\(\int \text {arcsinh}(a+b x) \, dx\) [61]

Optimal result

Integrand size = 6, antiderivative size = 34 \[ \int \text {arcsinh}(a+b x) \, dx=-\frac {\sqrt {1+(a+b x)^2}}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5858, 5772, 267} \[ \int \text {arcsinh}(a+b x) \, dx=\frac {(a+b x) \text {arcsinh}(a+b x)}{b}-\frac {\sqrt {(a+b x)^2+1}}{b} \]

[In]

Int[ArcSinh[a + b*x],x]

[Out]

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

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5858

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSinh[x])^n, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int \text {arcsinh}(x) \, dx,x,a+b x)}{b} \\ & = \frac {(a+b x) \text {arcsinh}(a+b x)}{b}-\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\sqrt {1+(a+b x)^2}}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)}{b} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(78\) vs. \(2(34)=68\).

Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.29 \[ \int \text {arcsinh}(a+b x) \, dx=x \text {arcsinh}(a+b x)-\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}+2 a \text {arctanh}\left (\frac {b x}{\sqrt {1+a^2}-\sqrt {1+a^2+2 a b x+b^2 x^2}}\right )}{b} \]

[In]

Integrate[ArcSinh[a + b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}}{b}\)	\(31\)
default	\(\frac {\left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}}{b}\)	\(31\)
parts	\(x \,\operatorname {arcsinh}\left (b x +a \right )-b \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )\)	\(84\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.68 \[ \int \text {arcsinh}(a+b x) \, dx=\frac {{\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b} \]

[In]

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

[Out]

((b*x + a)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/b

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \text {arcsinh}(a+b x) \, dx=\begin {cases} \frac {a \operatorname {asinh}{\left (a + b x \right )}}{b} + x \operatorname {asinh}{\left (a + b x \right )} - \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {asinh}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a*asinh(a + b*x)/b + x*asinh(a + b*x) - sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/b, Ne(b, 0)), (x*asinh
(a), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \text {arcsinh}(a+b x) \, dx=\frac {{\left (b x + a\right )} \operatorname {arsinh}\left (b x + a\right ) - \sqrt {{\left (b x + a\right )}^{2} + 1}}{b} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (32) = 64\).

Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.71 \[ \int \text {arcsinh}(a+b x) \, dx=-b {\left (\frac {a \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |}\right )}{b {\left | b \right |}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{2}}\right )} + x \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right ) \]

[In]

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

[Out]

-b*(a*log(-a*b - (x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*abs(b))/(b*abs(b)) + sqrt(b^2*x^2 + 2*a*b*x +
a^2 + 1)/b^2) + x*log(b*x + a + sqrt((b*x + a)^2 + 1))

Mupad [B] (verification not implemented)

Time = 2.93 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.24 \[ \int \text {arcsinh}(a+b x) \, dx=x\,\mathrm {asinh}\left (a+b\,x\right )-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{b}+\frac {a\,\ln \left (\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}+\frac {x\,b^2+a\,b}{\sqrt {b^2}}\right )}{\sqrt {b^2}} \]

[In]

int(asinh(a + b*x),x)

[Out]

x*asinh(a + b*x) - (a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2)/b + (a*log((a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2) + (a*b +
b^2*x)/(b^2)^(1/2)))/(b^2)^(1/2)