Integrand size = 10, antiderivative size = 203 \[ \int x \text {arcsinh}(a+b x)^3 \, dx=\frac {6 a \sqrt {1+(a+b x)^2}}{b^2}-\frac {3 (a+b x) \sqrt {1+(a+b x)^2}}{8 b^2}+\frac {3 \text {arcsinh}(a+b x)}{8 b^2}-\frac {6 a (a+b x) \text {arcsinh}(a+b x)}{b^2}+\frac {3 (a+b x)^2 \text {arcsinh}(a+b x)}{4 b^2}+\frac {3 a \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b^2}-\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{4 b^2}+\frac {\text {arcsinh}(a+b x)^3}{4 b^2}-\frac {a^2 \text {arcsinh}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {arcsinh}(a+b x)^3 \]
3/8*arcsinh(b*x+a)/b^2-6*a*(b*x+a)*arcsinh(b*x+a)/b^2+3/4*(b*x+a)^2*arcsin h(b*x+a)/b^2+1/4*arcsinh(b*x+a)^3/b^2-1/2*a^2*arcsinh(b*x+a)^3/b^2+1/2*x^2 *arcsinh(b*x+a)^3+6*a*(1+(b*x+a)^2)^(1/2)/b^2-3/8*(b*x+a)*(1+(b*x+a)^2)^(1 /2)/b^2+3*a*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)/b^2-3/4*(b*x+a)*arcsinh(b *x+a)^2*(1+(b*x+a)^2)^(1/2)/b^2
Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.64 \[ \int x \text {arcsinh}(a+b x)^3 \, dx=\frac {3 (15 a-b x) \sqrt {1+a^2+2 a b x+b^2 x^2}+\left (3-42 a^2-36 a b x+6 b^2 x^2\right ) \text {arcsinh}(a+b x)+6 (3 a-b x) \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^2+\left (2-4 a^2+4 b^2 x^2\right ) \text {arcsinh}(a+b x)^3}{8 b^2} \]
(3*(15*a - b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (3 - 42*a^2 - 36*a*b*x + 6*b^2*x^2)*ArcSinh[a + b*x] + 6*(3*a - b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^ 2*x^2]*ArcSinh[a + b*x]^2 + (2 - 4*a^2 + 4*b^2*x^2)*ArcSinh[a + b*x]^3)/(8 *b^2)
Time = 0.58 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6274, 25, 27, 6243, 6258, 3042, 3798, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x \text {arcsinh}(a+b x)^3 \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int x \text {arcsinh}(a+b x)^3d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -x \text {arcsinh}(a+b x)^3d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -b x \text {arcsinh}(a+b x)^3d(a+b x)}{b^2}\) |
| \(\Big \downarrow \) 6243 |
| \(\displaystyle -\frac {\frac {3}{2} \int \frac {b^2 x^2 \text {arcsinh}(a+b x)^2}{\sqrt {(a+b x)^2+1}}d(a+b x)-\frac {1}{2} b^2 x^2 \text {arcsinh}(a+b x)^3}{b^2}\) |
| \(\Big \downarrow \) 6258 |
| \(\displaystyle -\frac {\frac {3}{2} \int b^2 x^2 \text {arcsinh}(a+b x)^2d\text {arcsinh}(a+b x)-\frac {1}{2} b^2 x^2 \text {arcsinh}(a+b x)^3}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {1}{2} b^2 x^2 \text {arcsinh}(a+b x)^3+\frac {3}{2} \int \text {arcsinh}(a+b x)^2 (a+i \sin (i \text {arcsinh}(a+b x)))^2d\text {arcsinh}(a+b x)}{b^2}\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle -\frac {\frac {3}{2} \int \left (a^2 \text {arcsinh}(a+b x)^2+(a+b x)^2 \text {arcsinh}(a+b x)^2-2 a (a+b x) \text {arcsinh}(a+b x)^2\right )d\text {arcsinh}(a+b x)-\frac {1}{2} b^2 x^2 \text {arcsinh}(a+b x)^3}{b^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {1}{3} a^2 \text {arcsinh}(a+b x)^3-\frac {1}{6} \text {arcsinh}(a+b x)^3-2 a \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2-\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)+4 a (a+b x) \text {arcsinh}(a+b x)+\frac {1}{4} (-a-b x)-4 a \sqrt {(a+b x)^2+1}+\frac {1}{4} (a+b x) \sqrt {(a+b x)^2+1}\right )-\frac {1}{2} b^2 x^2 \text {arcsinh}(a+b x)^3}{b^2}\) |
-((-1/2*(b^2*x^2*ArcSinh[a + b*x]^3) + (3*((-a - b*x)/4 - 4*a*Sqrt[1 + (a + b*x)^2] + ((a + b*x)*Sqrt[1 + (a + b*x)^2])/4 + 4*a*(a + b*x)*ArcSinh[a + b*x] - ((a + b*x)^2*ArcSinh[a + b*x])/2 - 2*a*Sqrt[1 + (a + b*x)^2]*ArcS inh[a + b*x]^2 + ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/2 - ArcSinh[a + b*x]^3/6 + (a^2*ArcSinh[a + b*x]^3)/3))/2)/b^2)
3.1.76.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ m, 0] || NeQ[a^2 - b^2, 0])
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^( n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n , 0] && NeQ[m, -1]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/S qrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d]) Subst[I nt[(a + b*x)^n*(c*f + g*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b , c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && GtQ[d, 0] && (Gt Q[m, 0] || IGtQ[n, 0])
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.12 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arcsinh}\left (b x +a \right )^{3} \left (1+\left (b x +a \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )}{4}-\frac {\operatorname {arcsinh}\left (b x +a \right )^{3}}{4}+\frac {3 \,\operatorname {arcsinh}\left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right )}{4}-\frac {3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (b x +a \right )}{8}-a \left (\operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )-3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}\right )}{b^{2}}\) | \(169\) |
| default | \(\frac {\frac {\operatorname {arcsinh}\left (b x +a \right )^{3} \left (1+\left (b x +a \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )}{4}-\frac {\operatorname {arcsinh}\left (b x +a \right )^{3}}{4}+\frac {3 \,\operatorname {arcsinh}\left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right )}{4}-\frac {3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (b x +a \right )}{8}-a \left (\operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )-3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}\right )}{b^{2}}\) | \(169\) |
1/b^2*(1/2*arcsinh(b*x+a)^3*(1+(b*x+a)^2)-3/4*arcsinh(b*x+a)^2*(1+(b*x+a)^ 2)^(1/2)*(b*x+a)-1/4*arcsinh(b*x+a)^3+3/4*arcsinh(b*x+a)*(1+(b*x+a)^2)-3/8 *(b*x+a)*(1+(b*x+a)^2)^(1/2)-3/8*arcsinh(b*x+a)-a*(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)))
Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.89 \[ \int x \text {arcsinh}(a+b x)^3 \, dx=\frac {2 \, {\left (2 \, b^{2} x^{2} - 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x - 3 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 3 \, {\left (2 \, b^{2} x^{2} - 12 \, a b x - 14 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x - 15 \, a\right )}}{8 \, b^{2}} \]
1/8*(2*(2*b^2*x^2 - 2*a^2 + 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3 - 6*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b*x - 3*a)*log(b*x + a + sq rt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2 + 3*(2*b^2*x^2 - 12*a*b*x - 14*a^2 + 1) *log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 3*sqrt(b^2*x^2 + 2*a*b *x + a^2 + 1)*(b*x - 15*a))/b^2
Time = 0.29 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.22 \[ \int x \text {arcsinh}(a+b x)^3 \, dx=\begin {cases} - \frac {a^{2} \operatorname {asinh}^{3}{\left (a + b x \right )}}{2 b^{2}} - \frac {21 a^{2} \operatorname {asinh}{\left (a + b x \right )}}{4 b^{2}} - \frac {9 a x \operatorname {asinh}{\left (a + b x \right )}}{2 b} + \frac {9 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4 b^{2}} + \frac {45 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{8 b^{2}} + \frac {x^{2} \operatorname {asinh}^{3}{\left (a + b x \right )}}{2} + \frac {3 x^{2} \operatorname {asinh}{\left (a + b x \right )}}{4} - \frac {3 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4 b} - \frac {3 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{8 b} + \frac {\operatorname {asinh}^{3}{\left (a + b x \right )}}{4 b^{2}} + \frac {3 \operatorname {asinh}{\left (a + b x \right )}}{8 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {asinh}^{3}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \]
Piecewise((-a**2*asinh(a + b*x)**3/(2*b**2) - 21*a**2*asinh(a + b*x)/(4*b* *2) - 9*a*x*asinh(a + b*x)/(2*b) + 9*a*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1 )*asinh(a + b*x)**2/(4*b**2) + 45*a*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/( 8*b**2) + x**2*asinh(a + b*x)**3/2 + 3*x**2*asinh(a + b*x)/4 - 3*x*sqrt(a* *2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**2/(4*b) - 3*x*sqrt(a**2 + 2* a*b*x + b**2*x**2 + 1)/(8*b) + asinh(a + b*x)**3/(4*b**2) + 3*asinh(a + b* x)/(8*b**2), Ne(b, 0)), (x**2*asinh(a)**3/2, True))
\[ \int x \text {arcsinh}(a+b x)^3 \, dx=\int { x \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]
1/2*x^2*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3 - integrate(3/2 *(b^3*x^4 + 2*a*b^2*x^3 + (a^2*b + b)*x^2 + (b^2*x^3 + a*b*x^2)*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/(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)
\[ \int x \text {arcsinh}(a+b x)^3 \, dx=\int { x \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int x \text {arcsinh}(a+b x)^3 \, dx=\int x\,{\mathrm {asinh}\left (a+b\,x\right )}^3 \,d x \]