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

Optimal. Leaf size=67 \[ \frac{1}{2} x \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )+\frac{\sqrt{\pi } \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c}-\frac{1}{4} \sqrt{\pi } b c x^2 \]

[Out]

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

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Rubi [A]  time = 0.0593713, antiderivative size = 111, normalized size of antiderivative = 1.66, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5682, 5675, 30} \[ \frac{1}{2} x \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )+\frac{\sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt{c^2 x^2+1}}-\frac{b c x^2 \sqrt{\pi c^2 x^2+\pi }}{4 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*x^2*Sqrt[Pi + c^2*Pi*x^2])/(4*Sqrt[1 + c^2*x^2]) + (x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/2 + (S
qrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2*x^2])

Rule 5682

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

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
]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.1366, size = 69, normalized size = 1.03 \[ \frac{\sqrt{\pi } \left (2 \sinh ^{-1}(c x) \left (2 a+b \sinh \left (2 \sinh ^{-1}(c x)\right )\right )+4 a c x \sqrt{c^2 x^2+1}+2 b \sinh ^{-1}(c x)^2-b \cosh \left (2 \sinh ^{-1}(c x)\right )\right )}{8 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Pi]*(4*a*c*x*Sqrt[1 + c^2*x^2] + 2*b*ArcSinh[c*x]^2 - b*Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(2*a + b*S
inh[2*ArcSinh[c*x]])))/(8*c)

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Maple [B]  time = 0.041, size = 112, normalized size = 1.7 \begin{align*}{\frac{ax}{2}\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}+{\frac{a\pi }{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\sqrt{\pi }{\it Arcsinh} \left ( cx \right ) x}{2}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{bc{x}^{2}\sqrt{\pi }}{4}}+{\frac{b\sqrt{\pi } \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{4\,c}}-{\frac{b\sqrt{\pi }}{4\,c}} \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]

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

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

Exception raised: RuntimeError

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \sqrt{\pi } \left (\int a \sqrt{c^{2} x^{2} + 1}\, dx + \int b \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}\, dx\right ) \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]

sqrt(pi)*(Integral(a*sqrt(c**2*x**2 + 1), x) + Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x), x))

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