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

Optimal. Leaf size=39 \[ a x-\frac{b \sqrt{(c+d x)^2+1}}{d}+\frac{b (c+d x) \sinh ^{-1}(c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0227585, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5863, 5653, 261} \[ a x-\frac{b \sqrt{(c+d x)^2+1}}{d}+\frac{b (c+d x) \sinh ^{-1}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcSinh[c + d*x],x]

[Out]

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

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]

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0281951, size = 50, normalized size = 1.28 \[ a x-\frac{b \left (\sqrt{c^2+2 c d x+d^2 x^2+1}-c \sinh ^{-1}(c+d x)\right )}{d}+b x \sinh ^{-1}(c+d x) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcSinh[c + d*x],x]

[Out]

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

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Maple [A]  time = 0.002, size = 36, normalized size = 0.9 \begin{align*} ax+{\frac{b}{d} \left ( \left ( dx+c \right ){\it Arcsinh} \left ( dx+c \right ) -\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.11765, size = 47, normalized size = 1.21 \begin{align*} a x + \frac{{\left ({\left (d x + c\right )} \operatorname{arsinh}\left (d x + c\right ) - \sqrt{{\left (d x + c\right )}^{2} + 1}\right )} b}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.42392, size = 154, normalized size = 3.95 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arcsinh(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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Sympy [A]  time = 0.183411, size = 51, normalized size = 1.31 \begin{align*} a x + b \left (\begin{cases} \frac{c \operatorname{asinh}{\left (c + d x \right )}}{d} + x \operatorname{asinh}{\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{asinh}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.39628, size = 134, normalized size = 3.44 \begin{align*} -{\left (d{\left (\frac{c \log \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 )}{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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*(c*log(-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