\(\int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx\) [5]

Optimal result

Integrand size = 10, antiderivative size = 63 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=-\frac {b \text {csch}^{-1}(a+b x)}{a}-\frac {\text {csch}^{-1}(a+b x)}{x}+\frac {2 b \text {arctanh}\left (\frac {a+\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )}{\sqrt {1+a^2}}\right )}{a \sqrt {1+a^2}} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(63)=126\).

Time = 0.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.24 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=-\frac {\text {csch}^{-1}(a+b x)}{x}-\frac {b \left (\sqrt {1+a^2} \text {arcsinh}\left (\frac {1}{a+b x}\right )+\log (x)-\log \left (1+a^2+a b x+a \sqrt {1+a^2} \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}+\sqrt {1+a^2} b x \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{a \sqrt {1+a^2}} \] Input:

Integrate[ArcCsch[a + b*x]/x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6876, 5992, 3042, 4270, 3042, 3139, 1083, 219}

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.

\(\displaystyle \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx\)

\(\Big \downarrow \) 6876

\(\displaystyle -b \int \frac {(a+b x)^2 \sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)}{b^2 x^2}d\text {csch}^{-1}(a+b x)\)

\(\Big \downarrow \) 5992

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4270

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

\(\Big \downarrow \) 3042

\(\displaystyle -b \left (-\frac {\int \frac {1}{i a \sin \left (i \text {csch}^{-1}(a+b x)\right )+1}d\text {csch}^{-1}(a+b x)}{a}+\frac {\text {csch}^{-1}(a+b x)}{a}+\frac {\text {csch}^{-1}(a+b x)}{b x}\right )\)

\(\Big \downarrow \) 3139

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

\(\Big \downarrow \) 1083

\(\displaystyle -b \left (\frac {4 \int \frac {1}{4 \left (a^2+1\right )-\left (-2 a-2 \tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )\right )^2}d\left (-2 a-2 \tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )\right )}{a}+\frac {\text {csch}^{-1}(a+b x)}{a}+\frac {\text {csch}^{-1}(a+b x)}{b x}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle -b \left (\frac {2 \text {arctanh}\left (\frac {-2 \tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )-2 a}{2 \sqrt {a^2+1}}\right )}{a \sqrt {a^2+1}}+\frac {\text {csch}^{-1}(a+b x)}{a}+\frac {\text {csch}^{-1}(a+b x)}{b x}\right )\)

Input:

Int[ArcCsch[a + b*x]/x^2,x]
 

Output:

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

Defintions of rubi rules used

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

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

rule 3042

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

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[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 4270

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

rule 5992

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] + Simp[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 6876

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(126\) vs. \(2(57)=114\).

Time = 0.47 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.02

method	result	size
derivativedivides	\(b \left (-\frac {\operatorname {arccsch}\left (b x +a \right )}{b x}-\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \sqrt {a^{2}+1}-\ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right )\right )}{\sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \sqrt {a^{2}+1}}\right )\)	\(127\)
default	\(b \left (-\frac {\operatorname {arccsch}\left (b x +a \right )}{b x}-\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \sqrt {a^{2}+1}-\ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right )\right )}{\sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \sqrt {a^{2}+1}}\right )\)	\(127\)
parts	\(-\frac {\operatorname {arccsch}\left (b x +a \right )}{x}-\frac {b \sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}}\right ) \sqrt {a^{2}+1}-\ln \left (\frac {2 a^{2}+2+2 b x a +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}}{x}\right )\right )}{\sqrt {\frac {b^{2} x^{2}+2 b x a +a^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \sqrt {a^{2}+1}}\)	\(152\)

Input:

int(arccsch(b*x+a)/x^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (57) = 114\).

Time = 0.12 (sec) , antiderivative size = 343, normalized size of antiderivative = 5.44 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=-\frac {{\left (a^{2} + 1\right )} b x \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + 1\right ) - {\left (a^{2} + 1\right )} b x \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - 1\right ) - \sqrt {a^{2} + 1} b x \log \left (-\frac {a^{2} b x + a^{3} + {\left (a b x + a^{2} + {\left (a b x + a^{2}\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1\right )} \sqrt {a^{2} + 1} + {\left (a^{3} + {\left (a^{2} + 1\right )} b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + a}{x}\right ) + {\left (a^{3} + a\right )} \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right )}{{\left (a^{3} + a\right )} x} \] Input:

integrate(arccsch(b*x+a)/x^2,x, algorithm="fricas")
 

Output:

-((a^2 + 1)*b*x*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^2 + 1)*b*x*log(-b*x + (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)) - a - 1) - sqrt 
(a^2 + 1)*b*x*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^3 + a)*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)*x)
 

Sympy [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=\int \frac {\operatorname {acsch}{\left (a + b x \right )}}{x^{2}}\, dx \] Input:

integrate(acsch(b*x+a)/x**2,x)
 

Output:

Integral(acsch(a + b*x)/x**2, x)
 

Maxima [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{x^{2}} \,d x } \] Input:

integrate(arccsch(b*x+a)/x^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{x^{2}} \,d x } \] Input:

integrate(arccsch(b*x+a)/x^2,x, algorithm="giac")
 

Output:

integrate(arccsch(b*x + a)/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=\int \frac {\mathrm {asinh}\left (\frac {1}{a+b\,x}\right )}{x^2} \,d x \] Input:

int(asinh(1/(a + b*x))/x^2,x)
                                                                                    
                                                                                    
 

Output:

int(asinh(1/(a + b*x))/x^2, x)
 

Reduce [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=\int \frac {\mathit {acsch} \left (b x +a \right )}{x^{2}}d x \] Input:

int(acsch(b*x+a)/x^2,x)
 

Output:

int(acsch(a + b*x)/x**2,x)