\(\int (a+b \text {arcsinh}(c+d x)) \, dx\) [119]

Optimal result

Integrand size = 10, antiderivative size = 39 \[ \int (a+b \text {arcsinh}(c+d x)) \, dx=a x-\frac {b \sqrt {1+(c+d x)^2}}{d}+\frac {b (c+d x) \text {arcsinh}(c+d x)}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5858, 5772, 267} \[ \int (a+b \text {arcsinh}(c+d x)) \, dx=a x+\frac {b (c+d x) \text {arcsinh}(c+d x)}{d}-\frac {b \sqrt {(c+d x)^2+1}}{d} \]

[In]

Int[a + b*ArcSinh[c + d*x],x]

[Out]

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(39)=78\).

Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.13 \[ \int (a+b \text {arcsinh}(c+d x)) \, dx=a x+b x \text {arcsinh}(c+d x)-\frac {b \left (\sqrt {1+c^2+2 c d x+d^2 x^2}+2 c \text {arctanh}\left (\frac {d x}{\sqrt {1+c^2}-\sqrt {1+c^2+2 c d x+d^2 x^2}}\right )\right )}{d} \]

[In]

Integrate[a + b*ArcSinh[c + d*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92

method	result	size
default	\(a x +\frac {b \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\)	\(36\)
parts	\(a x +\frac {b \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\)	\(36\)
derivativedivides	\(\frac {\left (d x +c \right ) a +b \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.67 \[ \int (a+b \text {arcsinh}(c+d x)) \, dx=\frac {a d x + {\left (b d x + b c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} b}{d} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.31 \[ \int (a+b \text {arcsinh}(c+d x)) \, dx=a x + b \left (\begin {cases} \frac {c \operatorname {asinh}{\left (c + d x \right )}}{d} + x \operatorname {asinh}{\left (c + d x \right )} - \frac {\sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} & \text {for}\: d \neq 0 \\x \operatorname {asinh}{\left (c \right )} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int (a+b \text {arcsinh}(c+d x)) \, dx=a x + \frac {{\left ({\left (d x + c\right )} \operatorname {arsinh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} + 1}\right )} b}{d} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (37) = 74\).

Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.54 \[ \int (a+b \text {arcsinh}(c+d x)) \, dx=-{\left (d {\left (\frac {c \log \left (-c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} {\left | d \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {{\left (d x + c\right )}^{2} + 1}\right )\right )} b + a x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.98 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.18 \[ \int (a+b \text {arcsinh}(c+d x)) \, dx=a\,x+b\,x\,\mathrm {asinh}\left (c+d\,x\right )-\frac {b\,\sqrt {c^2+2\,c\,d\,x+d^2\,x^2+1}}{d}+\frac {b\,c\,d^2\,\ln \left (\sqrt {c^2+2\,c\,d\,x+d^2\,x^2+1}+\frac {x\,d^2+c\,d}{\sqrt {d^2}}\right )}{{\left (d^2\right )}^{3/2}} \]

[In]

int(a + b*asinh(c + d*x),x)

[Out]

a*x + b*x*asinh(c + d*x) - (b*(c^2 + d^2*x^2 + 2*c*d*x + 1)^(1/2))/d + (b*c*d^2*log((c^2 + d^2*x^2 + 2*c*d*x +
1)^(1/2) + (c*d + d^2*x)/(d^2)^(1/2)))/(d^2)^(3/2)