∫ (a + b sinh−1(c + dx)) dx [119]

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

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Rubi [A]
time = 0.02, antiderivative size = 39, 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 = {5858, 5772, 267} \begin {gather*} a x-\frac {b \sqrt {(c+d x)^2+1}}{d}+\frac {b (c+d x) \sinh ^{-1}(c+d x)}{d} \end {gather*}

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

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 \left (a+b \sinh ^{-1}(c+d x)\right ) \, dx &=a x+b \int \sinh ^{-1}(c+d x) \, dx\\ &=a x+\frac {b \text {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 \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(39)=78\).
time = 0.25, size = 140, normalized size = 3.59 \begin {gather*} a x+b x \sinh ^{-1}(c+d x)-\frac {b \left (2 d \sqrt {1+c^2+2 c d x+d^2 x^2}+c \left (d+\sqrt {d^2}\right ) \log \left (-c-\sqrt {d^2} x+\sqrt {1+c^2+2 c d x+d^2 x^2}\right )+c \left (-d+\sqrt {d^2}\right ) \log \left (c-\sqrt {d^2} x+\sqrt {1+c^2+2 c d x+d^2 x^2}\right )\right )}{2 d^2} \end {gather*}

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

x + Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]] + c*(-d + Sqrt[d^2])*Log[c - Sqrt[d^2]*x + Sqrt[1 + c^2 + 2*c*d*x + d^2

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method	result	size

default	\(a x +\frac {b \left (\left (d x +c \right ) \arcsinh \left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\)	\(36\)
derivativedivides	\(\frac {\left (d x +c \right ) a +b \left (\left (d x +c \right ) \arcsinh \left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\)	\(41\)

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Maxima [A]
time = 0.25, size = 35, normalized size = 0.90 \begin {gather*} 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 {gather*}

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Fricas [A]
time = 0.34, size = 65, normalized size = 1.67 \begin {gather*} \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 {gather*}

(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)

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Sympy [A]
time = 0.10, size = 51, normalized size = 1.31 \begin {gather*} 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 {gather*}

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)),

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (37) = 74\).
time = 0.41, size = 99, normalized size = 2.54 \begin {gather*} -{\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 {gather*}

-(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 +

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Mupad [B]
time = 0.48, size = 85, normalized size = 2.18 \begin {gather*} a\,x+b\,x\,\mathrm {asinh}\left (c+d\,x\right )-\frac {b\,\sqrt {c^2+2\,c\,d\,x+d^2\,x^2+1}}{d}+\frac {b\,c\,d^2\,\ln \left (\sqrt {c^2+2\,c\,d\,x+d^2\,x^2+1}+\frac {x\,d^2+c\,d}{\sqrt {d^2}}\right )}{{\left (d^2\right )}^{3/2}} \end {gather*}

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

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3.2.19 \(\int (a+b \sinh ^{-1}(c+d x)) \, dx\) [119]