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

Optimal result

Integrand size = 8, antiderivative size = 75 \[ \int x \text {csch}^{-1}(a+b x) \, dx=\frac {(a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 b^2}-\frac {a^2 \text {csch}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {csch}^{-1}(a+b x)-\frac {a \text {arctanh}\left (\sqrt {1+\frac {1}{(a+b x)^2}}\right )}{b^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.47 \[ \int x \text {csch}^{-1}(a+b x) \, dx=\frac {(a+b x) \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}+b^2 x^2 \text {csch}^{-1}(a+b x)-a^2 \text {arcsinh}\left (\frac {1}{a+b x}\right )-2 a \log \left ((a+b x) \left (1+\sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{2 b^2} \] Input:

Integrate[x*ArcCsch[a + b*x],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.625, Rules used = {6876, 25, 5992, 3042, 4260, 25, 26, 3042, 25, 26, 4254, 24, 4257}

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.

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

\(\Big \downarrow \) 6876

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5992

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4260

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4254

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 4257

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

Input:

Int[x*ArcCsch[a + b*x],x]
 

Output:

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

Defintions of rubi rules used

rule 24

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

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

rule 4254

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, 
 d}, x] && IGtQ[n/2, 0]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

rule 4260

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

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 [A] (verified)

Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {arccsch}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\operatorname {arccsch}\left (b x +a \right ) a \left (b x +a \right )-\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (2 a \,\operatorname {arcsinh}\left (b x +a \right )-\sqrt {\left (b x +a \right )^{2}+1}\right )}{2 \left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}}}{b^{2}}\)	\(97\)
default	\(\frac {\frac {\operatorname {arccsch}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\operatorname {arccsch}\left (b x +a \right ) a \left (b x +a \right )-\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (2 a \,\operatorname {arcsinh}\left (b x +a \right )-\sqrt {\left (b x +a \right )^{2}+1}\right )}{2 \left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}}}{b^{2}}\)	\(97\)
parts	\(\frac {x^{2} \operatorname {arccsch}\left (b x +a \right )}{2}-\frac {\sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}\, \left (a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}}\right ) \sqrt {b^{2}}+2 a \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) b -\sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}\, \sqrt {b^{2}}\right )}{2 b^{2} \sqrt {\frac {b^{2} x^{2}+2 b x a +a^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \sqrt {b^{2}}}\)	\(177\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (65) = 130\).

Time = 0.10 (sec) , antiderivative size = 285, normalized size of antiderivative = 3.80 \[ \int x \text {csch}^{-1}(a+b x) \, dx=\frac {b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - a^{2} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + 1\right ) + a^{2} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - 1\right ) + 2 \, a \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a\right ) + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \, b^{2}} \] Input:

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

Output:

1/2*(b^2*x^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^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^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) + 2*a*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) + (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)))/b^2
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

integrate(x*arccsch(b*x + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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