∫ x(a+b sinh−1(cx))_____________√ π + c2πx2 dx [84]

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

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

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Rubi [A]
time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 = {5798, 8} \begin {gather*} \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} \end {gather*}

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

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

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)

^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

\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*}

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Mathematica [A]
time = 0.05, size = 49, normalized size = 1.17 \begin {gather*} \frac {-b c x+a \sqrt {1+c^2 x^2}+b \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{c^2 \sqrt {\pi }} \end {gather*}

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

(-(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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method	result	size

default	\(\frac {a \sqrt {\pi \,c^{2} x^{2}+\pi }}{\pi \,c^{2}}+\frac {b \left (\arcsinh \left (c x \right ) x^{2} c^{2}+\arcsinh \left (c x \right )-\sqrt {c^{2} x^{2}+1}\, c x \right )}{c^{2} \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}}\)	\(72\)

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

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)*x^2*c^2+arcsinh(c*x)-(c^2*x^2+1)

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Maxima [A]
time = 0.33, size = 55, normalized size = 1.31 \begin {gather*} -\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 {gather*}

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

-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] Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (38) = 76\).
time = 0.37, size = 96, normalized size = 2.29 \begin {gather*} \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 {gather*}

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

(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(

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Sympy [A]
time = 1.39, size = 60, normalized size = 1.43 \begin {gather*} \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 {gather*}

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

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \end {gather*}

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

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

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3.1.84 \(\int \frac {x (a+b \sinh ^{-1}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx\) [84]