3.1.86 \(\int \cosh ^{-1}(a+b x) \, dx\) [86]

Optimal. Leaf size=41 \[ -\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{b}+\frac {(a+b x) \cosh ^{-1}(a+b x)}{b} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5995, 5879, 75} \begin {gather*} \frac {(a+b x) \cosh ^{-1}(a+b x)}{b}-\frac {\sqrt {a+b x-1} \sqrt {a+b x+1}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a + b*x],x]

[Out]

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

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5995

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

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 59, normalized size = 1.44 \begin {gather*} x \cosh ^{-1}(a+b x)-\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}-2 a \tanh ^{-1}\left (\sqrt {\frac {-1+a+b x}{1+a+b x}}\right )}{b} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a + b*x],x]

[Out]

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

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Maple [A]
time = 0.42, size = 36, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {\left (b x +a \right ) \mathrm {arccosh}\left (b x +a \right )-\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{b}\)	\(36\)
default	\(\frac {\left (b x +a \right ) \mathrm {arccosh}\left (b x +a \right )-\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{b}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.25, size = 30, normalized size = 0.73 \begin {gather*} \frac {{\left (b x + a\right )} \operatorname {arcosh}\left (b x + a\right ) - \sqrt {{\left (b x + a\right )}^{2} - 1}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.42, size = 57, normalized size = 1.39 \begin {gather*} \frac {{\left (b x + a\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}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(b*x+a),x, algorithm="fricas")

[Out]

((b*x + a)*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

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Sympy [A]
time = 0.07, size = 46, normalized size = 1.12 \begin {gather*} \begin {cases} \frac {a \operatorname {acosh}{\left (a + b x \right )}}{b} + x \operatorname {acosh}{\left (a + b x \right )} - \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} - 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {acosh}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(b*x+a),x)

[Out]

Piecewise((a*acosh(a + b*x)/b + x*acosh(a + b*x) - sqrt(a**2 + 2*a*b*x + b**2*x**2 - 1)/b, Ne(b, 0)), (x*acosh
(a), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (37) = 74\).
time = 0.43, size = 93, normalized size = 2.27 \begin {gather*} -b {\left (\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 )}{b {\left | b \right |}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{b^{2}}\right )} + x \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(b*x+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.00, size = 266, normalized size = 6.49 \begin {gather*} x\,\mathrm {acosh}\left (a+b\,x\right )-\frac {\frac {4\,a\,\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}{b\,\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}+\frac {4\,a\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^3}{b\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^3}-\frac {8\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2\,\sqrt {a-1}\,\sqrt {a+1}}{b\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}}{\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^4}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^4}-\frac {2\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}+1}+\frac {4\,a\,\mathrm {atanh}\left (\frac {\sqrt {a-1}-\sqrt {a+b\,x-1}}{\sqrt {a+1}-\sqrt {a+b\,x+1}}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a + b*x),x)

[Out]

x*acosh(a + b*x) - ((4*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/2)))/(b*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))) + (4
*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^3)/(b*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^3) - (8*((a - 1)^(1/2) -
(a + b*x - 1)^(1/2))^2*(a - 1)^(1/2)*(a + 1)^(1/2))/(b*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2))/(((a - 1)^(1/
2) - (a + b*x - 1)^(1/2))^4/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4 - (2*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))
^2)/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2 + 1) + (4*a*atanh(((a - 1)^(1/2) - (a + b*x - 1)^(1/2))/((a + 1)^(
1/2) - (a + b*x + 1)^(1/2))))/b

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