3.97 \(\int (a+b \cosh ^{-1}(c+d x)) \, dx\)

Optimal. Leaf size=46 \[ a x-\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{d}+\frac {b (c+d x) \cosh ^{-1}(c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5864, 5654, 74} \[ a x-\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{d}+\frac {b (c+d x) \cosh ^{-1}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcCosh[c + d*x],x]

[Out]

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

Rule 74

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 5654

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 5864

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

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Mathematica [A]  time = 0.05, size = 61, normalized size = 1.33 \[ a x-\frac {b \left (\sqrt {c+d x-1} \sqrt {c+d x+1}-2 c \sinh ^{-1}\left (\frac {\sqrt {c+d x-1}}{\sqrt {2}}\right )\right )}{d}+b x \cosh ^{-1}(c+d x) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcCosh[c + d*x],x]

[Out]

a*x + b*x*ArcCosh[c + d*x] - (b*(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 2*c*ArcSinh[Sqrt[-1 + c + d*x]/Sqrt[2]
]))/d

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fricas [A]  time = 0.60, size = 65, normalized size = 1.41 \[ \frac {a d x + {\left (b d x + b c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} b}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.90, size = 100, normalized size = 2.17 \[ -{\left (d {\left (\frac {c \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {{\left (d x + c\right )}^{2} - 1}\right )\right )} b + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 41, normalized size = 0.89 \[ a x +\frac {b \left (\left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*arccosh(d*x+c),x)

[Out]

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

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maxima [A]  time = 0.34, size = 35, normalized size = 0.76 \[ a x + \frac {{\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} b}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.96, size = 272, normalized size = 5.91 \[ a\,x+b\,x\,\mathrm {acosh}\left (c+d\,x\right )-\frac {b\,\left (\frac {4\,c\,\left (\sqrt {c-1}-\sqrt {c+d\,x-1}\right )}{d\,\left (\sqrt {c+1}-\sqrt {c+d\,x+1}\right )}+\frac {4\,c\,{\left (\sqrt {c-1}-\sqrt {c+d\,x-1}\right )}^3}{d\,{\left (\sqrt {c+1}-\sqrt {c+d\,x+1}\right )}^3}-\frac {8\,{\left (\sqrt {c-1}-\sqrt {c+d\,x-1}\right )}^2\,\sqrt {c-1}\,\sqrt {c+1}}{d\,{\left (\sqrt {c+1}-\sqrt {c+d\,x+1}\right )}^2}\right )}{\frac {{\left (\sqrt {c-1}-\sqrt {c+d\,x-1}\right )}^4}{{\left (\sqrt {c+1}-\sqrt {c+d\,x+1}\right )}^4}-\frac {2\,{\left (\sqrt {c-1}-\sqrt {c+d\,x-1}\right )}^2}{{\left (\sqrt {c+1}-\sqrt {c+d\,x+1}\right )}^2}+1}+\frac {4\,b\,c\,\mathrm {atanh}\left (\frac {\sqrt {c-1}-\sqrt {c+d\,x-1}}{\sqrt {c+1}-\sqrt {c+d\,x+1}}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*acosh(c + d*x),x)

[Out]

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

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sympy [A]  time = 0.15, size = 51, normalized size = 1.11 \[ a x + b \left (\begin {cases} \frac {c \operatorname {acosh}{\left (c + d x \right )}}{d} + x \operatorname {acosh}{\left (c + d x \right )} - \frac {\sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{d} & \text {for}\: d \neq 0 \\x \operatorname {acosh}{\relax (c )} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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