\(\int \text {arcsinh}(a+b x)^3 \, dx\) [77]

Optimal result

Integrand size = 8, antiderivative size = 78 \[ \int \text {arcsinh}(a+b x)^3 \, dx=-\frac {6 \sqrt {1+(a+b x)^2}}{b}+\frac {6 (a+b x) \text {arcsinh}(a+b x)}{b}-\frac {3 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)^3}{b} \]

[Out]

6*(b*x+a)*arcsinh(b*x+a)/b+(b*x+a)*arcsinh(b*x+a)^3/b-6*(1+(b*x+a)^2)^(1/2)/b-3*arcsinh(b*x+a)^2*(1+(b*x+a)^2)
^(1/2)/b

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5858, 5772, 5798, 267} \[ \int \text {arcsinh}(a+b x)^3 \, dx=\frac {(a+b x) \text {arcsinh}(a+b x)^3}{b}-\frac {3 \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2}{b}+\frac {6 (a+b x) \text {arcsinh}(a+b x)}{b}-\frac {6 \sqrt {(a+b x)^2+1}}{b} \]

[In]

Int[ArcSinh[a + b*x]^3,x]

[Out]

(-6*Sqrt[1 + (a + b*x)^2])/b + (6*(a + b*x)*ArcSinh[a + b*x])/b - (3*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)
/b + ((a + b*x)*ArcSinh[a + b*x]^3)/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 5798

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^p], Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e
, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

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}\left (\int \text {arcsinh}(x)^3 \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \text {arcsinh}(a+b x)^3}{b}-\frac {3 \text {Subst}\left (\int \frac {x \text {arcsinh}(x)^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b} \\ & = -\frac {3 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)^3}{b}+\frac {6 \text {Subst}(\int \text {arcsinh}(x) \, dx,x,a+b x)}{b} \\ & = \frac {6 (a+b x) \text {arcsinh}(a+b x)}{b}-\frac {3 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)^3}{b}-\frac {6 \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b} \\ & = -\frac {6 \sqrt {1+(a+b x)^2}}{b}+\frac {6 (a+b x) \text {arcsinh}(a+b x)}{b}-\frac {3 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)^3}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90 \[ \int \text {arcsinh}(a+b x)^3 \, dx=\frac {-6 \sqrt {1+(a+b x)^2}+6 (a+b x) \text {arcsinh}(a+b x)-3 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2+(a+b x) \text {arcsinh}(a+b x)^3}{b} \]

[In]

Integrate[ArcSinh[a + b*x]^3,x]

[Out]

(-6*Sqrt[1 + (a + b*x)^2] + 6*(a + b*x)*ArcSinh[a + b*x] - 3*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2 + (a + b
*x)*ArcSinh[a + b*x]^3)/b

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {\operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )-3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}}{b}\)	\(67\)
default	\(\frac {\operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )-3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}}{b}\)	\(67\)

[In]

int(arcsinh(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/b*(arcsinh(b*x+a)^3*(b*x+a)-3*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)+6*(b*x+a)*arcsinh(b*x+a)-6*(1+(b*x+a)^2)^
(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.78 \[ \int \text {arcsinh}(a+b x)^3 \, dx=\frac {{\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 6 \, {\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b} \]

[In]

integrate(arcsinh(b*x+a)^3,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.40 \[ \int \text {arcsinh}(a+b x)^3 \, dx=\begin {cases} \frac {a \operatorname {asinh}^{3}{\left (a + b x \right )}}{b} + \frac {6 a \operatorname {asinh}{\left (a + b x \right )}}{b} + x \operatorname {asinh}^{3}{\left (a + b x \right )} + 6 x \operatorname {asinh}{\left (a + b x \right )} - \frac {3 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{b} - \frac {6 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {asinh}^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(asinh(b*x+a)**3,x)

[Out]

Piecewise((a*asinh(a + b*x)**3/b + 6*a*asinh(a + b*x)/b + x*asinh(a + b*x)**3 + 6*x*asinh(a + b*x) - 3*sqrt(a*
*2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**2/b - 6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/b, Ne(b, 0)), (x*as
inh(a)**3, True))

Maxima [F]

\[ \int \text {arcsinh}(a+b x)^3 \, dx=\int { \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]

[In]

integrate(arcsinh(b*x+a)^3,x, algorithm="maxima")

[Out]

x*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3 - integrate(3*(b^3*x^3 + 2*a*b^2*x^2 + (a^2*b + b)*x + sq
rt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b^2*x^2 + a*b*x))*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2/(b^3*x^3
+ 3*a*b^2*x^2 + a^3 + (3*a^2*b + b)*x + (b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + a), x)

Giac [F]

\[ \int \text {arcsinh}(a+b x)^3 \, dx=\int { \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]

[In]

integrate(arcsinh(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(arcsinh(b*x + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \text {arcsinh}(a+b x)^3 \, dx=\int {\mathrm {asinh}\left (a+b\,x\right )}^3 \,d x \]

[In]

int(asinh(a + b*x)^3,x)

[Out]

int(asinh(a + b*x)^3, x)