3.15 \(\int (a+b \sinh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=46 \[ -\frac{2 b \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}+x \left (a+b \sinh ^{-1}(c x)\right )^2+2 b^2 x \]

[Out]

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

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Rubi [A]  time = 0.0633947, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5653, 5717, 8} \[ -\frac{2 b \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}+x \left (a+b \sinh ^{-1}(c x)\right )^2+2 b^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c*x])^2,x]

[Out]

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

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 5717

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0585261, size = 74, normalized size = 1.61 \[ x \left (a^2+2 b^2\right )-\frac{2 a b \sqrt{c^2 x^2+1}}{c}+\frac{2 b \sinh ^{-1}(c x) \left (a c x-b \sqrt{c^2 x^2+1}\right )}{c}+b^2 x \sinh ^{-1}(c x)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSinh[c*x])^2,x]

[Out]

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

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Maple [A]  time = 0.002, size = 72, normalized size = 1.6 \begin{align*}{\frac{1}{c} \left ( cx{a}^{2}+{b}^{2} \left ( \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx-2\,{\it Arcsinh} \left ( cx \right ) \sqrt{{c}^{2}{x}^{2}+1}+2\,cx \right ) +2\,ab \left ({\it Arcsinh} \left ( cx \right ) cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))^2,x)

[Out]

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

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Maxima [A]  time = 1.07637, size = 97, normalized size = 2.11 \begin{align*} b^{2} x \operatorname{arsinh}\left (c x\right )^{2} + 2 \, b^{2}{\left (x - \frac{\sqrt{c^{2} x^{2} + 1} \operatorname{arsinh}\left (c x\right )}{c}\right )} + a^{2} x + \frac{2 \,{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} a b}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.35053, size = 212, normalized size = 4.61 \begin{align*} \frac{b^{2} c x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} +{\left (a^{2} + 2 \, b^{2}\right )} c x - 2 \, \sqrt{c^{2} x^{2} + 1} a b + 2 \,{\left (a b c x - \sqrt{c^{2} x^{2} + 1} b^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.319157, size = 82, normalized size = 1.78 \begin{align*} \begin{cases} a^{2} x + 2 a b x \operatorname{asinh}{\left (c x \right )} - \frac{2 a b \sqrt{c^{2} x^{2} + 1}}{c} + b^{2} x \operatorname{asinh}^{2}{\left (c x \right )} + 2 b^{2} x - \frac{2 b^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{c} & \text{for}\: c \neq 0 \\a^{2} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(c*x))**2,x)

[Out]

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

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Giac [B]  time = 1.57865, size = 150, normalized size = 3.26 \begin{align*} 2 \,{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{c}\right )} a b +{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \, c{\left (\frac{x}{c} - \frac{\sqrt{c^{2} x^{2} + 1} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{2}}\right )}\right )} b^{2} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

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