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

Optimal. Leaf size=76 \[ \frac{\left (1-2 a^2\right ) \sinh ^{-1}(a+b x)}{4 b^2}+\frac{3 a \sqrt{(a+b x)^2+1}}{4 b^2}+\frac{1}{2} x^2 \sinh ^{-1}(a+b x)-\frac{x \sqrt{(a+b x)^2+1}}{4 b} \]

[Out]

(3*a*Sqrt[1 + (a + b*x)^2])/(4*b^2) - (x*Sqrt[1 + (a + b*x)^2])/(4*b) + ((1 - 2*a^2)*ArcSinh[a + b*x])/(4*b^2)
+ (x^2*ArcSinh[a + b*x])/2

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Rubi [A]  time = 0.0661435, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5865, 5801, 743, 641, 215} \[ \frac{\left (1-2 a^2\right ) \sinh ^{-1}(a+b x)}{4 b^2}+\frac{3 a \sqrt{(a+b x)^2+1}}{4 b^2}+\frac{1}{2} x^2 \sinh ^{-1}(a+b x)-\frac{x \sqrt{(a+b x)^2+1}}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[x*ArcSinh[a + b*x],x]

[Out]

(3*a*Sqrt[1 + (a + b*x)^2])/(4*b^2) - (x*Sqrt[1 + (a + b*x)^2])/(4*b) + ((1 - 2*a^2)*ArcSinh[a + b*x])/(4*b^2)
+ (x^2*ArcSinh[a + b*x])/2

Rule 5865

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

Rule 5801

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)
*(a + b*ArcSinh[c*x])^n)/(e*(m + 1)), x] - Dist[(b*c*n)/(e*(m + 1)), Int[((d + e*x)^(m + 1)*(a + b*ArcSinh[c*x
])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 743

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2,
0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int x \sinh ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b}+\frac{x}{b}\right ) \sinh ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{1}{2} x^2 \sinh ^{-1}(a+b x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{b}+\frac{x}{b}\right )^2}{\sqrt{1+x^2}} \, dx,x,a+b x\right )\\ &=-\frac{x \sqrt{1+(a+b x)^2}}{4 b}+\frac{1}{2} x^2 \sinh ^{-1}(a+b x)-\frac{1}{4} \operatorname{Subst}\left (\int \frac{-\frac{1-2 a^2}{b^2}-\frac{3 a x}{b^2}}{\sqrt{1+x^2}} \, dx,x,a+b x\right )\\ &=\frac{3 a \sqrt{1+(a+b x)^2}}{4 b^2}-\frac{x \sqrt{1+(a+b x)^2}}{4 b}+\frac{1}{2} x^2 \sinh ^{-1}(a+b x)+\frac{\left (1-2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,a+b x\right )}{4 b^2}\\ &=\frac{3 a \sqrt{1+(a+b x)^2}}{4 b^2}-\frac{x \sqrt{1+(a+b x)^2}}{4 b}+\frac{\left (1-2 a^2\right ) \sinh ^{-1}(a+b x)}{4 b^2}+\frac{1}{2} x^2 \sinh ^{-1}(a+b x)\\ \end{align*}

Mathematica [A]  time = 0.0404693, size = 60, normalized size = 0.79 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2+1} (3 a-b x)+\left (-2 a^2+2 b^2 x^2+1\right ) \sinh ^{-1}(a+b x)}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*ArcSinh[a + b*x],x]

[Out]

((3*a - b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (1 - 2*a^2 + 2*b^2*x^2)*ArcSinh[a + b*x])/(4*b^2)

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Maple [A]  time = 0.004, size = 74, normalized size = 1. \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{{\it Arcsinh} \left ( bx+a \right ) \left ( bx+a \right ) ^{2}}{2}}-{\it Arcsinh} \left ( bx+a \right ) a \left ( bx+a \right ) -{\frac{bx+a}{4}\sqrt{1+ \left ( bx+a \right ) ^{2}}}+{\frac{{\it Arcsinh} \left ( bx+a \right ) }{4}}+a\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^2*(1/2*arcsinh(b*x+a)*(b*x+a)^2-arcsinh(b*x+a)*a*(b*x+a)-1/4*(b*x+a)*(1+(b*x+a)^2)^(1/2)+1/4*arcsinh(b*x+a
)+a*(1+(b*x+a)^2)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.52007, size = 178, normalized size = 2.34 \begin{align*} \frac{{\left (2 \, b^{2} x^{2} - 2 \, a^{2} + 1\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}{\left (b x - 3 \, a\right )}}{4 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((2*b^2*x^2 - 2*a^2 + 1)*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*x - 3*a))/b^2

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Sympy [A]  time = 0.365054, size = 104, normalized size = 1.37 \begin{align*} \begin{cases} - \frac{a^{2} \operatorname{asinh}{\left (a + b x \right )}}{2 b^{2}} + \frac{3 a \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{4 b^{2}} + \frac{x^{2} \operatorname{asinh}{\left (a + b x \right )}}{2} - \frac{x \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{4 b} + \frac{\operatorname{asinh}{\left (a + b x \right )}}{4 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \operatorname{asinh}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*asinh(a + b*x)/(2*b**2) + 3*a*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(4*b**2) + x**2*asinh(a +
b*x)/2 - x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(4*b) + asinh(a + b*x)/(4*b**2), Ne(b, 0)), (x**2*asinh(a)/2,
True))

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Giac [A]  time = 1.29715, size = 150, normalized size = 1.97 \begin{align*} \frac{1}{2} \, x^{2} \log \left (b x + a + \sqrt{{\left (b x + a\right )}^{2} + 1}\right ) - \frac{1}{4} \,{\left (\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{\left (\frac{x}{b^{2}} - \frac{3 \, a}{b^{3}}\right )} - \frac{{\left (2 \, a^{2} - 1\right )} \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^{2}{\left | b \right |}}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*log(b*x + a + sqrt((b*x + a)^2 + 1)) - 1/4*(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(x/b^2 - 3*a/b^3) - (2*a
^2 - 1)*log(-a*b - (x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*abs(b))/(b^2*abs(b)))*b