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

Optimal result

Integrand size = 10, antiderivative size = 110 \[ \int x^2 \text {csch}^{-1}(a+b x) \, dx=-\frac {5 a (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{6 b^3}+\frac {x (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{6 b^2}+\frac {a^3 \text {csch}^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {csch}^{-1}(a+b x)-\frac {\left (1-6 a^2\right ) \text {arctanh}\left (\sqrt {1+\frac {1}{(a+b x)^2}}\right )}{6 b^3} \] Output:

-5/6*a*(b*x+a)*(1+1/(b*x+a)^2)^(1/2)/b^3+1/6*x*(b*x+a)*(1+1/(b*x+a)^2)^(1/ 
2)/b^2+1/3*a^3*arccsch(b*x+a)/b^3+1/3*x^3*arccsch(b*x+a)-1/6*(-6*a^2+1)*ar 
ctanh((1+1/(b*x+a)^2)^(1/2))/b^3
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.17 \[ \int x^2 \text {csch}^{-1}(a+b x) \, dx=\frac {\left (-5 a^2-4 a b x+b^2 x^2\right ) \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}+2 b^3 x^3 \text {csch}^{-1}(a+b x)+2 a^3 \text {arcsinh}\left (\frac {1}{a+b x}\right )+\left (-1+6 a^2\right ) \log \left ((a+b x) \left (1+\sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{6 b^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6876, 5992, 3042, 4269, 2009}

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.

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

\(\Big \downarrow \) 6876

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

\(\Big \downarrow \) 5992

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4269

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4269

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*C 
ot[c + d*x]*((a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1))), x] + Simp[1/(n - 1) 
   Int[(a + b*Csc[c + d*x])^(n - 3)*Simp[a^3*(n - 1) + (b*(b^2*(n - 2) + 3* 
a^2*(n - 1)))*Csc[c + d*x] + (a*b^2*(3*n - 4))*Csc[c + d*x]^2, x], x], x] / 
; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
 

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 = 170, normalized size of antiderivative = 1.55

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arccsch}\left (b x +a \right ) a^{3}}{3}+\operatorname {arccsch}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arccsch}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arccsch}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (2 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right )+6 a^{2} \operatorname {arcsinh}\left (b x +a \right )-6 a \sqrt {\left (b x +a \right )^{2}+1}+\left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}+1}-\operatorname {arcsinh}\left (b x +a \right )\right )}{6 \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{3}}\)	\(170\)
default	\(\frac {-\frac {\operatorname {arccsch}\left (b x +a \right ) a^{3}}{3}+\operatorname {arccsch}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arccsch}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arccsch}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (2 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right )+6 a^{2} \operatorname {arcsinh}\left (b x +a \right )-6 a \sqrt {\left (b x +a \right )^{2}+1}+\left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}+1}-\operatorname {arcsinh}\left (b x +a \right )\right )}{6 \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{3}}\)	\(170\)
parts	\(\frac {x^{3} \operatorname {arccsch}\left (b x +a \right )}{3}+\frac {\sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}\, \left (2 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}}\right ) \sqrt {b^{2}}+6 \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) a^{2} b +x \sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}\, b \sqrt {b^{2}}-5 \sqrt {b^{2}}\, \sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}\, a -\ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 b x a +a^{2}+1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) b \right )}{6 b^{3} \sqrt {\frac {b^{2} x^{2}+2 b x a +a^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \sqrt {b^{2}}}\)	\(252\)

Input:

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

Output:

1/b^3*(-1/3*arccsch(b*x+a)*a^3+arccsch(b*x+a)*a^2*(b*x+a)-arccsch(b*x+a)*a 
*(b*x+a)^2+1/3*arccsch(b*x+a)*(b*x+a)^3+1/6*((b*x+a)^2+1)^(1/2)*(2*a^3*arc 
tanh(1/((b*x+a)^2+1)^(1/2))+6*a^2*arcsinh(b*x+a)-6*a*((b*x+a)^2+1)^(1/2)+( 
b*x+a)*((b*x+a)^2+1)^(1/2)-arcsinh(b*x+a))/(((b*x+a)^2+1)/(b*x+a)^2)^(1/2) 
/(b*x+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (94) = 188\).

Time = 0.15 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.78 \[ \int x^2 \text {csch}^{-1}(a+b x) \, dx=\frac {2 \, b^{3} x^{3} \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) + 2 \, a^{3} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + 1\right ) - 2 \, a^{3} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - 1\right ) - {\left (6 \, a^{2} - 1\right )} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a\right ) + {\left (b^{2} x^{2} - 4 \, a b x - 5 \, a^{2}\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{6 \, b^{3}} \] Input:

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

Output:

1/6*(2*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)/(b*x + a)) + 2*a^3*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^3*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) - (6*a^2 - 1)*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^2*x^2 - 4*a*b*x - 5*a^2)*sqrt 
((b^2*x^2 + 2*a*b*x + a^2 + 1)/(b^2*x^2 + 2*a*b*x + a^2)))/b^3
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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