∫ sinh−1(a + bx) dx [61]

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

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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 = {5858, 5772, 267} \begin {gather*} \frac {(a+b x) \sinh ^{-1}(a+b x)}{b}-\frac {\sqrt {(a+b x)^2+1}}{b} \end {gather*}

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

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

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]

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

\begin {align*} \int \sinh ^{-1}(a+b x) \, dx &=\frac {\text {Subst}\left (\int \sinh ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sinh ^{-1}(a+b x)}{b}-\frac {\text {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 [B] Leaf count is larger than twice the leaf count of optimal. \(135\) vs. \(2(34)=68\).
time = 0.22, size = 135, normalized size = 3.97 \begin {gather*} x \sinh ^{-1}(a+b x)-\frac {2 b \sqrt {1+a^2+2 a b x+b^2 x^2}+a \left (b+\sqrt {b^2}\right ) \log \left (-a-\sqrt {b^2} x+\sqrt {1+a^2+2 a b x+b^2 x^2}\right )+a \left (-b+\sqrt {b^2}\right ) \log \left (a-\sqrt {b^2} x+\sqrt {1+a^2+2 a b x+b^2 x^2}\right )}{2 b^2} \end {gather*}

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

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

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method	result	size

derivativedivides	\(\frac {\left (b x +a \right ) \arcsinh \left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}}{b}\)	\(31\)
default	\(\frac {\left (b x +a \right ) \arcsinh \left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}}{b}\)	\(31\)

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Maxima [A]
time = 0.26, size = 30, normalized size = 0.88 \begin {gather*} \frac {{\left (b x + a\right )} \operatorname {arsinh}\left (b x + a\right ) - \sqrt {{\left (b x + a\right )}^{2} + 1}}{b} \end {gather*}

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

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Fricas [A]
time = 0.40, size = 57, normalized size = 1.68 \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*}

((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 = 1.35 \begin {gather*} \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}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (32) = 64\).
time = 0.41, size = 92, normalized size = 2.71 \begin {gather*} -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 ) \end {gather*}

-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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Mupad [B]
time = 0.45, size = 76, normalized size = 2.24 \begin {gather*} 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}} \end {gather*}

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 +

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3.1.61 \(\int \sinh ^{-1}(a+b x) \, dx\) [61]