{3}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} + 64 \, a^{3} + 24 \, a b^{2} - 24 \,{\left (3 \, a b^{2} c^{4} x^{4} - 8 \, a b^{2} +{\left (3 \, b^{3} c^{3} x^{3} + 2 \, b^{3} c x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 3 \,{\left (3 \,{\left (8 \, a^{2} b + 5 \, b^{3}\right )} c^{4} x^{4} - 24 \, b^{3} c^{2} x^{2} - 64 \, a^{2} b - 8 \, b^{3} + 16 \,{\left (3 \, a b^{2} c^{3} x^{3} + 2 \, a b^{2} c x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 3 \,{\left (3 \,{\left (8 \, a^{2} b + 5 \, b^{3}\right )} c^{3} x^{3} + 2 \,{\left (8 \, a^{2} b + b^{3}\right )} c x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{256 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))^3/x^5,x, algorithm="fricas")

[Out]

-1/256*(72*a*b^2*c^2*x^2 - 8*(3*b^3*c^4*x^4 - 8*b^3)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x))^3 + 6
4*a^3 + 24*a*b^2 - 24*(3*a*b^2*c^4*x^4 - 8*a*b^2 + (3*b^3*c^3*x^3 + 2*b^3*c*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))
*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x))^2 - 3*(3*(8*a^2*b + 5*b^3)*c^4*x^4 - 24*b^3*c^2*x^2 - 64*
a^2*b - 8*b^3 + 16*(3*a*b^2*c^3*x^3 + 2*a*b^2*c*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))*log((c*x*sqrt(-(c^2*x^2 - 1
)/(c^2*x^2)) + 1)/(c*x)) - 3*(3*(8*a^2*b + 5*b^3)*c^3*x^3 + 2*(8*a^2*b + b^3)*c*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^
2)))/x^4

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asech}{\left (c x \right )}\right )^{3}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asech(c*x))**3/x**5,x)

[Out]

Integral((a + b*asech(c*x))**3/x**5, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}^{3}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))^3/x^5,x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)^3/x^5, x)