3.277 \(\int e^{\cosh ^{-1}(a+b x)} x \, dx\)

Optimal. Leaf size=67 \[ -\frac {a e^{2 \cosh ^{-1}(a+b x)}}{4 b^2}+\frac {a \cosh ^{-1}(a+b x)}{2 b^2}+\frac {e^{-\cosh ^{-1}(a+b x)}}{4 b^2}+\frac {e^{3 \cosh ^{-1}(a+b x)}}{12 b^2} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5899, 2282, 12, 1628} \[ -\frac {a e^{2 \cosh ^{-1}(a+b x)}}{4 b^2}+\frac {a \cosh ^{-1}(a+b x)}{2 b^2}+\frac {e^{-\cosh ^{-1}(a+b x)}}{4 b^2}+\frac {e^{3 \cosh ^{-1}(a+b x)}}{12 b^2} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCosh[a + b*x]*x,x]

[Out]

1/(4*b^2*E^ArcCosh[a + b*x]) - (a*E^(2*ArcCosh[a + b*x]))/(4*b^2) + E^(3*ArcCosh[a + b*x])/(12*b^2) + (a*ArcCo
sh[a + b*x])/(2*b^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(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]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 5899

Int[(f_)^(ArcCosh[(a_.) + (b_.)*(x_)]^(n_.)*(c_.))*(x_)^(m_.), x_Symbol] :> Dist[1/b, Subst[Int[(-(a/b) + Cosh
[x]/b)^m*f^(c*x^n)*Sinh[x], x], x, ArcCosh[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int e^{\cosh ^{-1}(a+b x)} x \, dx &=\frac {\operatorname {Subst}\left (\int e^x \left (-\frac {a}{b}+\frac {\cosh (x)}{b}\right ) \sinh (x) \, dx,x,\cosh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right ) \left (-1+2 a x-x^2\right )}{4 b x^2} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right ) \left (-1+2 a x-x^2\right )}{x^2} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{4 b^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{x^2}+\frac {2 a}{x}-2 a x+x^2\right ) \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{4 b^2}\\ &=\frac {e^{-\cosh ^{-1}(a+b x)}}{4 b^2}-\frac {a e^{2 \cosh ^{-1}(a+b x)}}{4 b^2}+\frac {e^{3 \cosh ^{-1}(a+b x)}}{12 b^2}+\frac {a \cosh ^{-1}(a+b x)}{2 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 93, normalized size = 1.39 \[ \frac {1}{6} \left (\frac {\sqrt {a+b x-1} \sqrt {a+b x+1} \left (-a^2+a b x+2 b^2 x^2-2\right )}{b^2}+\frac {3 a \log \left (\sqrt {a+b x-1} \sqrt {a+b x+1}+a+b x\right )}{b^2}+3 a x^2+2 b x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCosh[a + b*x]*x,x]

[Out]

(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

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fricas [A]  time = 0.62, size = 88, normalized size = 1.31 \[ \frac {2 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} + {\left (2 \, b^{2} x^{2} + a b x - a^{2} - 2\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - 3 \, a \log \left (-b x + \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a\right )}{6 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))*x,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.51, size = 280, normalized size = 4.18 \[ \frac {2 \, b^{2} x^{3} + {\left (\sqrt {b x + a + 1} \sqrt {b x + a - 1} {\left ({\left (b x + a + 1\right )} {\left (\frac {2 \, {\left (b x + a + 1\right )}}{b^{2}} - \frac {6 \, a b^{6} + 7 \, b^{6}}{b^{8}}\right )} + \frac {3 \, {\left (2 \, a^{2} b^{6} + 6 \, a b^{6} + 3 \, b^{6}\right )}}{b^{8}}\right )} + \frac {6 \, {\left (2 \, a^{2} + 2 \, a + 1\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{b^{2}}\right )} b + \frac {3 \, {\left ({\left (b x + a + 1\right )}^{2} - 2 \, {\left (b x + a + 1\right )} a - 2 \, b x - 2 \, a - 2\right )} a}{b} + \frac {3 \, {\left (\sqrt {b x + a + 1} \sqrt {b x + a - 1} {\left (b x - a - 2\right )} - 2 \, {\left (2 \, a + 1\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )\right )} a}{b} + \frac {3 \, {\left (\sqrt {b x + a + 1} \sqrt {b x + a - 1} {\left (b x - a - 2\right )} - 2 \, {\left (2 \, a + 1\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )\right )}}{b}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))*x,x, algorithm="giac")

[Out]

1/6*(2*b^2*x^3 + (sqrt(b*x + a + 1)*sqrt(b*x + a - 1)*((b*x + a + 1)*(2*(b*x + a + 1)/b^2 - (6*a*b^6 + 7*b^6)/
b^8) + 3*(2*a^2*b^6 + 6*a*b^6 + 3*b^6)/b^8) + 6*(2*a^2 + 2*a + 1)*log(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))/b
^2)*b + 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 + 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

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maple [C]  time = 0.01, size = 194, normalized size = 2.90 \[ \frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b ) x^{2} b^{2}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b ) x a b -\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b ) a^{2}-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b )+3 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b )+b x +a \right ) \mathrm {csgn}\relax (b )\right ) a \right ) \mathrm {csgn}\relax (b )}{6 b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}+\frac {b \,x^{3}}{3}+\frac {a \,x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))*x,x)

[Out]

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)*csgn(b)*x^2*b^2+(b^2*x^2+2*a*b*x+a^2-1)^(
1/2)*csgn(b)*x*a*b-(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

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maxima [A]  time = 0.31, size = 177, normalized size = 2.64 \[ \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} + \frac {a^{3} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a x}{2 \, b} - \frac {{\left (a^{2} - 1\right )} a \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a^{2}}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{\frac {3}{2}}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))*x,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 16.72, size = 852, normalized size = 12.72 \[ \frac {a\,x^2}{2}-\frac {\frac {2\,a\,\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}{b^2\,\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^3\,\left (42\,a-\frac {160\,a^3}{3}\right )}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^3}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^9\,\left (42\,a-\frac {160\,a^3}{3}\right )}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^9}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^5\,\left (212\,a-288\,a^3\right )}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^5}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^7\,\left (212\,a-288\,a^3\right )}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^7}+\frac {2\,a\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{11}}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{11}}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2\,\left (8\,a^2-8\right )\,\sqrt {a-1}\,\sqrt {a+1}}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{10}\,\left (8\,a^2-8\right )\,\sqrt {a-1}\,\sqrt {a+1}}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{10}}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^4\,\left (160\,a^2-32\right )\,\sqrt {a-1}\,\sqrt {a+1}}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^4}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^8\,\left (160\,a^2-32\right )\,\sqrt {a-1}\,\sqrt {a+1}}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^8}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^6\,\left (\frac {1040\,a^2}{3}-\frac {272}{3}\right )\,\sqrt {a-1}\,\sqrt {a+1}}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^6}}{\frac {15\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^4}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^4}-\frac {6\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}-\frac {20\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^6}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^6}+\frac {15\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^8}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^8}-\frac {6\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{10}}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{10}}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{12}}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{12}}+1}+\frac {b\,x^3}{3}+\frac {2\,a\,\mathrm {atanh}\left (\frac {\sqrt {a-1}-\sqrt {a+b\,x-1}}{\sqrt {a+1}-\sqrt {a+b\,x+1}}\right )}{b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + (a + b*x - 1)^(1/2)*(a + b*x + 1)^(1/2) + b*x),x)

[Out]

(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*((10
40*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 - (20*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^6)/((a + 1)^(1/2) -
(a + b*x + 1)^(1/2))^6 + (15*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^8)/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^8
- (6*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^10)/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^10 + ((a - 1)^(1/2) - (a
+ b*x - 1)^(1/2))^12/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^12 + 1) + (b*x^3)/3 + (2*a*atanh(((a - 1)^(1/2) -
(a + b*x - 1)^(1/2))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))))/b^2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b x + \sqrt {a + b x - 1} \sqrt {a + b x + 1}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+(b*x+a-1)**(1/2)*(b*x+a+1)**(1/2))*x,x)

[Out]

Integral(x*(a + b*x + sqrt(a + b*x - 1)*sqrt(a + b*x + 1)), x)

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