3.255 \(\int \frac{(a+b \sinh ^{-1}(c x))^2}{\sqrt{\pi +c^2 \pi x^2}} \, dx\)

Optimal. Leaf size=25 \[ \frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 \sqrt{\pi } b c} \]

[Out]

(a + b*ArcSinh[c*x])^3/(3*b*c*Sqrt[Pi])

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Rubi [A]  time = 0.0502889, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {5675} \[ \frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 \sqrt{\pi } b c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c*x])^2/Sqrt[Pi + c^2*Pi*x^2],x]

[Out]

(a + b*ArcSinh[c*x])^3/(3*b*c*Sqrt[Pi])

Rule 5675

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

Rubi steps

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

Mathematica [A]  time = 0.0240984, size = 25, normalized size = 1. \[ \frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 \sqrt{\pi } b c} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSinh[c*x])^2/Sqrt[Pi + c^2*Pi*x^2],x]

[Out]

(a + b*ArcSinh[c*x])^3/(3*b*c*Sqrt[Pi])

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Maple [B]  time = 0.063, size = 72, normalized size = 2.9 \begin{align*}{{a}^{2}\ln \left ({\pi \,{c}^{2}x{\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+\sqrt{\pi \,{c}^{2}{x}^{2}+\pi } \right ){\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+{\frac{{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{3}}{3\,c\sqrt{\pi }}}+{\frac{ab \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{c\sqrt{\pi }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))^2/(Pi*c^2*x^2+Pi)^(1/2),x)

[Out]

a^2*ln(Pi*x*c^2/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+1/3*b^2/c/Pi^(1/2)*arcsinh(c*x)^3+a*b*arc
sinh(c*x)^2/c/Pi^(1/2)

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Maxima [B]  time = 1.363, size = 230, normalized size = 9.2 \begin{align*} \frac{b^{2} \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right ) \operatorname{arsinh}\left (c x\right )^{2}}{\sqrt{\pi c^{2}}} + \frac{2 \, a b \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right ) \operatorname{arsinh}\left (c x\right )}{\sqrt{\pi c^{2}}} + \frac{1}{3} \,{\left (\frac{\operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )^{3}}{\sqrt{\pi c^{2}}} - \frac{3 \, \sqrt{c^{2}} \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )^{2} \operatorname{arsinh}\left (c x\right )}{\sqrt{\pi c^{2}} c}\right )} b^{2} - \frac{a b \sqrt{c^{2}} \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )^{2}}{\sqrt{\pi c^{2}} c} + \frac{a^{2} \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{\pi c^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2/(pi*c^2*x^2+pi)^(1/2),x, algorithm="maxima")

[Out]

b^2*arcsinh(c^2*x/sqrt(c^2))*arcsinh(c*x)^2/sqrt(pi*c^2) + 2*a*b*arcsinh(c^2*x/sqrt(c^2))*arcsinh(c*x)/sqrt(pi
*c^2) + 1/3*(arcsinh(c^2*x/sqrt(c^2))^3/sqrt(pi*c^2) - 3*sqrt(c^2)*arcsinh(c^2*x/sqrt(c^2))^2*arcsinh(c*x)/(sq
rt(pi*c^2)*c))*b^2 - a*b*sqrt(c^2)*arcsinh(c^2*x/sqrt(c^2))^2/(sqrt(pi*c^2)*c) + a^2*arcsinh(c^2*x/sqrt(c^2))/
sqrt(pi*c^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}{\sqrt{\pi + \pi c^{2} x^{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2/(pi*c^2*x^2+pi)^(1/2),x, algorithm="fricas")

[Out]

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/sqrt(pi + pi*c^2*x^2), x)

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Sympy [A]  time = 3.12885, size = 88, normalized size = 3.52 \begin{align*} \begin{cases} a^{2} \left (\begin{cases} \frac{\sqrt{- \frac{1}{c^{2}}} \operatorname{asin}{\left (x \sqrt{- c^{2}} \right )}}{\sqrt{\pi }} & \text{for}\: \pi c^{2} < 0 \\\frac{\sqrt{\frac{1}{c^{2}}} \operatorname{asinh}{\left (x \sqrt{c^{2}} \right )}}{\sqrt{\pi }} & \text{for}\: \pi c^{2} > 0 \end{cases}\right ) & \text{for}\: b = 0 \\\frac{a^{2} x}{\sqrt{\pi }} & \text{for}\: c = 0 \\\frac{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{3}}{3 \sqrt{\pi } b c} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(c*x))**2/(pi*c**2*x**2+pi)**(1/2),x)

[Out]

Piecewise((a**2*Piecewise((sqrt(-1/c**2)*asin(x*sqrt(-c**2))/sqrt(pi), pi*c**2 < 0), (sqrt(c**(-2))*asinh(x*sq
rt(c**2))/sqrt(pi), pi*c**2 > 0)), Eq(b, 0)), (a**2*x/sqrt(pi), Eq(c, 0)), ((a + b*asinh(c*x))**3/(3*sqrt(pi)*
b*c), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt{\pi + \pi c^{2} x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2/(pi*c^2*x^2+pi)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)^2/sqrt(pi + pi*c^2*x^2), x)