3.3.78 \(\int e^{\cosh ^{-1}(a+b x)} \, dx\) [278]

Optimal. Leaf size=31 \[ \frac {e^{2 \cosh ^{-1}(a+b x)}}{4 b}-\frac {\cosh ^{-1}(a+b x)}{2 b} \]

[Out]

1/4*(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))^2/b-1/2*arccosh(b*x+a)/b

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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6014, 2320, 12, 14} \begin {gather*} \frac {e^{2 \cosh ^{-1}(a+b x)}}{4 b}-\frac {\cosh ^{-1}(a+b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcCosh[a + b*x],x]

[Out]

E^(2*ArcCosh[a + b*x])/(4*b) - ArcCosh[a + b*x]/(2*b)

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2320

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 6014

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

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(31)=62\).
time = 0.02, size = 69, normalized size = 2.23 \begin {gather*} \frac {(a+b x) \left (a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}\right )-\log \left (a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCosh[a + b*x],x]

[Out]

((a + b*x)*(a + b*x + Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]) - Log[a + b*x + Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*
x]])/(2*b)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(146\) vs. \(2(41)=82\).
time = 0.09, size = 147, normalized size = 4.74

method	result	size
default	\(a x +\frac {b \,x^{2}}{2}+\frac {\sqrt {b x +a -1}\, \left (b x +a +1\right )^{\frac {3}{2}}}{2 b}-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{2 b}-\frac {\sqrt {\left (b x +a -1\right ) \left (b x +a +1\right )}\, \ln \left (\frac {\frac {b \left (1+a \right )}{2}+\frac {\left (-1+a \right ) b}{2}+b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+\left (b \left (1+a \right )+\left (-1+a \right ) b \right ) x +\left (1+a \right ) \left (-1+a \right )}\right )}{2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, \sqrt {b^{2}}}\)	\(147\)

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,method=_RETURNVERBOSE)

[Out]

a*x+1/2*b*x^2+1/2/b*(b*x+a-1)^(1/2)*(b*x+a+1)^(3/2)-1/2*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/b-1/2*((b*x+a-1)*(b*x+
a+1))^(1/2)/(b*x+a+1)^(1/2)/(b*x+a-1)^(1/2)*ln((1/2*b*(1+a)+1/2*(-1+a)*b+b^2*x)/(b^2)^(1/2)+(b^2*x^2+(b*(1+a)+
(-1+a)*b)*x+(1+a)*(-1+a))^(1/2))/(b^2)^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (41) = 82\).
time = 0.29, size = 143, normalized size = 4.61 \begin {gather*} \frac {1}{2} \, b x^{2} + a x - \frac {a^{2} \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} + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} x + \frac {{\left (a^{2} - 1\right )} \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} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a}{2 \, b} \end {gather*}

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, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 66, normalized size = 2.13 \begin {gather*} \frac {b^{2} x^{2} + 2 \, a b x + \sqrt {b x + a + 1} {\left (b x + a\right )} \sqrt {b x + a - 1} + \log \left (-b x + \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a\right )}{2 \, b} \end {gather*}

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, algorithm="fricas")

[Out]

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

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

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)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (41) = 82\).
time = 0.40, size = 151, normalized size = 4.87 \begin {gather*} \frac {1}{2} \, b x^{2} + a x + \frac {\sqrt {b x + a + 1} \sqrt {b x + a - 1} {\left (b x - a - 2\right )} + 2 \, {\left (\sqrt {b x + a + 1} \sqrt {b x + a - 1} + 2 \, \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )\right )} a - 2 \, {\left (2 \, a + 1\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right ) + 2 \, \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 4 \, \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{2 \, b} \end {gather*}

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, algorithm="giac")

[Out]

1/2*b*x^2 + a*x + 1/2*(sqrt(b*x + a + 1)*sqrt(b*x + a - 1)*(b*x - a - 2) + 2*(sqrt(b*x + a + 1)*sqrt(b*x + a -
1) + 2*log(sqrt(b*x + a + 1) - sqrt(b*x + a - 1)))*a - 2*(2*a + 1)*log(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))
+ 2*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + 4*log(sqrt(b*x + a + 1) - sqrt(b*x + a - 1)))/b

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Mupad [B]
time = 0.17, size = 79, normalized size = 2.55 \begin {gather*} a\,x+\frac {b\,x^2}{2}-\frac {\ln \left (a+\sqrt {a+b\,x-1}\,\sqrt {a+b\,x+1}+b\,x\right )}{2\,b}+\frac {x\,\sqrt {a+b\,x-1}\,\sqrt {a+b\,x+1}}{2}+\frac {a\,\sqrt {a+b\,x-1}\,\sqrt {a+b\,x+1}}{2\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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