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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6457, 5577, 3867, 3855, 3852, 8} \[ \int x^2 \text {csch}^{-1}(a+b x) \, dx=\frac {a^3 \text {csch}^{-1}(a+b x)}{3 b^3}-\frac {\left (1-6 a^2\right ) \text {arctanh}\left (\sqrt {\frac {1}{(a+b x)^2}+1}\right )}{6 b^3}-\frac {5 a (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}{6 b^3}+\frac {x (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}{6 b^2}+\frac {1}{3} x^3 \text {csch}^{-1}(a+b x) \]

[In]

Int[x^2*ArcCsch[a + b*x],x]

[Out]

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

Rule 8

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

Rule 3852

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

Rule 3855

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

Rule 3867

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*Cot[c + d*x]*((a + b*Csc[c + d*x])^(
n - 2)/(d*(n - 1))), x] + Dist[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 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}& = -\frac {\text {Subst}\left (\int x \coth (x) \text {csch}(x) (-a+\text {csch}(x))^2 \, dx,x,\text {csch}^{-1}(a+b x)\right )}{b^3} \\ & = \frac {1}{3} x^3 \text {csch}^{-1}(a+b x)-\frac {\text {Subst}\left (\int (-a+\text {csch}(x))^3 \, dx,x,\text {csch}^{-1}(a+b x)\right )}{3 b^3} \\ & = \frac {x (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{6 b^2}+\frac {1}{3} x^3 \text {csch}^{-1}(a+b x)-\frac {\text {Subst}\left (\int \left (-2 a^3-\left (1-6 a^2\right ) \text {csch}(x)-5 a \text {csch}^2(x)\right ) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{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 {(5 a) \text {Subst}\left (\int \text {csch}^2(x) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{6 b^3}+\frac {\left (1-6 a^2\right ) \text {Subst}\left (\int \text {csch}(x) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{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}-\frac {(5 i a) \text {Subst}\left (\int 1 \, dx,x,-i (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}\right )}{6 b^3} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.17 \[ \int x^2 \text {csch}^{-1}(a+b x) \, dx=\frac {\left (-5 a^2-4 a b x+b^2 x^2\right ) \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}+2 b^3 x^3 \text {csch}^{-1}(a+b x)+2 a^3 \text {arcsinh}\left (\frac {1}{a+b x}\right )+\left (-1+6 a^2\right ) \log \left ((a+b x) \left (1+\sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{6 b^3} \]

[In]

Integrate[x^2*ArcCsch[a + b*x],x]

[Out]

((-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)

Maple [A] (verified)

Time = 0.32 (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	\(\frac {x^{3} \operatorname {arccsch}\left (b x +a \right )}{3}+\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \left (2 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right ) \sqrt {b^{2}}+6 \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) a^{2} b +x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b \sqrt {b^{2}}-5 \sqrt {b^{2}}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a -\ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) b \right )}{6 b^{3} \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \sqrt {b^{2}}}\)	\(252\)

[In]

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

[Out]

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*arctanh(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 = 0.28 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.78 \[ \int x^2 \text {csch}^{-1}(a+b x) \, dx=\frac {2 \, b^{3} x^{3} \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) + 2 \, a^{3} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + 1\right ) - 2 \, a^{3} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - 1\right ) - {\left (6 \, a^{2} - 1\right )} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a\right ) + {\left (b^{2} x^{2} - 4 \, a b x - 5 \, a^{2}\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{6 \, b^{3}} \]

[In]

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

[Out]

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]

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

[In]

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

[Out]

Integral(x**2*acsch(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

-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]

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

[In]

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

[Out]

integrate(x^2*arccsch(b*x + a), x)

Mupad [F(-1)]

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

[In]

int(x^2*asinh(1/(a + b*x)),x)

[Out]

int(x^2*asinh(1/(a + b*x)), x)