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

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Rubi [A]  time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 8

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

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*}

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Mathematica [A]  time = 0.06, 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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fricas [B]  time = 0.45, size = 96, normalized size = 2.09 \[ \frac {b^{2} c x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + {\left (a^{2} + 2 \, b^{2}\right )} c x - 2 \, \sqrt {c^{2} x^{2} + 1} a b + 2 \, {\left (a b c x - \sqrt {c^{2} x^{2} + 1} b^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c} \]

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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giac [B]  time = 0.78, size = 111, normalized size = 2.41 \[ 2 \, {\left (x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{c}\right )} a b + {\left (x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, c {\left (\frac {x}{c} - \frac {\sqrt {c^{2} x^{2} + 1} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2}}\right )}\right )} b^{2} + a^{2} x \]

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

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maple [A]  time = 0.00, size = 72, normalized size = 1.57 \[ \frac {a^{2} c x +b^{2} \left (\arcsinh \left (c x \right )^{2} c x -2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+2 a b \left (\arcsinh \left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a^2*c*x+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 = 0.33, size = 72, normalized size = 1.57 \[ b^{2} x \operatorname {arsinh}\left (c x\right )^{2} + 2 \, b^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b}{c} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c*x))^2,x)

[Out]

int((a + b*asinh(c*x))^2, x)

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sympy [A]  time = 0.28, size = 82, normalized size = 1.78 \[ \begin {cases} a^{2} x + 2 a b x \operatorname {asinh}{\left (c x \right )} - \frac {2 a b \sqrt {c^{2} x^{2} + 1}}{c} + b^{2} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} x - \frac {2 b^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} & \text {for}\: c \neq 0 \\a^{2} x & \text {otherwise} \end {cases} \]

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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