\(\int (a+b \text {arcsinh}(c x)) \, dx\) [610]

Optimal result

Integrand size = 8, antiderivative size = 30 \[ \int (a+b \text {arcsinh}(c x)) \, dx=a x-\frac {b \sqrt {1+c^2 x^2}}{c}+b x \text {arcsinh}(c x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5772, 267} \[ \int (a+b \text {arcsinh}(c x)) \, dx=a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c} \]

[In]

Int[a + b*ArcSinh[c*x],x]

[Out]

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

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]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int (a+b \text {arcsinh}(c x)) \, dx=a x-\frac {b \sqrt {1+c^2 x^2}}{c}+b x \text {arcsinh}(c x) \]

[In]

Integrate[a + b*ArcSinh[c*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03

method	result	size
default	\(a x +\frac {b \left (\operatorname {arcsinh}\left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}\right )}{c}\)	\(31\)
parts	\(a x +\frac {b \left (\operatorname {arcsinh}\left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}\right )}{c}\)	\(31\)
derivativedivides	\(\frac {c x a +b \left (\operatorname {arcsinh}\left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}\right )}{c}\)	\(33\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int (a+b \text {arcsinh}(c x)) \, dx=\frac {b c x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + a c x - \sqrt {c^{2} x^{2} + 1} b}{c} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int (a+b \text {arcsinh}(c x)) \, dx=a x + b \left (\begin {cases} x \operatorname {asinh}{\left (c x \right )} - \frac {\sqrt {c^{2} x^{2} + 1}}{c} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int (a+b \text {arcsinh}(c x)) \, dx=a x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b}{c} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int (a+b \text {arcsinh}(c x)) \, dx={\left (x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{c}\right )} b + a x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.58 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int (a+b \text {arcsinh}(c x)) \, dx=a\,x-\frac {b\,\sqrt {c^2\,x^2+1}}{c}+b\,x\,\mathrm {asinh}\left (c\,x\right ) \]

[In]

int(a + b*asinh(c*x),x)

[Out]

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