\(\int \frac {\text {sech}^{-1}(a+b x)}{x^4} \, dx\) [8]

Optimal result

Integrand size = 10, antiderivative size = 197 \[ \int \frac {\text {sech}^{-1}(a+b x)}{x^4} \, dx=\frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a \left (1-a^2\right ) x^2}-\frac {\left (2-5 a^2\right ) b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a^2 \left (1-a^2\right )^2 x}-\frac {b^3 \text {sech}^{-1}(a+b x)}{3 a^3}-\frac {\text {sech}^{-1}(a+b x)}{3 x^3}+\frac {\left (2-5 a^2+6 a^4\right ) b^3 \text {arctanh}\left (\frac {\sqrt {1+a} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{3 a^3 \left (1-a^2\right )^{5/2}} \]

[Out]

-1/3*b^3*arcsech(b*x+a)/a^3-1/3*arcsech(b*x+a)/x^3+1/3*(6*a^4-5*a^2+2)*b^3*arctanh((1+a)^(1/2)*tanh(1/2*arcsec
h(b*x+a))/(1-a)^(1/2))/a^3/(-a^2+1)^(5/2)+1/6*b*(b*x+a+1)*((-b*x-a+1)/(b*x+a+1))^(1/2)/a/(-a^2+1)/x^2-1/6*(-5*
a^2+2)*b^2*(b*x+a+1)*((-b*x-a+1)/(b*x+a+1))^(1/2)/a^2/(-a^2+1)^2/x

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6456, 5576, 3870, 4145, 4004, 3916, 2738, 214} \[ \int \frac {\text {sech}^{-1}(a+b x)}{x^4} \, dx=-\frac {b^3 \text {sech}^{-1}(a+b x)}{3 a^3}-\frac {\left (2-5 a^2\right ) b^2 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{6 a^2 \left (1-a^2\right )^2 x}+\frac {b \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{6 a \left (1-a^2\right ) x^2}+\frac {\left (6 a^4-5 a^2+2\right ) b^3 \text {arctanh}\left (\frac {\sqrt {a+1} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{3 a^3 \left (1-a^2\right )^{5/2}}-\frac {\text {sech}^{-1}(a+b x)}{3 x^3} \]

[In]

Int[ArcSech[a + b*x]/x^4,x]

[Out]

(b*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(6*a*(1 - a^2)*x^2) - ((2 - 5*a^2)*b^2*Sqrt[(1 - a - b*x)/
(1 + a + b*x)]*(1 + a + b*x))/(6*a^2*(1 - a^2)^2*x) - (b^3*ArcSech[a + b*x])/(3*a^3) - ArcSech[a + b*x]/(3*x^3
) + ((2 - 5*a^2 + 6*a^4)*b^3*ArcTanh[(Sqrt[1 + a]*Tanh[ArcSech[a + b*x]/2])/Sqrt[1 - a]])/(3*a^3*(1 - a^2)^(5/
2))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3870

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[c + d*x]*((a + b*Csc[c + d*x])^(n +
1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Dist[1/(a*(n + 1)*(a^2 - b^2)), Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^
2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 4145

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)
*(a^2 - b^2))), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 5576

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

Rule 6456

Int[((a_.) + ArcSech[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[-(d^(m + 1)
)^(-1), Subst[Int[(a + b*x)^p*Sech[x]*Tanh[x]*(d*e - c*f + f*Sech[x])^m, x], x, ArcSech[c + d*x]], x] /; FreeQ
[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\left (b^3 \text {Subst}\left (\int \frac {x \text {sech}(x) \tanh (x)}{(-a+\text {sech}(x))^4} \, dx,x,\text {sech}^{-1}(a+b x)\right )\right ) \\ & = -\frac {\text {sech}^{-1}(a+b x)}{3 x^3}+\frac {1}{3} b^3 \text {Subst}\left (\int \frac {1}{(-a+\text {sech}(x))^3} \, dx,x,\text {sech}^{-1}(a+b x)\right ) \\ & = \frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a \left (1-a^2\right ) x^2}-\frac {\text {sech}^{-1}(a+b x)}{3 x^3}-\frac {b^3 \text {Subst}\left (\int \frac {2 \left (1-a^2\right )-2 a \text {sech}(x)-\text {sech}^2(x)}{(-a+\text {sech}(x))^2} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{6 a \left (1-a^2\right )} \\ & = \frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a \left (1-a^2\right ) x^2}-\frac {\left (2-5 a^2\right ) b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a^2 \left (1-a^2\right )^2 x}-\frac {\text {sech}^{-1}(a+b x)}{3 x^3}+\frac {b^3 \text {Subst}\left (\int \frac {2 \left (1-a^2\right )^2-a \left (1-4 a^2\right ) \text {sech}(x)}{-a+\text {sech}(x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{6 a^2 \left (1-a^2\right )^2} \\ & = \frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a \left (1-a^2\right ) x^2}-\frac {\left (2-5 a^2\right ) b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a^2 \left (1-a^2\right )^2 x}-\frac {b^3 \text {sech}^{-1}(a+b x)}{3 a^3}-\frac {\text {sech}^{-1}(a+b x)}{3 x^3}+\frac {\left (\left (2-5 a^2+6 a^4\right ) b^3\right ) \text {Subst}\left (\int \frac {\text {sech}(x)}{-a+\text {sech}(x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{6 a^3 \left (1-a^2\right )^2} \\ & = \frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a \left (1-a^2\right ) x^2}-\frac {\left (2-5 a^2\right ) b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a^2 \left (1-a^2\right )^2 x}-\frac {b^3 \text {sech}^{-1}(a+b x)}{3 a^3}-\frac {\text {sech}^{-1}(a+b x)}{3 x^3}+\frac {\left (\left (2-5 a^2+6 a^4\right ) b^3\right ) \text {Subst}\left (\int \frac {1}{1-a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{6 a^3 \left (1-a^2\right )^2} \\ & = \frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a \left (1-a^2\right ) x^2}-\frac {\left (2-5 a^2\right ) b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a^2 \left (1-a^2\right )^2 x}-\frac {b^3 \text {sech}^{-1}(a+b x)}{3 a^3}-\frac {\text {sech}^{-1}(a+b x)}{3 x^3}+\frac {\left (\left (2-5 a^2+6 a^4\right ) b^3\right ) \text {Subst}\left (\int \frac {1}{1-a-(1+a) x^2} \, dx,x,\tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}{3 a^3 \left (1-a^2\right )^2} \\ & = \frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a \left (1-a^2\right ) x^2}-\frac {\left (2-5 a^2\right ) b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 a^2 \left (1-a^2\right )^2 x}-\frac {b^3 \text {sech}^{-1}(a+b x)}{3 a^3}-\frac {\text {sech}^{-1}(a+b x)}{3 x^3}+\frac {\left (2-5 a^2+6 a^4\right ) b^3 \text {arctanh}\left (\frac {\sqrt {1+a} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{3 a^3 \left (1-a^2\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.87 \[ \int \frac {\text {sech}^{-1}(a+b x)}{x^4} \, dx=\frac {1}{6} \left (\frac {b \sqrt {-\frac {-1+a+b x}{1+a+b x}} \left (a-a^4-a b x-2 b x (1+b x)+a^3 (-1+4 b x)+a^2 \left (1+5 b x+5 b^2 x^2\right )\right )}{(-1+a)^2 a^2 (1+a)^2 x^2}-\frac {2 \text {sech}^{-1}(a+b x)}{x^3}-\frac {\left (2-5 a^2+6 a^4\right ) b^3 \log (x)}{a^3 \left (1-a^2\right )^{5/2}}+\frac {2 b^3 \log (a+b x)}{a^3}-\frac {2 b^3 \log \left (1+\sqrt {-\frac {-1+a+b x}{1+a+b x}}+a \sqrt {-\frac {-1+a+b x}{1+a+b x}}+b x \sqrt {-\frac {-1+a+b x}{1+a+b x}}\right )}{a^3}+\frac {\left (2-5 a^2+6 a^4\right ) b^3 \log \left (1-a^2-a b x+\sqrt {1-a^2} \sqrt {-\frac {-1+a+b x}{1+a+b x}}+a \sqrt {1-a^2} \sqrt {-\frac {-1+a+b x}{1+a+b x}}+\sqrt {1-a^2} b x \sqrt {-\frac {-1+a+b x}{1+a+b x}}\right )}{a^3 \left (1-a^2\right )^{5/2}}\right ) \]

[In]

Integrate[ArcSech[a + b*x]/x^4,x]

[Out]

((b*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]*(a - a^4 - a*b*x - 2*b*x*(1 + b*x) + a^3*(-1 + 4*b*x) + a^2*(1 + 5*b
*x + 5*b^2*x^2)))/((-1 + a)^2*a^2*(1 + a)^2*x^2) - (2*ArcSech[a + b*x])/x^3 - ((2 - 5*a^2 + 6*a^4)*b^3*Log[x])
/(a^3*(1 - a^2)^(5/2)) + (2*b^3*Log[a + b*x])/a^3 - (2*b^3*Log[1 + Sqrt[-((-1 + a + b*x)/(1 + a + b*x))] + a*S
qrt[-((-1 + a + b*x)/(1 + a + b*x))] + b*x*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]])/a^3 + ((2 - 5*a^2 + 6*a^4)*
b^3*Log[1 - a^2 - a*b*x + Sqrt[1 - a^2]*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))] + a*Sqrt[1 - a^2]*Sqrt[-((-1 + a
+ b*x)/(1 + a + b*x))] + Sqrt[1 - a^2]*b*x*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]])/(a^3*(1 - a^2)^(5/2)))/6

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.97 (sec) , antiderivative size = 608, normalized size of antiderivative = 3.09

method	result	size
parts	\(-\frac {\operatorname {arcsech}\left (b x +a \right )}{3 x^{3}}-\frac {b \sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \operatorname {csgn}\left (b \right )^{2} \left (2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{6} b^{2} x^{2}+6 \sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right ) a^{4} b^{2} x^{2}-6 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{4} b^{2} x^{2}-5 \sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right ) a^{2} b^{2} x^{2}-5 a^{5} b x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+6 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{2} b^{2} x^{2}+a^{6} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+2 \sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right ) b^{2} x^{2}+7 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3} b x -2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) b^{2} x^{2}-2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{4}-2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b x +\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2}\right )}{6 x^{2} \left (a^{2}-1\right )^{2} \left (-1+a \right ) \left (1+a \right ) a^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\)	\(608\)
derivativedivides	\(\text {Expression too large to display}\)	\(1027\)
default	\(\text {Expression too large to display}\)	\(1027\)

[In]

int(arcsech(b*x+a)/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*arcsech(b*x+a)/x^3-1/6*b*(-(b*x+a-1)/(b*x+a))^(1/2)*(b*x+a)*((b*x+a+1)/(b*x+a))^(1/2)*csgn(b)^2*(2*arctan
h(1/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))*a^6*b^2*x^2+6*(-a^2+1)^(1/2)*ln(2*(-a*b*x+(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x
-a^2+1)^(1/2)-a^2+1)/x)*a^4*b^2*x^2-6*arctanh(1/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))*a^4*b^2*x^2-5*(-a^2+1)^(1/2)*l
n(2*(-a*b*x+(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-a^2+1)/x)*a^2*b^2*x^2-5*a^5*b*x*(-b^2*x^2-2*a*b*x-a^
2+1)^(1/2)+6*arctanh(1/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))*a^2*b^2*x^2+a^6*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+2*(-a^2+
1)^(1/2)*ln(2*(-a*b*x+(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-a^2+1)/x)*b^2*x^2+7*(-b^2*x^2-2*a*b*x-a^2+
1)^(1/2)*a^3*b*x-2*arctanh(1/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))*b^2*x^2-2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a^4-2*(-
b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a*b*x+(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a^2)/x^2/(a^2-1)^2/(-1+a)/(1+a)/a^3/(-b^2*x^
2-2*a*b*x-a^2+1)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (167) = 334\).

Time = 0.33 (sec) , antiderivative size = 987, normalized size of antiderivative = 5.01 \[ \int \frac {\text {sech}^{-1}(a+b x)}{x^4} \, dx=\left [-\frac {{\left (6 \, a^{4} - 5 \, a^{2} + 2\right )} \sqrt {-a^{2} + 1} b^{3} x^{3} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 4 \, a^{2} - 2 \, {\left (a b^{2} x^{2} + a^{3} + {\left (2 \, a^{2} - 1\right )} b x - a\right )} \sqrt {-a^{2} + 1} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 2}{x^{2}}\right ) + 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b^{3} x^{3} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) - 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b^{3} x^{3} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) + 4 \, {\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - 2 \, {\left ({\left (5 \, a^{5} - 7 \, a^{3} + 2 \, a\right )} b^{3} x^{3} + {\left (4 \, a^{6} - 5 \, a^{4} + a^{2}\right )} b^{2} x^{2} - {\left (a^{7} - 2 \, a^{5} + a^{3}\right )} b x\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{12 \, {\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} x^{3}}, -\frac {{\left (6 \, a^{4} - 5 \, a^{2} + 2\right )} \sqrt {a^{2} - 1} b^{3} x^{3} \arctan \left (\frac {{\left (a b^{2} x^{2} + a^{3} + {\left (2 \, a^{2} - 1\right )} b x - a\right )} \sqrt {a^{2} - 1} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b^{3} x^{3} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) - {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b^{3} x^{3} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) + 2 \, {\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - {\left ({\left (5 \, a^{5} - 7 \, a^{3} + 2 \, a\right )} b^{3} x^{3} + {\left (4 \, a^{6} - 5 \, a^{4} + a^{2}\right )} b^{2} x^{2} - {\left (a^{7} - 2 \, a^{5} + a^{3}\right )} b x\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{6 \, {\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} x^{3}}\right ] \]

[In]

integrate(arcsech(b*x+a)/x^4,x, algorithm="fricas")

[Out]

[-1/12*((6*a^4 - 5*a^2 + 2)*sqrt(-a^2 + 1)*b^3*x^3*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 4*a^2
- 2*(a*b^2*x^2 + a^3 + (2*a^2 - 1)*b*x - a)*sqrt(-a^2 + 1)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*
b*x + a^2)) + 2)/x^2) + 2*(a^6 - 3*a^4 + 3*a^2 - 1)*b^3*x^3*log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)
/(b^2*x^2 + 2*a*b*x + a^2)) + 1)/x) - 2*(a^6 - 3*a^4 + 3*a^2 - 1)*b^3*x^3*log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a*
b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) - 1)/x) + 4*(a^9 - 3*a^7 + 3*a^5 - a^3)*log(((b*x + a)*sqrt(-(b^2*x^
2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) + 1)/(b*x + a)) - 2*((5*a^5 - 7*a^3 + 2*a)*b^3*x^3 + (4*a^6
- 5*a^4 + a^2)*b^2*x^2 - (a^7 - 2*a^5 + a^3)*b*x)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2
)))/((a^9 - 3*a^7 + 3*a^5 - a^3)*x^3), -1/6*((6*a^4 - 5*a^2 + 2)*sqrt(a^2 - 1)*b^3*x^3*arctan((a*b^2*x^2 + a^3
+ (2*a^2 - 1)*b*x - a)*sqrt(a^2 - 1)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2))/((a^2 - 1
)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) + (a^6 - 3*a^4 + 3*a^2 - 1)*b^3*x^3*log(((b*x + a)*sqrt(-(b^2*
x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) + 1)/x) - (a^6 - 3*a^4 + 3*a^2 - 1)*b^3*x^3*log(((b*x + a)
*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) - 1)/x) + 2*(a^9 - 3*a^7 + 3*a^5 - a^3)*log(((
b*x + a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) + 1)/(b*x + a)) - ((5*a^5 - 7*a^3 + 2*
a)*b^3*x^3 + (4*a^6 - 5*a^4 + a^2)*b^2*x^2 - (a^7 - 2*a^5 + a^3)*b*x)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2
*x^2 + 2*a*b*x + a^2)))/((a^9 - 3*a^7 + 3*a^5 - a^3)*x^3)]

Sympy [F]

\[ \int \frac {\text {sech}^{-1}(a+b x)}{x^4} \, dx=\int \frac {\operatorname {asech}{\left (a + b x \right )}}{x^{4}}\, dx \]

[In]

integrate(asech(b*x+a)/x**4,x)

[Out]

Integral(asech(a + b*x)/x**4, x)

Maxima [F]

\[ \int \frac {\text {sech}^{-1}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )}{x^{4}} \,d x } \]

[In]

integrate(arcsech(b*x+a)/x^4,x, algorithm="maxima")

[Out]

1/3*(6*a^4*b^3 - 3*a^2*b^3 + b^3)*log(x)/(a^9 - 3*a^7 + 3*a^5 - a^3) - 1/6*((a^6*b^3 - 3*a^5*b^3 + 3*a^4*b^3 -
a^3*b^3)*x^3*log(b*x + a + 1) + (a^6*b^3 + 3*a^5*b^3 + 3*a^4*b^3 + a^3*b^3)*x^3*log(-b*x - a + 1) - 2*(3*a^5*
b^2 - 4*a^3*b^2 + a*b^2)*x^2 + (a^6*b - 2*a^4*b + a^2*b)*x + 2*(a^9 - 3*a^7 + 3*a^5 - a^3)*log(sqrt(b*x + a +
1)*sqrt(-b*x - a + 1)*b*x + sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*a + b*x + a) - 2*(a^9 - 3*a^7 + 3*a^5 + (a^6*
b^3 - 3*a^4*b^3 + 3*a^2*b^3 - b^3)*x^3 - a^3)*log(b*x + a) - 2*(a^9 - 3*a^7 + 3*a^5 - a^3)*log(b*x + a))/((a^9
- 3*a^7 + 3*a^5 - a^3)*x^3) - integrate(1/3*(b^2*x + a*b)/(b^2*x^5 + 2*a*b*x^4 + (a^2 - 1)*x^3 + (b^2*x^5 + 2
*a*b*x^4 + (a^2 - 1)*x^3)*e^(1/2*log(b*x + a + 1) + 1/2*log(-b*x - a + 1))), x)

Giac [F]

\[ \int \frac {\text {sech}^{-1}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )}{x^{4}} \,d x } \]

[In]

integrate(arcsech(b*x+a)/x^4,x, algorithm="giac")

[Out]

integrate(arcsech(b*x + a)/x^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {sech}^{-1}(a+b x)}{x^4} \, dx=\int \frac {\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}{x^4} \,d x \]

[In]

int(acosh(1/(a + b*x))/x^4,x)

[Out]

int(acosh(1/(a + b*x))/x^4, x)