3.61 \(\int \sinh ^{-1}(a+b x) \, dx\)

Optimal. Leaf size=34 \[ \frac {(a+b x) \sinh ^{-1}(a+b x)}{b}-\frac {\sqrt {(a+b x)^2+1}}{b} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 34, 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 = {5863, 5653, 261} \[ \frac {(a+b x) \sinh ^{-1}(a+b x)}{b}-\frac {\sqrt {(a+b x)^2+1}}{b} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a + b*x],x]

[Out]

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5653

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5863

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 40, normalized size = 1.18 \[ \frac {(a+b x) \sinh ^{-1}(a+b x)-\sqrt {a^2+2 a b x+b^2 x^2+1}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a + b*x],x]

[Out]

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

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fricas [A]  time = 0.54, size = 57, normalized size = 1.68 \[ \frac {{\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(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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giac [B]  time = 0.36, size = 92, normalized size = 2.71 \[ -b {\left (\frac {a \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |}\right )}{b {\left | b \right |}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{2}}\right )} + x \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*(a*log(-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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maple [A]  time = 0.00, size = 31, normalized size = 0.91 \[ \frac {\left (b x +a \right ) \arcsinh \left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(b*x+a),x)

[Out]

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

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maxima [A]  time = 0.39, size = 30, normalized size = 0.88 \[ \frac {{\left (b x + a\right )} \operatorname {arsinh}\left (b x + a\right ) - \sqrt {{\left (b x + a\right )}^{2} + 1}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.45, size = 76, normalized size = 2.24 \[ x\,\mathrm {asinh}\left (a+b\,x\right )-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{b}+\frac {a\,\ln \left (\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}+\frac {x\,b^2+a\,b}{\sqrt {b^2}}\right )}{\sqrt {b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a + b*x),x)

[Out]

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

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sympy [A]  time = 0.16, size = 46, normalized size = 1.35 \[ \begin {cases} \frac {a \operatorname {asinh}{\left (a + b x \right )}}{b} + x \operatorname {asinh}{\left (a + b x \right )} - \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {asinh}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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