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

Optimal. Leaf size=42 \[ \frac{\sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{\pi c^2}-\frac{b x}{\sqrt{\pi } c} \]

[Out]

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

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Rubi [A]  time = 0.0646801, antiderivative size = 64, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5717, 8} \[ \frac{\sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{\pi c^2}-\frac{b x \sqrt{c^2 x^2+1}}{c \sqrt{\pi c^2 x^2+\pi }} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0792243, size = 49, normalized size = 1.17 \[ \frac{a \sqrt{c^2 x^2+1}+b \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-b c x}{\sqrt{\pi } c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 72, normalized size = 1.7 \begin{align*}{\frac{a}{\pi \,{c}^{2}}\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}+{\frac{b}{{c}^{2}\sqrt{\pi }} \left ({\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}+{\it Arcsinh} \left ( cx \right ) -cx\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.2497, size = 74, normalized size = 1.76 \begin{align*} -\frac{b x}{\sqrt{\pi } c} + \frac{\sqrt{\pi + \pi c^{2} x^{2}} b \operatorname{arsinh}\left (c x\right )}{\pi c^{2}} + \frac{\sqrt{\pi + \pi c^{2} x^{2}} a}{\pi c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.39019, size = 213, normalized size = 5.07 \begin{align*} \frac{\sqrt{\pi + \pi c^{2} x^{2}}{\left (b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + \sqrt{\pi + \pi c^{2} x^{2}}{\left (a c^{2} x^{2} - \sqrt{c^{2} x^{2} + 1} b c x + a\right )}}{\pi c^{4} x^{2} + \pi c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(pi + pi*c^2*x^2)*(b*c^2*x^2 + b)*log(c*x + sqrt(c^2*x^2 + 1)) + sqrt(pi + pi*c^2*x^2)*(a*c^2*x^2 - sqrt(
c^2*x^2 + 1)*b*c*x + a))/(pi*c^4*x^2 + pi*c^2)

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Sympy [A]  time = 2.2557, size = 60, normalized size = 1.43 \begin{align*} \frac{a \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: c^{2} = 0 \\\frac{\sqrt{c^{2} x^{2} + 1}}{c^{2}} & \text{otherwise} \end{cases}\right )}{\sqrt{\pi }} + \frac{b \left (\begin{cases} - \frac{x}{c} + \frac{\sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{c^{2}} & \text{for}\: c \neq 0 \\0 & \text{otherwise} \end{cases}\right )}{\sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} x}{\sqrt{\pi + \pi c^{2} x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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