Integrand size = 8, antiderivative size = 76 \[ \int x \text {arcsinh}(a+b x) \, dx=\frac {3 a \sqrt {1+(a+b x)^2}}{4 b^2}-\frac {x \sqrt {1+(a+b x)^2}}{4 b}+\frac {\left (1-2 a^2\right ) \text {arcsinh}(a+b x)}{4 b^2}+\frac {1}{2} x^2 \text {arcsinh}(a+b x) \]
1/4*(-2*a^2+1)*arcsinh(b*x+a)/b^2+1/2*x^2*arcsinh(b*x+a)+3/4*a*(1+(b*x+a)^ 2)^(1/2)/b^2-1/4*x*(1+(b*x+a)^2)^(1/2)/b
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79 \[ \int x \text {arcsinh}(a+b x) \, dx=\frac {(3 a-b x) \sqrt {1+a^2+2 a b x+b^2 x^2}+\left (1-2 a^2+2 b^2 x^2\right ) \text {arcsinh}(a+b x)}{4 b^2} \]
((3*a - b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (1 - 2*a^2 + 2*b^2*x^2)*A rcSinh[a + b*x])/(4*b^2)
Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6274, 25, 27, 6243, 497, 25, 455, 222}
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.
\(\displaystyle \int x \text {arcsinh}(a+b x) \, dx\) |
\(\Big \downarrow \) 6274 |
\(\displaystyle \frac {\int x \text {arcsinh}(a+b x)d(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int -x \text {arcsinh}(a+b x)d(a+b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int -b x \text {arcsinh}(a+b x)d(a+b x)}{b^2}\) |
\(\Big \downarrow \) 6243 |
\(\displaystyle -\frac {\frac {1}{2} \int \frac {b^2 x^2}{\sqrt {(a+b x)^2+1}}d(a+b x)-\frac {1}{2} b^2 x^2 \text {arcsinh}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 497 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} \int -\frac {-2 a^2+3 (a+b x) a+1}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} b x \sqrt {(a+b x)^2+1}\right )-\frac {1}{2} b^2 x^2 \text {arcsinh}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} b x \sqrt {(a+b x)^2+1}-\frac {1}{2} \int \frac {-2 a^2+3 (a+b x) a+1}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )-\frac {1}{2} b^2 x^2 \text {arcsinh}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} \left (-\left (1-2 a^2\right ) \int \frac {1}{\sqrt {(a+b x)^2+1}}d(a+b x)-3 a \sqrt {(a+b x)^2+1}\right )+\frac {1}{2} b x \sqrt {(a+b x)^2+1}\right )-\frac {1}{2} b^2 x^2 \text {arcsinh}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} \left (-\left (1-2 a^2\right ) \text {arcsinh}(a+b x)-3 a \sqrt {(a+b x)^2+1}\right )+\frac {1}{2} b x \sqrt {(a+b x)^2+1}\right )-\frac {1}{2} b^2 x^2 \text {arcsinh}(a+b x)}{b^2}\) |
-((-1/2*(b^2*x^2*ArcSinh[a + b*x]) + ((b*x*Sqrt[1 + (a + b*x)^2])/2 + (-3* a*Sqrt[1 + (a + b*x)^2] - (1 - 2*a^2)*ArcSinh[a + b*x])/2)/2)/b^2)
3.1.60.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
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_) + (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 = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )-\frac {\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (b x +a \right )}{4}+a \sqrt {1+\left (b x +a \right )^{2}}}{b^{2}}\) | \(74\) |
default | \(\frac {\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )-\frac {\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (b x +a \right )}{4}+a \sqrt {1+\left (b x +a \right )^{2}}}{b^{2}}\) | \(74\) |
parts | \(\frac {x^{2} \operatorname {arcsinh}\left (b x +a \right )}{2}-\frac {b \left (\frac {x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}-\frac {3 a \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{2 b}-\frac {\left (a^{2}+1\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{2}\) | \(170\) |
1/b^2*(1/2*arcsinh(b*x+a)*(b*x+a)^2-arcsinh(b*x+a)*a*(b*x+a)-1/4*(b*x+a)*( 1+(b*x+a)^2)^(1/2)+1/4*arcsinh(b*x+a)+a*(1+(b*x+a)^2)^(1/2))
Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.99 \[ \int x \text {arcsinh}(a+b x) \, dx=\frac {{\left (2 \, b^{2} x^{2} - 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x - 3 \, a\right )}}{4 \, b^{2}} \]
1/4*((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)) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b*x - 3*a))/b^2
Time = 0.18 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.37 \[ \int x \text {arcsinh}(a+b x) \, dx=\begin {cases} - \frac {a^{2} \operatorname {asinh}{\left (a + b x \right )}}{2 b^{2}} + \frac {3 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{4 b^{2}} + \frac {x^{2} \operatorname {asinh}{\left (a + b x \right )}}{2} - \frac {x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{4 b} + \frac {\operatorname {asinh}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {asinh}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \]
Piecewise((-a**2*asinh(a + b*x)/(2*b**2) + 3*a*sqrt(a**2 + 2*a*b*x + b**2* x**2 + 1)/(4*b**2) + x**2*asinh(a + b*x)/2 - x*sqrt(a**2 + 2*a*b*x + b**2* x**2 + 1)/(4*b) + asinh(a + b*x)/(4*b**2), Ne(b, 0)), (x**2*asinh(a)/2, Tr ue))
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (64) = 128\).
Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.96 \[ \int x \text {arcsinh}(a+b x) \, dx=\frac {1}{2} \, x^{2} \operatorname {arsinh}\left (b x + a\right ) - \frac {1}{4} \, b {\left (\frac {3 \, a^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{b^{2}} - \frac {{\left (a^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{3}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{b^{3}}\right )} \]
1/2*x^2*arcsinh(b*x + a) - 1/4*b*(3*a^2*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^ 2*b^2 + 4*(a^2 + 1)*b^2))/b^3 + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*x/b^2 - (a^2 + 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^3 - 3*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a/b^3)
Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.46 \[ \int x \text {arcsinh}(a+b x) \, dx=\frac {1}{2} \, x^{2} \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right ) - \frac {1}{4} \, {\left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (\frac {x}{b^{2}} - \frac {3 \, a}{b^{3}}\right )} - \frac {{\left (2 \, a^{2} - 1\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |}\right )}{b^{2} {\left | b \right |}}\right )} b \]
1/2*x^2*log(b*x + a + sqrt((b*x + a)^2 + 1)) - 1/4*(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(x/b^2 - 3*a/b^3) - (2*a^2 - 1)*log(-a*b - (x*abs(b) - sqrt(b^ 2*x^2 + 2*a*b*x + a^2 + 1))*abs(b))/(b^2*abs(b)))*b
Timed out. \[ \int x \text {arcsinh}(a+b x) \, dx=\int x\,\mathrm {asinh}\left (a+b\,x\right ) \,d x \]