\(\int \frac {\text {csch}^{-1}(a+b x)}{x^3} \, dx\) [6]

Optimal result

Integrand size = 10, antiderivative size = 114 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^3} \, dx=\frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}-\frac {\left (1+2 a^2\right ) b^2 \text {arctanh}\left (\frac {a+\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )}{\sqrt {1+a^2}}\right )}{a^2 \left (1+a^2\right )^{3/2}} \]

[Out]

1/2*b^2*arccsch(b*x+a)/a^2-1/2*arccsch(b*x+a)/x^2-(2*a^2+1)*b^2*arctanh((a+tanh(1/2*arccsch(b*x+a)))/(a^2+1)^(
1/2))/a^2/(a^2+1)^(3/2)+1/2*b*(b*x+a)*(1+1/(b*x+a)^2)^(1/2)/a/(a^2+1)/x

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 114, 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 = {6457, 5577, 3870, 4004, 3916, 2739, 632, 212} \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^3} \, dx=-\frac {\left (2 a^2+1\right ) b^2 \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )+a}{\sqrt {a^2+1}}\right )}{a^2 \left (a^2+1\right )^{3/2}}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}+\frac {b (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}{2 a \left (a^2+1\right ) x}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2} \]

[In]

Int[ArcCsch[a + b*x]/x^3,x]

[Out]

(b*(a + b*x)*Sqrt[1 + (a + b*x)^(-2)])/(2*a*(1 + a^2)*x) + (b^2*ArcCsch[a + b*x])/(2*a^2) - ArcCsch[a + b*x]/(
2*x^2) - ((1 + 2*a^2)*b^2*ArcTanh[(a + Tanh[ArcCsch[a + b*x]/2])/Sqrt[1 + a^2]])/(a^2*(1 + a^2)^(3/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*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 5577

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

Rule 6457

Int[((a_.) + ArcCsch[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[-(d^(m + 1)
)^(-1), Subst[Int[(a + b*x)^p*Csch[x]*Coth[x]*(d*e - c*f + f*Csch[x])^m, x], x, ArcCsch[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^2 \text {Subst}\left (\int \frac {x \coth (x) \text {csch}(x)}{(-a+\text {csch}(x))^3} \, dx,x,\text {csch}^{-1}(a+b x)\right )\right ) \\ & = -\frac {\text {csch}^{-1}(a+b x)}{2 x^2}+\frac {1}{2} b^2 \text {Subst}\left (\int \frac {1}{(-a+\text {csch}(x))^2} \, dx,x,\text {csch}^{-1}(a+b x)\right ) \\ & = \frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}+\frac {b^2 \text {Subst}\left (\int \frac {-1-a^2-a \text {csch}(x)}{-a+\text {csch}(x)} \, dx,x,\text {csch}^{-1}(a+b x)\right )}{2 a \left (1+a^2\right )} \\ & = \frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1+2 a^2\right ) b^2\right ) \text {Subst}\left (\int \frac {\text {csch}(x)}{-a+\text {csch}(x)} \, dx,x,\text {csch}^{-1}(a+b x)\right )}{2 a^2 \left (1+a^2\right )} \\ & = \frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1+2 a^2\right ) b^2\right ) \text {Subst}\left (\int \frac {1}{1-a \sinh (x)} \, dx,x,\text {csch}^{-1}(a+b x)\right )}{2 a^2 \left (1+a^2\right )} \\ & = \frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1+2 a^2\right ) b^2\right ) \text {Subst}\left (\int \frac {1}{1-2 a x-x^2} \, dx,x,\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )\right )}{a^2 \left (1+a^2\right )} \\ & = \frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}+\frac {\left (2 \left (1+2 a^2\right ) b^2\right ) \text {Subst}\left (\int \frac {1}{4 \left (1+a^2\right )-x^2} \, dx,x,-2 a-2 \tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )\right )}{a^2 \left (1+a^2\right )} \\ & = \frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}-\frac {\left (1+2 a^2\right ) b^2 \text {arctanh}\left (\frac {a+\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )}{\sqrt {1+a^2}}\right )}{a^2 \left (1+a^2\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.93 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^3} \, dx=\frac {1}{2} \left (\frac {b (a+b x) \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}}{a \left (1+a^2\right ) x}-\frac {\text {csch}^{-1}(a+b x)}{x^2}+\frac {b^2 \text {arcsinh}\left (\frac {1}{a+b x}\right )}{a^2}+\frac {\left (1+2 a^2\right ) b^2 \log (x)}{a^2 \left (1+a^2\right )^{3/2}}-\frac {\left (1+2 a^2\right ) b^2 \log \left (1+a^2+a b x+a \sqrt {1+a^2} \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}+\sqrt {1+a^2} b x \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )}{a^2 \left (1+a^2\right )^{3/2}}\right ) \]

[In]

Integrate[ArcCsch[a + b*x]/x^3,x]

[Out]

((b*(a + b*x)*Sqrt[(1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2])/(a*(1 + a^2)*x) - ArcCsch[a + b*x]/x^2 + (b^2*A
rcSinh[(a + b*x)^(-1)])/a^2 + ((1 + 2*a^2)*b^2*Log[x])/(a^2*(1 + a^2)^(3/2)) - ((1 + 2*a^2)*b^2*Log[1 + a^2 +
a*b*x + a*Sqrt[1 + a^2]*Sqrt[(1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2] + Sqrt[1 + a^2]*b*x*Sqrt[(1 + a^2 + 2*
a*b*x + b^2*x^2)/(a + b*x)^2]])/(a^2*(1 + a^2)^(3/2)))/2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(317\) vs. \(2(100)=200\).

Time = 0.66 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.79

method	result	size
parts	\(-\frac {\operatorname {arccsch}\left (b x +a \right )}{2 x^{2}}+\frac {b \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \left (\left (a^{2}+1\right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right ) a^{2} b x -2 \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a^{4} b x +b \,\operatorname {arctanh}\left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right ) x \left (a^{2}+1\right )^{\frac {3}{2}}+\left (a^{2}+1\right )^{\frac {3}{2}} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a -3 \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a^{2} b x -b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) x \right )}{2 \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}+1\right )^{\frac {5}{2}} x}\)	\(318\)
derivativedivides	\(b^{2} \left (-\frac {\operatorname {arccsch}\left (b x +a \right )}{2 b^{2} x^{2}}+\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (-\operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} a^{3}+\operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} a^{2} \left (b x +a \right )+2 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right ) a^{5}-2 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right ) a^{4} \left (b x +a \right )+\sqrt {\left (b x +a \right )^{2}+1}\, \left (a^{2}+1\right )^{\frac {3}{2}} a -\operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} a +\operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} \left (b x +a \right )+3 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right ) a^{3}-3 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right ) a^{2} \left (b x +a \right )+a \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right )-\ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right ) \left (b x +a \right )\right )}{2 \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}+1\right )^{\frac {5}{2}} b x}\right )\)	\(456\)
default	\(b^{2} \left (-\frac {\operatorname {arccsch}\left (b x +a \right )}{2 b^{2} x^{2}}+\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (-\operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} a^{3}+\operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} a^{2} \left (b x +a \right )+2 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right ) a^{5}-2 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right ) a^{4} \left (b x +a \right )+\sqrt {\left (b x +a \right )^{2}+1}\, \left (a^{2}+1\right )^{\frac {3}{2}} a -\operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} a +\operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} \left (b x +a \right )+3 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right ) a^{3}-3 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right ) a^{2} \left (b x +a \right )+a \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right )-\ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right ) \left (b x +a \right )\right )}{2 \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}+1\right )^{\frac {5}{2}} b x}\right )\)	\(456\)

[In]

int(arccsch(b*x+a)/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*arccsch(b*x+a)/x^2+1/2*b*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*((a^2+1)^(3/2)*arctanh(1/(b^2*x^2+2*a*b*x+a^2+1)^(
1/2))*a^2*b*x-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*x+b*arctanh(1/(b^2*x^2
+2*a*b*x+a^2+1)^(1/2))*x*(a^2+1)^(3/2)+(a^2+1)^(3/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a-3*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^2*b*x-b*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)*x)/((b^2*x^2+2*a*b*x+a^2+1)/(b*x+a)^2)^(1/2)/(b*x+a)/a^2/(a^2+1)^(5/2)/x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (100) = 200\).

Time = 0.28 (sec) , antiderivative size = 461, normalized size of antiderivative = 4.04 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^3} \, dx=\frac {{\left (2 \, a^{2} + 1\right )} \sqrt {a^{2} + 1} b^{2} x^{2} \log \left (-\frac {a^{2} b x + a^{3} - {\left (a b x + a^{2} + {\left (a b x + a^{2}\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1\right )} \sqrt {a^{2} + 1} + {\left (a^{3} + {\left (a^{2} + 1\right )} b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + a}{x}\right ) + {\left (a^{4} + 2 \, a^{2} + 1\right )} b^{2} x^{2} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + 1\right ) - {\left (a^{4} + 2 \, a^{2} + 1\right )} b^{2} x^{2} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - 1\right ) + {\left (a^{3} + a\right )} b^{2} x^{2} - {\left (a^{6} + 2 \, a^{4} + a^{2}\right )} \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) + {\left ({\left (a^{3} + a\right )} b^{2} x^{2} + {\left (a^{4} + a^{2}\right )} b x\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \, {\left (a^{6} + 2 \, a^{4} + a^{2}\right )} x^{2}} \]

[In]

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

[Out]

1/2*((2*a^2 + 1)*sqrt(a^2 + 1)*b^2*x^2*log(-(a^2*b*x + a^3 - (a*b*x + a^2 + (a*b*x + a^2)*sqrt((b^2*x^2 + 2*a*
b*x + a^2 + 1)/(b^2*x^2 + 2*a*b*x + a^2)) + 1)*sqrt(a^2 + 1) + (a^3 + (a^2 + 1)*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)) + a)/x) + (a^4 + 2*a^2 + 1)*b^2*x^2*log(-b*x + (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)) - a + 1) - (a^4 + 2*a^2 + 1)*b^2*x^2*log(-b*x + (b*x + a)*sq
rt((b^2*x^2 + 2*a*b*x + a^2 + 1)/(b^2*x^2 + 2*a*b*x + a^2)) - a - 1) + (a^3 + a)*b^2*x^2 - (a^6 + 2*a^4 + a^2)
*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)) + ((a^3 + a)*b^2
*x^2 + (a^4 + a^2)*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^6 + 2*a^4 + a^2)*x^
2)

Sympy [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{x^3} \, dx=\int \frac {\operatorname {acsch}{\left (a + b x \right )}}{x^{3}}\, dx \]

[In]

integrate(acsch(b*x+a)/x**3,x)

[Out]

Integral(acsch(a + b*x)/x**3, x)

Maxima [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{x^{3}} \,d x } \]

[In]

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

[Out]

1/2*I*a*b^2*(log(I*(b^2*x + a*b)/b + 1) - log(-I*(b^2*x + a*b)/b + 1))/(a^4 + 2*a^2 + 1) + 1/2*(3*a^2*b^2 + b^
2)*log(x)/(a^6 + 2*a^4 + a^2) + 1/4*((a^4*b^2 - a^2*b^2)*x^2*log(b^2*x^2 + 2*a*b*x + a^2 + 1) + 2*(a^3*b + a*b
)*x + 2*(a^6 + 2*a^4 - (a^4*b^2 + 2*a^2*b^2 + b^2)*x^2 + a^2)*log(b*x + a) - 2*(a^6 + 2*a^4 + a^2)*log(sqrt(b^
2*x^2 + 2*a*b*x + a^2 + 1) + 1))/((a^6 + 2*a^4 + a^2)*x^2) - integrate(1/2*(b^2*x + a*b)/(b^2*x^4 + 2*a*b*x^3
+ (a^2 + 1)*x^2 + (b^2*x^4 + 2*a*b*x^3 + (a^2 + 1)*x^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)), x)

Giac [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(arccsch(b*x + a)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {asinh}\left (\frac {1}{a+b\,x}\right )}{x^3} \,d x \]

[In]

int(asinh(1/(a + b*x))/x^3,x)

[Out]

int(asinh(1/(a + b*x))/x^3, x)