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

Optimal result

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

1/6*b*((-b*x-a+1)/(b*x+a+1))^(1/2)*(b*x+a+1)/a/(-a^2+1)/x^2-1/6*(-5*a^2+2) 
*b^2*((-b*x-a+1)/(b*x+a+1))^(1/2)*(b*x+a+1)/a^2/(-a^2+1)^2/x-1/3*b^3*arcse 
ch(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*arcsech(b*x+a))/(1-a)^(1/2))/a^3/(-a^2+1)^(5/2)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.87 \[ \int \frac {\text {sech}^{-1}(a+b x)}{x^4} \, dx=\frac {1}{6} \left (\frac {b \sqrt {-\frac {-1+a+b x}{1+a+b x}} \left (a-a^4-a b x-2 b x (1+b x)+a^3 (-1+4 b x)+a^2 \left (1+5 b x+5 b^2 x^2\right )\right )}{(-1+a)^2 a^2 (1+a)^2 x^2}-\frac {2 \text {sech}^{-1}(a+b x)}{x^3}-\frac {\left (2-5 a^2+6 a^4\right ) b^3 \log (x)}{a^3 \left (1-a^2\right )^{5/2}}+\frac {2 b^3 \log (a+b x)}{a^3}-\frac {2 b^3 \log \left (1+\sqrt {-\frac {-1+a+b x}{1+a+b x}}+a \sqrt {-\frac {-1+a+b x}{1+a+b x}}+b x \sqrt {-\frac {-1+a+b x}{1+a+b x}}\right )}{a^3}+\frac {\left (2-5 a^2+6 a^4\right ) b^3 \log \left (1-a^2-a b x+\sqrt {1-a^2} \sqrt {-\frac {-1+a+b x}{1+a+b x}}+a \sqrt {1-a^2} \sqrt {-\frac {-1+a+b x}{1+a+b x}}+\sqrt {1-a^2} b x \sqrt {-\frac {-1+a+b x}{1+a+b x}}\right )}{a^3 \left (1-a^2\right )^{5/2}}\right ) \] Input:

Integrate[ArcSech[a + b*x]/x^4,x]
 

Output:

((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*Sqrt[-((-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
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.21, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {6875, 5991, 3042, 4272, 3042, 4548, 3042, 4407, 3042, 4318, 3042, 3138, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6875

\(\displaystyle -b^3 \int \frac {(a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{b^4 x^4}d\text {sech}^{-1}(a+b x)\)

\(\Big \downarrow \) 5991

\(\displaystyle -b^3 \left (\frac {1}{3} \int -\frac {1}{b^3 x^3}d\text {sech}^{-1}(a+b x)+\frac {\text {sech}^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^3 \left (\frac {\text {sech}^{-1}(a+b x)}{3 b^3 x^3}+\frac {1}{3} \int \frac {1}{\left (a-\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )\right )^3}d\text {sech}^{-1}(a+b x)\right )\)

\(\Big \downarrow \) 4272

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\int \frac {-(a+b x)^2-2 a (a+b x)+2 \left (1-a^2\right )}{b^2 x^2}d\text {sech}^{-1}(a+b x)}{2 a \left (1-a^2\right )}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\text {sech}^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^3 \left (\frac {\text {sech}^{-1}(a+b x)}{3 b^3 x^3}+\frac {1}{3} \left (-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) b^2 x^2}+\frac {\int \frac {-\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )^2-2 a \csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )+2 \left (1-a^2\right )}{\left (a-\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )\right )^2}d\text {sech}^{-1}(a+b x)}{2 a \left (1-a^2\right )}\right )\right )\)

\(\Big \downarrow \) 4548

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\int -\frac {2 \left (1-a^2\right )^2-a \left (1-4 a^2\right ) (a+b x)}{b x}d\text {sech}^{-1}(a+b x)}{a \left (1-a^2\right )}+\frac {\left (2-5 a^2\right ) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\text {sech}^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^3 \left (\frac {\text {sech}^{-1}(a+b x)}{3 b^3 x^3}+\frac {1}{3} \left (-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) b^2 x^2}+\frac {\frac {\left (2-5 a^2\right ) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}+\frac {\int \frac {2 \left (1-a^2\right )^2-a \left (1-4 a^2\right ) \csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )}{a-\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )}d\text {sech}^{-1}(a+b x)}{a \left (1-a^2\right )}}{2 a \left (1-a^2\right )}\right )\right )\)

\(\Big \downarrow \) 4407

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\frac {\left (6 a^4-5 a^2+2\right ) \int -\frac {a+b x}{b x}d\text {sech}^{-1}(a+b x)}{a}+\frac {2 \left (1-a^2\right )^2 \text {sech}^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}+\frac {\left (2-5 a^2\right ) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\text {sech}^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^3 \left (\frac {\text {sech}^{-1}(a+b x)}{3 b^3 x^3}+\frac {1}{3} \left (-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) b^2 x^2}+\frac {\frac {\left (2-5 a^2\right ) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}+\frac {\frac {2 \left (1-a^2\right )^2 \text {sech}^{-1}(a+b x)}{a}+\frac {\left (6 a^4-5 a^2+2\right ) \int \frac {\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )}{a-\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )}d\text {sech}^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}}{2 a \left (1-a^2\right )}\right )\right )\)

\(\Big \downarrow \) 4318

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\frac {2 \left (1-a^2\right )^2 \text {sech}^{-1}(a+b x)}{a}-\frac {\left (6 a^4-5 a^2+2\right ) \int \frac {1}{1-\frac {a}{a+b x}}d\text {sech}^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}+\frac {\left (2-5 a^2\right ) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\text {sech}^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^3 \left (\frac {\text {sech}^{-1}(a+b x)}{3 b^3 x^3}+\frac {1}{3} \left (-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) b^2 x^2}+\frac {\frac {\left (2-5 a^2\right ) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}+\frac {\frac {2 \left (1-a^2\right )^2 \text {sech}^{-1}(a+b x)}{a}-\frac {\left (6 a^4-5 a^2+2\right ) \int \frac {1}{1-a \sin \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )}d\text {sech}^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}}{2 a \left (1-a^2\right )}\right )\right )\)

\(\Big \downarrow \) 3138

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\frac {2 \left (1-a^2\right )^2 \text {sech}^{-1}(a+b x)}{a}-\frac {2 \left (6 a^4-5 a^2+2\right ) \int \frac {1}{-\left ((a+1) \tanh ^2\left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )-a+1}d\tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{a}}{a \left (1-a^2\right )}+\frac {\left (2-5 a^2\right ) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\text {sech}^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\left (2-5 a^2\right ) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}+\frac {\frac {2 \left (1-a^2\right )^2 \text {sech}^{-1}(a+b x)}{a}-\frac {2 \left (6 a^4-5 a^2+2\right ) \text {arctanh}\left (\frac {\sqrt {a+1} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{a \sqrt {1-a^2}}}{a \left (1-a^2\right )}}{2 a \left (1-a^2\right )}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\text {sech}^{-1}(a+b x)}{3 b^3 x^3}\right )\)

Input:

Int[ArcSech[a + b*x]/x^4,x]
 

Output:

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

Defintions of rubi rules used

rule 221

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3138

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ 
e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[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 4272

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] + Sim 
p[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] && Integ 
erQ[2*n]
 

rule 4318

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

rule 4407

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
 (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(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 4548

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] + Simp[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 5991

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] + Simp[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 6875

Int[((a_.) + ArcSech[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1)   Subst[Int[(a + b*x)^p*Sech[x]*T 
anh[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]
 

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

Input:

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

Output:

-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*arctanh(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*(-b*x*a+(-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)*ln(2*(-b*x*a+(-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*(-b*x*a+(-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/(a-1)/(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 = 0.16 (sec) , antiderivative size = 987, normalized size of antiderivative = 5.01 \[ \int \frac {\text {sech}^{-1}(a+b x)}{x^4} \, dx =\text {Too large to display} \] Input:

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

Output:

[-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)*s 
qrt(-(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)*sqr 
t(-(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...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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]

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

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

Output:

integrate(arcsech(b*x + a)/x^4, x)
 

Mupad [F(-1)]

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

int(acosh(1/(a + b*x))/x^4,x)
 

Output:

int(acosh(1/(a + b*x))/x^4, x)
 

Reduce [F]

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

int(asech(b*x+a)/x^4,x)
 

Output:

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