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

Optimal result

Integrand size = 10, antiderivative size = 153 \[ \int x^2 \text {sech}^{-1}(a+b x) \, dx=\frac {5 a \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 b^3}-\frac {x \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 b^2}+\frac {a^3 \text {sech}^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)-\frac {\left (1+6 a^2\right ) \arctan \left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{6 b^3} \]

[Out]

1/3*a^3*arcsech(b*x+a)/b^3+1/3*x^3*arcsech(b*x+a)-1/6*(6*a^2+1)*arctan((b*x+a+1)*((-b*x-a+1)/(b*x+a+1))^(1/2)/
(b*x+a))/b^3+5/6*a*(b*x+a+1)*((-b*x-a+1)/(b*x+a+1))^(1/2)/b^3-1/6*x*(b*x+a+1)*((-b*x-a+1)/(b*x+a+1))^(1/2)/b^2

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 153, 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 = {6456, 5576, 3867, 3855, 3852, 8} \[ \int x^2 \text {sech}^{-1}(a+b x) \, dx=\frac {a^3 \text {sech}^{-1}(a+b x)}{3 b^3}-\frac {\left (6 a^2+1\right ) \arctan \left (\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{6 b^3}+\frac {5 a \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{6 b^3}-\frac {x \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{6 b^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x) \]

[In]

Int[x^2*ArcSech[a + b*x],x]

[Out]

(5*a*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(6*b^3) - (x*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a +
b*x))/(6*b^2) + (a^3*ArcSech[a + b*x])/(3*b^3) + (x^3*ArcSech[a + b*x])/3 - ((1 + 6*a^2)*ArcTan[(Sqrt[(1 - a -
b*x)/(1 + a + b*x)]*(1 + a + b*x))/(a + b*x)])/(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 5576

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

Rule 6456

Int[((a_.) + ArcSech[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[-(d^(m + 1)
)^(-1), Subst[Int[(a + b*x)^p*Sech[x]*Tanh[x]*(d*e - c*f + f*Sech[x])^m, x], x, ArcSech[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 \text {sech}(x) (-a+\text {sech}(x))^2 \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^3} \\ & = \frac {1}{3} x^3 \text {sech}^{-1}(a+b x)-\frac {\text {Subst}\left (\int (-a+\text {sech}(x))^3 \, dx,x,\text {sech}^{-1}(a+b x)\right )}{3 b^3} \\ & = -\frac {x \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 b^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)-\frac {\text {Subst}\left (\int \left (-2 a^3+\left (1+6 a^2\right ) \text {sech}(x)-5 a \text {sech}^2(x)\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{6 b^3} \\ & = -\frac {x \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 b^2}+\frac {a^3 \text {sech}^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)+\frac {(5 a) \text {Subst}\left (\int \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{6 b^3}-\frac {\left (1+6 a^2\right ) \text {Subst}\left (\int \text {sech}(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{6 b^3} \\ & = -\frac {x \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 b^2}+\frac {a^3 \text {sech}^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)-\frac {\left (1+6 a^2\right ) \arctan \left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{6 b^3}+\frac {(5 i a) \text {Subst}\left (\int 1 \, dx,x,-i \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)\right )}{6 b^3} \\ & = \frac {5 a \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 b^3}-\frac {x \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 b^2}+\frac {a^3 \text {sech}^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)-\frac {\left (1+6 a^2\right ) \arctan \left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{6 b^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.31 \[ \int x^2 \text {sech}^{-1}(a+b x) \, dx=\frac {\sqrt {-\frac {-1+a+b x}{1+a+b x}} \left (5 a^2-b x (1+b x)+a (5+4 b x)\right )+2 b^3 x^3 \text {sech}^{-1}(a+b x)-2 a^3 \log (a+b x)+2 a^3 \log \left (1+\sqrt {-\frac {-1+a+b x}{1+a+b x}}+a \sqrt {-\frac {-1+a+b x}{1+a+b x}}+b x \sqrt {-\frac {-1+a+b x}{1+a+b x}}\right )+i \left (1+6 a^2\right ) \log \left (-2 i (a+b x)+2 \sqrt {-\frac {-1+a+b x}{1+a+b x}} (1+a+b x)\right )}{6 b^3} \]

[In]

Integrate[x^2*ArcSech[a + b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.24

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arcsech}\left (b x +a \right ) a^{3}}{3}+\operatorname {arcsech}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arcsech}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arcsech}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (2 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right )+6 a^{2} \arcsin \left (b x +a \right )+6 a \sqrt {1-\left (b x +a \right )^{2}}-\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\arcsin \left (b x +a \right )\right )}{6 \sqrt {1-\left (b x +a \right )^{2}}}}{b^{3}}\)	\(189\)
default	\(\frac {-\frac {\operatorname {arcsech}\left (b x +a \right ) a^{3}}{3}+\operatorname {arcsech}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arcsech}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arcsech}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (2 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right )+6 a^{2} \arcsin \left (b x +a \right )+6 a \sqrt {1-\left (b x +a \right )^{2}}-\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\arcsin \left (b x +a \right )\right )}{6 \sqrt {1-\left (b x +a \right )^{2}}}}{b^{3}}\)	\(189\)
parts	\(\frac {x^{3} \operatorname {arcsech}\left (b x +a \right )}{3}+\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) \operatorname {csgn}\left (b \right ) a^{3}-\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \operatorname {csgn}\left (b \right ) b x +5 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \operatorname {csgn}\left (b \right ) a +6 \arctan \left (\frac {\operatorname {csgn}\left (b \right ) \left (b x +a \right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{2}+\arctan \left (\frac {\operatorname {csgn}\left (b \right ) \left (b x +a \right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )\right ) \operatorname {csgn}\left (b \right )}{6 b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\)	\(233\)

[In]

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

[Out]

1/b^3*(-1/3*arcsech(b*x+a)*a^3+arcsech(b*x+a)*a^2*(b*x+a)-arcsech(b*x+a)*a*(b*x+a)^2+1/3*arcsech(b*x+a)*(b*x+a
)^3+1/6*(-(b*x+a-1)/(b*x+a))^(1/2)*(b*x+a)*((b*x+a+1)/(b*x+a))^(1/2)*(2*a^3*arctanh(1/(1-(b*x+a)^2)^(1/2))+6*a
^2*arcsin(b*x+a)+6*a*(1-(b*x+a)^2)^(1/2)-(b*x+a)*(1-(b*x+a)^2)^(1/2)+arcsin(b*x+a))/(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. 327 vs. \(2 (131) = 262\).

Time = 0.30 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.14 \[ \int x^2 \text {sech}^{-1}(a+b x) \, dx=\frac {2 \, b^{3} x^{3} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) + a^{3} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) - a^{3} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) - {\left (6 \, a^{2} + 1\right )} \arctan \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (b^{2} x^{2} - 4 \, a b x - 5 \, a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{6 \, b^{3}} \]

[In]

integrate(x^2*arcsech(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)) +
a^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)/x) - a^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)/x) - (6*a^2 + 1)*arctan((b^2*x^2 + 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))/(b^2*x^2 + 2*a*b*x + a^2 - 1)) - (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]

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

[In]

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

[Out]

Integral(x**2*asech(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(x^2*arcsech(b*x + a), x)

Mupad [F(-1)]

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

[In]

int(x^2*acosh(1/(a + b*x)),x)

[Out]

int(x^2*acosh(1/(a + b*x)), x)