∫ a+b sinh−1(cx)__ x2√ _____

Optimal. Leaf size=41 \[ -\frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\pi x}+\frac {b c \log (x)}{\sqrt {\pi }} \]

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

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Rubi [A]
time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5800, 29} \begin {gather*} \frac {b c \log (x)}{\sqrt {\pi }}-\frac {\sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{\pi x} \end {gather*}

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

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[

(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(

d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]

/; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

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

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Mathematica [A]
time = 0.07, size = 42, normalized size = 1.02 \begin {gather*} -\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi } x}+\frac {b c \log (x)}{\sqrt {\pi }} \end {gather*}

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(37)=74\).
time = 2.05, size = 84, normalized size = 2.05


method	result	size

default	\(-\frac {a \sqrt {\pi \,c^{2} x^{2}+\pi }}{\pi x}-\frac {b c \arcsinh \left (c x \right )}{\sqrt {\pi }}-\frac {b \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{\sqrt {\pi }\, x}+\frac {b c \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{\sqrt {\pi }}\)	\(84\)

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

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (37) = 74\).
time = 0.31, size = 101, normalized size = 2.46 \begin {gather*} -\frac {{\left (\sqrt {\pi } \left (-1\right )^{2 \, \pi + 2 \, \pi c^{2} x^{2}} \log \left (2 \, \pi c^{2} + \frac {2 \, \pi }{x^{2}}\right ) - \sqrt {\pi } \log \left (x^{2} + \frac {1}{c^{2}}\right )\right )} b c}{2 \, \pi } - \frac {\sqrt {\pi + \pi c^{2} x^{2}} b \operatorname {arsinh}\left (c x\right )}{\pi x} - \frac {\sqrt {\pi + \pi c^{2} x^{2}} a}{\pi x} \end {gather*}

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

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

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (37) = 74\).
time = 0.43, size = 132, normalized size = 3.22 \begin {gather*} \frac {\sqrt {\pi } b c x \log \left (\frac {\pi + \pi c^{2} x^{6} + \pi c^{2} x^{2} + \pi x^{4} + \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )}}{c^{2} x^{4} + x^{2}}\right ) - 2 \, \sqrt {\pi + \pi c^{2} x^{2}} b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, \sqrt {\pi + \pi c^{2} x^{2}} a}{2 \, \pi x} \end {gather*}

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

1/2*(sqrt(pi)*b*c*x*log((pi + pi*c^2*x^6 + pi*c^2*x^2 + pi*x^4 + sqrt(pi)*sqrt(pi + pi*c^2*x^2)*sqrt(c^2*x^2 +

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a}{x^{2} \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{2} \sqrt {c^{2} x^{2} + 1}}\, dx}{\sqrt {\pi }} \end {gather*}

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

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

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

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3.1.87 \(\int \frac {a+b \sinh ^{-1}(c x)}{x^2 \sqrt {\pi +c^2 \pi x^2}} \, dx\) [87]