Integrand size = 10, antiderivative size = 67 \[ \int e^{\text {arcsinh}(a+b x)} x \, dx=\frac {e^{-\text {arcsinh}(a+b x)}}{4 b^2}-\frac {a e^{2 \text {arcsinh}(a+b x)}}{4 b^2}+\frac {e^{3 \text {arcsinh}(a+b x)}}{12 b^2}-\frac {a \text {arcsinh}(a+b x)}{2 b^2} \] Output:
1/4/b^2/(b*x+a+(1+(b*x+a)^2)^(1/2))-1/4*a*(b*x+a+(1+(b*x+a)^2)^(1/2))^2/b^ 2+1/12*(b*x+a+(1+(b*x+a)^2)^(1/2))^3/b^2-1/2*a*arcsinh(b*x+a)/b^2
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09 \[ \int e^{\text {arcsinh}(a+b x)} x \, dx=\frac {1}{6} \left (3 a x^2+2 b x^3+\frac {\sqrt {1+a^2+2 a b x+b^2 x^2} \left (2-a^2+a b x+2 b^2 x^2\right )}{b^2}-\frac {3 a \text {arcsinh}(a+b x)}{b^2}\right ) \] Input:
Integrate[E^ArcSinh[a + b*x]*x,x]
Output:
(3*a*x^2 + 2*b*x^3 + (Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(2 - a^2 + a*b*x + 2*b^2*x^2))/b^2 - (3*a*ArcSinh[a + b*x])/b^2)/6
Time = 0.53 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6288, 25, 2720, 27, 2159, 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.
\(\displaystyle \int x e^{\text {arcsinh}(a+b x)} \, dx\) |
\(\Big \downarrow \) 6288 |
\(\displaystyle \frac {\int -e^{\text {arcsinh}(a+b x)} \left (\frac {a}{b}-\frac {a+b x}{b}\right ) \sqrt {(a+b x)^2+1}d\text {arcsinh}(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int e^{\text {arcsinh}(a+b x)} \left (\frac {a}{b}-\frac {a+b x}{b}\right ) \sqrt {(a+b x)^2+1}d\text {arcsinh}(a+b x)}{b}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {\int \frac {e^{-2 \text {arcsinh}(a+b x)} \left (2 e^{\text {arcsinh}(a+b x)} a-e^{2 \text {arcsinh}(a+b x)}+1\right ) \left (1+e^{2 \text {arcsinh}(a+b x)}\right )}{4 b}de^{\text {arcsinh}(a+b x)}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int e^{-2 \text {arcsinh}(a+b x)} \left (2 e^{\text {arcsinh}(a+b x)} a-e^{2 \text {arcsinh}(a+b x)}+1\right ) \left (1+e^{2 \text {arcsinh}(a+b x)}\right )de^{\text {arcsinh}(a+b x)}}{4 b^2}\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle -\frac {\int \left (2 e^{-\text {arcsinh}(a+b x)} a+2 e^{\text {arcsinh}(a+b x)} a+e^{-2 \text {arcsinh}(a+b x)}-e^{2 \text {arcsinh}(a+b x)}\right )de^{\text {arcsinh}(a+b x)}}{4 b^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a e^{2 \text {arcsinh}(a+b x)}-e^{-\text {arcsinh}(a+b x)}-\frac {1}{3} e^{3 \text {arcsinh}(a+b x)}+2 a \log \left (e^{\text {arcsinh}(a+b x)}\right )}{4 b^2}\) |
Input:
Int[E^ArcSinh[a + b*x]*x,x]
Output:
-1/4*(-E^(-ArcSinh[a + b*x]) + a*E^(2*ArcSinh[a + b*x]) - E^(3*ArcSinh[a + b*x])/3 + 2*a*Log[E^ArcSinh[a + b*x]])/b^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[(f_)^(ArcSinh[(a_.) + (b_.)*(x_)]^(n_.)*(c_.))*(x_)^(m_.), x_Symbol] :> Simp[1/b Subst[Int[(-a/b + Sinh[x]/b)^m*f^(c*x^n)*Cosh[x], x], x, ArcSin h[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Time = 0.18 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.16
method | result | size |
default | \(\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 b^{2}}-\frac {a \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{b}+\frac {b \,x^{3}}{3}+\frac {a \,x^{2}}{2}\) | \(145\) |
Input:
int((b*x+a+(1+(b*x+a)^2)^(1/2))*x,x,method=_RETURNVERBOSE)
Output:
1/3*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)/b^2-a/b*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^ 2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*ln((b^2*x+a*b)/(b ^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2))+1/3*b*x^3+1/2*a*x^2
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.39 \[ \int e^{\text {arcsinh}(a+b x)} x \, dx=\frac {2 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (2 \, b^{2} x^{2} + a b x - a^{2} + 2\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{6 \, b^{2}} \] Input:
integrate((b*x+a+(1+(b*x+a)^2)^(1/2))*x,x, algorithm="fricas")
Output:
1/6*(2*b^3*x^3 + 3*a*b^2*x^2 + 3*a*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + (2*b^2*x^2 + a*b*x - a^2 + 2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/b^2
Time = 1.02 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.94 \[ \int e^{\text {arcsinh}(a+b x)} x \, dx=\frac {a x^{2}}{2} + \frac {b x^{3}}{3} + \begin {cases} \frac {\left (- \frac {a \left (\frac {1}{3} - \frac {a^{2}}{6}\right )}{b} - \frac {a \left (a^{2} + 1\right )}{6 b}\right ) \log {\left (2 a b + 2 b^{2} x + 2 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \sqrt {b^{2}} \right )}}{\sqrt {b^{2}}} + \left (\frac {a x}{6 b} + \frac {x^{2}}{3} + \frac {\frac {1}{3} - \frac {a^{2}}{6}}{b^{2}}\right ) \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} & \text {for}\: b^{2} \neq 0 \\\frac {\frac {\left (- a^{2} - 1\right ) \left (a^{2} + 2 a b x + 1\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x + 1\right )^{\frac {5}{2}}}{5}}{2 a^{2} b^{2}} & \text {for}\: a b \neq 0 \\\frac {x^{2} \sqrt {a^{2} + 1}}{2} & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a+(1+(b*x+a)**2)**(1/2))*x,x)
Output:
a*x**2/2 + b*x**3/3 + Piecewise(((-a*(1/3 - a**2/6)/b - a*(a**2 + 1)/(6*b) )*log(2*a*b + 2*b**2*x + 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*sqrt(b**2) )/sqrt(b**2) + (a*x/(6*b) + x**2/3 + (1/3 - a**2/6)/b**2)*sqrt(a**2 + 2*a* b*x + b**2*x**2 + 1), Ne(b**2, 0)), (((-a**2 - 1)*(a**2 + 2*a*b*x + 1)**(3 /2)/3 + (a**2 + 2*a*b*x + 1)**(5/2)/5)/(2*a**2*b**2), Ne(a*b, 0)), (x**2*s qrt(a**2 + 1)/2, True))
Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (83) = 166\).
Time = 0.06 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.61 \[ \int e^{\text {arcsinh}(a+b x)} x \, dx=\frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} + \frac {a^{3} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{2 \, b} - \frac {{\left (a^{2} + 1\right )} a \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, b^{2}} \] Input:
integrate((b*x+a+(1+(b*x+a)^2)^(1/2))*x,x, algorithm="maxima")
Output:
1/3*b*x^3 + 1/2*a*x^2 + 1/2*a^3*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^2 - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*x/b - 1/2* (a^2 + 1)*a*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^ 2 - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2/b^2 + 1/3*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/b^2
Time = 0.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.60 \[ \int e^{\text {arcsinh}(a+b x)} x \, dx=\frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} + \frac {1}{6} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left ({\left (2 \, x + \frac {a}{b}\right )} x - \frac {a^{2} b - 2 \, b}{b^{3}}\right )} + \frac {a \log \left ({\left | -a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |} \right |}\right )}{2 \, b {\left | b \right |}} \] Input:
integrate((b*x+a+(1+(b*x+a)^2)^(1/2))*x,x, algorithm="giac")
Output:
1/3*b*x^3 + 1/2*a*x^2 + 1/6*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*((2*x + a/b) *x - (a^2*b - 2*b)/b^3) + 1/2*a*log(abs(-a*b - (x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*abs(b)))/(b*abs(b))
Timed out. \[ \int e^{\text {arcsinh}(a+b x)} x \, dx=\int x\,\left (a+\sqrt {{\left (a+b\,x\right )}^2+1}+b\,x\right ) \,d x \] Input:
int(x*(a + ((a + b*x)^2 + 1)^(1/2) + b*x),x)
Output:
int(x*(a + ((a + b*x)^2 + 1)^(1/2) + b*x), x)
Time = 0.15 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.10 \[ \int e^{\text {arcsinh}(a+b x)} x \, dx=\frac {-\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a b x +2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b^{2} x^{2}+2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}-3 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) a +3 a \,b^{2} x^{2}+2 b^{3} x^{3}}{6 b^{2}} \] Input:
int((b*x+a+(1+(b*x+a)^2)^(1/2))*x,x)
Output:
( - sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**2 + sqrt(a**2 + 2*a*b*x + b**2 *x**2 + 1)*a*b*x + 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*b**2*x**2 + 2*sq rt(a**2 + 2*a*b*x + b**2*x**2 + 1) - 3*log(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + a + b*x)*a + 3*a*b**2*x**2 + 2*b**3*x**3)/(6*b**2)