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

Optimal. Leaf size=25 \[ \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{\pi } b c} \]

[Out]

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

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Rubi [A]  time = 0.029716, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {5675} \[ \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{\pi } b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*ArcSinh[c*x])^2/(2*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{a+b \sinh ^{-1}(c x)}{\sqrt{\pi +c^2 \pi x^2}} \, dx &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt{\pi }}\\ \end{align*}

Mathematica [A]  time = 0.0170171, size = 25, normalized size = 1. \[ \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{\pi } b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.037, size = 53, normalized size = 2.1 \begin{align*}{a\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 \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{2\,c\sqrt{\pi }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.15035, size = 103, normalized size = 4.12 \begin{align*} \frac{b \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right ) \operatorname{arsinh}\left (c x\right )}{\sqrt{\pi c^{2}}} - \frac{b \sqrt{c^{2}} \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )^{2}}{2 \, \sqrt{\pi c^{2}} c} + \frac{a \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))/(pi*c^2*x^2+pi)^(1/2),x, algorithm="maxima")

[Out]

b*arcsinh(c^2*x/sqrt(c^2))*arcsinh(c*x)/sqrt(pi*c^2) - 1/2*b*sqrt(c^2)*arcsinh(c^2*x/sqrt(c^2))^2/(sqrt(pi*c^2
)*c) + a*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 \operatorname{arsinh}\left (c x\right ) + a}{\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))/(pi*c^2*x^2+pi)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.32607, size = 85, normalized size = 3.4 \begin{align*} \begin{cases} a \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 x}{\sqrt{\pi }} & \text{for}\: c = 0 \\\frac{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{2 \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))/(pi*c**2*x**2+pi)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsinh}\left (c x\right ) + a}{\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))/(pi*c^2*x^2+pi)^(1/2),x, algorithm="giac")

[Out]

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