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

Optimal result

Integrand size = 8, antiderivative size = 107 \[ \int x \text {sech}^{-1}(a+b x) \, dx=-\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{2 b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)+\frac {a \arctan \left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6456, 5576, 3858, 3855, 3852, 8} \[ \int x \text {sech}^{-1}(a+b x) \, dx=-\frac {a^2 \text {sech}^{-1}(a+b x)}{2 b^2}+\frac {a \arctan \left (\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{b^2}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x) \]

[In]

Int[x*ArcSech[a + b*x],x]

[Out]

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

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 3858

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

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)) \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2} \\ & = \frac {1}{2} x^2 \text {sech}^{-1}(a+b x)-\frac {\text {Subst}\left (\int (-a+\text {sech}(x))^2 \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 b^2} \\ & = -\frac {a^2 \text {sech}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)-\frac {\text {Subst}\left (\int \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 b^2}+\frac {a \text {Subst}\left (\int \text {sech}(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2} \\ & = -\frac {a^2 \text {sech}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)+\frac {a \arctan \left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{b^2}-\frac {i \text {Subst}\left (\int 1 \, dx,x,-i \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)\right )}{2 b^2} \\ & = -\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{2 b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)+\frac {a \arctan \left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{b^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.64 \[ \int x \text {sech}^{-1}(a+b x) \, dx=\frac {-\sqrt {-\frac {-1+a+b x}{1+a+b x}} (1+a+b x)+b^2 x^2 \text {sech}^{-1}(a+b x)+a^2 \log (a+b x)-a^2 \log \left (1+\sqrt {-\frac {-1+a+b x}{1+a+b x}}+a \sqrt {-\frac {-1+a+b x}{1+a+b x}}+b x \sqrt {-\frac {-1+a+b x}{1+a+b x}}\right )-2 i a \log \left (-2 i (a+b x)+2 \sqrt {-\frac {-1+a+b x}{1+a+b x}} (1+a+b x)\right )}{2 b^2} \]

[In]

Integrate[x*ArcSech[a + b*x],x]

[Out]

(-(Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]*(1 + a + b*x)) + b^2*x^2*ArcSech[a + b*x] + a^2*Log[a + b*x] - a^2*Lo
g[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))]] - (2*I)*a*Log[(-2*I)*(a + b*x) + 2*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]*(1 + a + b*x)])/
(2*b^2)

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(\frac {-\operatorname {arcsech}\left (b x +a \right ) a \left (b x +a \right )+\frac {\operatorname {arcsech}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (2 a \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \sqrt {1-\left (b x +a \right )^{2}}}}{b^{2}}\)	\(111\)
default	\(\frac {-\operatorname {arcsech}\left (b x +a \right ) a \left (b x +a \right )+\frac {\operatorname {arcsech}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (2 a \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \sqrt {1-\left (b x +a \right )^{2}}}}{b^{2}}\)	\(111\)
parts	\(\frac {x^{2} \operatorname {arcsech}\left (b x +a \right )}{2}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (\operatorname {csgn}\left (b \right ) \operatorname {arctanh}\left (\frac {1}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{2}+\operatorname {csgn}\left (b \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+2 \arctan \left (\frac {\operatorname {csgn}\left (b \right ) \left (b x +a \right )}{\sqrt {-\left (b x +a -1\right ) \left (b x +a +1\right )}}\right ) a \right ) \operatorname {csgn}\left (b \right )}{2 b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\)	\(163\)

[In]

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

[Out]

1/b^2*(-arcsech(b*x+a)*a*(b*x+a)+1/2*arcsech(b*x+a)*(b*x+a)^2-1/2*(-(b*x+a-1)/(b*x+a))^(1/2)*(b*x+a)*((b*x+a+1
)/(b*x+a))^(1/2)*(2*a*arcsin(b*x+a)+(1-(b*x+a)^2)^(1/2))/(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. 308 vs. \(2 (93) = 186\).

Time = 0.28 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.88 \[ \int x \text {sech}^{-1}(a+b x) \, dx=\frac {2 \, b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - a^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) + a^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) + 4 \, a \arctan \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, {\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \, b^{2}} \]

[In]

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

[Out]

1/4*(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 + 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^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)/x) + 4*a*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)) - 2*(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]

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

[In]

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

[Out]

Integral(x*asech(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

1/4*(2*b^2*x^2*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^2*x^2*log(b*x + a) - (a^2 + 2*a + 1)*log(b*x + a + 1) - 2*(b^2*x^2 - a^2)*log(b*x + a) - (a^2 - 2*a +
1)*log(-b*x - a + 1))/b^2 + integrate(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)*e^(1/2*log(b*x + a + 1) + 1/2*log(-b*x - a + 1)) - 1), x)

Giac [F]

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

[In]

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

[Out]

integrate(x*arcsech(b*x + a), x)

Mupad [F(-1)]

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

[In]

int(x*acosh(1/(a + b*x)),x)

[Out]

int(x*acosh(1/(a + b*x)), x)