Integrand size = 10, antiderivative size = 67 \[ \int e^{\text {arccosh}(a+b x)} x \, dx=\frac {e^{-\text {arccosh}(a+b x)}}{4 b^2}-\frac {a e^{2 \text {arccosh}(a+b x)}}{4 b^2}+\frac {e^{3 \text {arccosh}(a+b x)}}{12 b^2}+\frac {a \text {arccosh}(a+b x)}{2 b^2} \] Output:
1/4/b^2/(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))-1/4*a*(b*x+a+(b*x+a-1)^(1/ 2)*(b*x+a+1)^(1/2))^2/b^2+1/12*(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))^3/b ^2+1/2*a*arccosh(b*x+a)/b^2
Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.39 \[ \int e^{\text {arccosh}(a+b x)} x \, dx=\frac {1}{6} \left (3 a x^2+2 b x^3+\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (-2-a^2+a b x+2 b^2 x^2\right )}{b^2}+\frac {3 a \log \left (a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{b^2}\right ) \] Input:
Integrate[E^ArcCosh[a + b*x]*x,x]
Output:
(3*a*x^2 + 2*b*x^3 + (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(-2 - a^2 + a*b *x + 2*b^2*x^2))/b^2 + (3*a*Log[a + b*x + Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]])/b^2)/6
Time = 0.54 (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 = {6430, 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.
\(\displaystyle \int x e^{\text {arccosh}(a+b x)} \, dx\) |
\(\Big \downarrow \) 6430 |
\(\displaystyle \frac {\int -e^{\text {arccosh}(a+b x)} \sqrt {\frac {a+b x-1}{a+b x+1}} (a+b x+1) \left (\frac {a}{b}-\frac {a+b x}{b}\right )d\text {arccosh}(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int e^{\text {arccosh}(a+b x)} \sqrt {\frac {a+b x-1}{a+b x+1}} (a+b x+1) \left (\frac {a}{b}-\frac {a+b x}{b}\right )d\text {arccosh}(a+b x)}{b}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {\int \frac {e^{-2 \text {arccosh}(a+b x)} \left (1-e^{2 \text {arccosh}(a+b x)}\right ) \left (-2 e^{\text {arccosh}(a+b x)} a+e^{2 \text {arccosh}(a+b x)}+1\right )}{4 b}de^{\text {arccosh}(a+b x)}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int e^{-2 \text {arccosh}(a+b x)} \left (1-e^{2 \text {arccosh}(a+b x)}\right ) \left (-2 e^{\text {arccosh}(a+b x)} a+e^{2 \text {arccosh}(a+b x)}+1\right )de^{\text {arccosh}(a+b x)}}{4 b^2}\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle -\frac {\int \left (-2 e^{-\text {arccosh}(a+b x)} a+2 e^{\text {arccosh}(a+b x)} a+e^{-2 \text {arccosh}(a+b x)}-e^{2 \text {arccosh}(a+b x)}\right )de^{\text {arccosh}(a+b x)}}{4 b^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a e^{2 \text {arccosh}(a+b x)}-e^{-\text {arccosh}(a+b x)}-\frac {1}{3} e^{3 \text {arccosh}(a+b x)}-2 a \log \left (e^{\text {arccosh}(a+b x)}\right )}{4 b^2}\) |
Input:
Int[E^ArcCosh[a + b*x]*x,x]
Output:
-1/4*(-E^(-ArcCosh[a + b*x]) + a*E^(2*ArcCosh[a + b*x]) - E^(3*ArcCosh[a + b*x])/3 - 2*a*Log[E^ArcCosh[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_)^(ArcCosh[(a_.) + (b_.)*(x_)]^(n_.)*(c_.))*(x_)^(m_.), x_Symbol] :> Simp[1/b Subst[Int[(-a/b + Cosh[x]/b)^m*f^(c*x^n)*Sinh[x], x], x, ArcCos h[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.18 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.90
method | result | size |
default | \(-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right ) b^{2} x^{2}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right ) a b x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right ) a^{2}+2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right )-3 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right )+b x +a \right ) \operatorname {csgn}\left (b \right )\right ) a \right ) \operatorname {csgn}\left (b \right )}{6 b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}+\frac {b \,x^{3}}{3}+\frac {a \,x^{2}}{2}\) | \(194\) |
Input:
int((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))*x,x,method=_RETURNVERBOSE)
Output:
-1/6*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)*(-2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csg n(b)*b^2*x^2-(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)*a*b*x+(b^2*x^2+2*a*b*x+ a^2-1)^(1/2)*csgn(b)*a^2+2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)-3*ln(((b^ 2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)+b*x+a)*csgn(b))*a)*csgn(b)/b^2/(b^2*x^2 +2*a*b*x+a^2-1)^(1/2)+1/3*b*x^3+1/2*a*x^2
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31 \[ \int e^{\text {arccosh}(a+b x)} x \, dx=\frac {2 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} + {\left (2 \, b^{2} x^{2} + a b x - a^{2} - 2\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - 3 \, a \log \left (-b x + \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a\right )}{6 \, b^{2}} \] Input:
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))*x,x, algorithm="fricas")
Output:
1/6*(2*b^3*x^3 + 3*a*b^2*x^2 + (2*b^2*x^2 + a*b*x - a^2 - 2)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) - 3*a*log(-b*x + sqrt(b*x + a + 1)*sqrt(b*x + a - 1 ) - a))/b^2
\[ \int e^{\text {arccosh}(a+b x)} x \, dx=\int x \left (a + b x + \sqrt {a + b x - 1} \sqrt {a + b x + 1}\right )\, dx \] Input:
integrate((b*x+a+(b*x+a-1)**(1/2)*(b*x+a+1)**(1/2))*x,x)
Output:
Integral(x*(a + b*x + sqrt(a + b*x - 1)*sqrt(a + b*x + 1)), x)
Time = 0.10 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.64 \[ \int e^{\text {arccosh}(a+b x)} x \, dx=\frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} + \frac {a^{3} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a x}{2 \, b} - \frac {{\left (a^{2} - 1\right )} a \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a^{2}}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{\frac {3}{2}}}{3 \, b^{2}} \] Input:
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))*x,x, algorithm="maxima")
Output:
1/3*b*x^3 + 1/2*a*x^2 + 1/2*a^3*log(2*b^2*x + 2*a*b + 2*sqrt(b^2*x^2 + 2*a *b*x + a^2 - 1)*b)/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*log(2*b^2*x + 2*a*b + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*b)/ 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
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (101) = 202\).
Time = 0.16 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.63 \[ \int e^{\text {arccosh}(a+b x)} x \, dx=\frac {2 \, b^{2} x^{3} + \frac {3 \, {\left ({\left (b x + a + 1\right )}^{2} - 2 \, {\left (b x + a + 1\right )} a - 2 \, b x - 2 \, a - 2\right )} a}{b} + \frac {3 \, {\left (\sqrt {b x + a + 1} \sqrt {b x + a - 1} {\left (b x - a - 2\right )} - 2 \, {\left (2 \, a + 1\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )\right )} a}{b} + \frac {{\left ({\left (2 \, b x - 4 \, a - 5\right )} {\left (b x + a + 1\right )} + 6 \, a^{2} + 18 \, a + 9\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 6 \, {\left (2 \, a^{2} + 2 \, a + 1\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{b} + \frac {3 \, {\left (\sqrt {b x + a + 1} \sqrt {b x + a - 1} {\left (b x - a - 2\right )} - 2 \, {\left (2 \, a + 1\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )\right )}}{b}}{6 \, b} \] Input:
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))*x,x, algorithm="giac")
Output:
1/6*(2*b^2*x^3 + 3*((b*x + a + 1)^2 - 2*(b*x + a + 1)*a - 2*b*x - 2*a - 2) *a/b + 3*(sqrt(b*x + a + 1)*sqrt(b*x + a - 1)*(b*x - a - 2) - 2*(2*a + 1)* log(sqrt(b*x + a + 1) - sqrt(b*x + a - 1)))*a/b + (((2*b*x - 4*a - 5)*(b*x + a + 1) + 6*a^2 + 18*a + 9)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + 6*(2*a ^2 + 2*a + 1)*log(sqrt(b*x + a + 1) - sqrt(b*x + a - 1)))/b + 3*(sqrt(b*x + a + 1)*sqrt(b*x + a - 1)*(b*x - a - 2) - 2*(2*a + 1)*log(sqrt(b*x + a + 1) - sqrt(b*x + a - 1)))/b)/b
Time = 72.91 (sec) , antiderivative size = 852, normalized size of antiderivative = 12.72 \[ \int e^{\text {arccosh}(a+b x)} x \, dx =\text {Too large to display} \] Input:
int(x*(a + (a + b*x - 1)^(1/2)*(a + b*x + 1)^(1/2) + b*x),x)
Output:
(a*x^2)/2 - ((2*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/2)))/(b^2*((a + 1)^(1/ 2) - (a + b*x + 1)^(1/2))) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^3*(42* a - (160*a^3)/3))/(b^2*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^3) + (((a - 1 )^(1/2) - (a + b*x - 1)^(1/2))^9*(42*a - (160*a^3)/3))/(b^2*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^9) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^5*(212 *a - 288*a^3))/(b^2*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^5) + (((a - 1)^( 1/2) - (a + b*x - 1)^(1/2))^7*(212*a - 288*a^3))/(b^2*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^7) + (2*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^11)/(b^2 *((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^11) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^2*(8*a^2 - 8)*(a - 1)^(1/2)*(a + 1)^(1/2))/(b^2*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^10*(8*a^ 2 - 8)*(a - 1)^(1/2)*(a + 1)^(1/2))/(b^2*((a + 1)^(1/2) - (a + b*x + 1)^(1 /2))^10) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^4*(160*a^2 - 32)*(a - 1) ^(1/2)*(a + 1)^(1/2))/(b^2*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^8*(160*a^2 - 32)*(a - 1)^(1/2)*(a + 1)^ (1/2))/(b^2*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^8) + (((a - 1)^(1/2) - ( a + b*x - 1)^(1/2))^6*((1040*a^2)/3 - 272/3)*(a - 1)^(1/2)*(a + 1)^(1/2))/ (b^2*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^6))/((15*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^4)/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4 - (6*((a - 1)^( 1/2) - (a + b*x - 1)^(1/2))^2)/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2 ...
Time = 0.16 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.82 \[ \int e^{\text {arccosh}(a+b x)} x \, dx=\frac {-\sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a^{2}+\sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a b x +2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, b^{2} x^{2}-2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}+6 \,\mathrm {log}\left (\frac {\sqrt {b x +a -1}+\sqrt {b x +a +1}}{\sqrt {2}}\right ) a +3 a \,b^{2} x^{2}+2 b^{3} x^{3}}{6 b^{2}} \] Input:
int((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))*x,x)
Output:
( - sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**2 + sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a*b*x + 2*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*b**2*x**2 - 2*sqrt( a + b*x + 1)*sqrt(a + b*x - 1) + 6*log((sqrt(a + b*x - 1) + sqrt(a + b*x + 1))/sqrt(2))*a + 3*a*b**2*x**2 + 2*b**3*x**3)/(6*b**2)