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

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

[Out]

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

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Rubi [A]  time = 0.0882457, antiderivative size = 63, normalized size of antiderivative = 1.54, 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 = {5723, 29} \[ \frac{b c \sqrt{c^2 x^2+1} \log (x)}{\sqrt{\pi c^2 x^2+\pi }}-\frac{\sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{\pi x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5723

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*d^IntPart[p]*(d + e
*x^2)^FracPart[p])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*Arc
Sinh[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]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\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*}

Mathematica [A]  time = 0.108567, size = 42, normalized size = 1.02 \[ \frac{b c \log (x)}{\sqrt{\pi }}-\frac{\sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{\pi } x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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]  time = 0.066, size = 84, normalized size = 2.1 \begin{align*} -{\frac{a}{\pi \,x}\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}-{\frac{bc{\it Arcsinh} \left ( cx \right ) }{\sqrt{\pi }}}-{\frac{b{\it Arcsinh} \left ( cx \right ) }{\sqrt{\pi }x}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{bc}{\sqrt{\pi }}\ln \left ( \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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/
2)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)

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Maxima [B]  time = 1.17426, size = 150, normalized size = 3.66 \begin{align*} \frac{{\left (\pi c^{2} \sqrt{\frac{1}{\pi c^{4}}} \log \left (x^{2} + \frac{1}{c^{2}}\right ) - \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 )\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(pi*c^2*sqrt(1/(pi*c^4))*log(x^2 + 1/c^2) - sqrt(pi)*(-1)^(2*pi + 2*pi*c^2*x^2)*log(2*pi*c^2 + 2*pi/x^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]  time = 2.8408, size = 319, normalized size = 7.78 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*
x^2)*a)/(pi*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsinh}\left (c x\right ) + a}{\sqrt{\pi + \pi c^{2} x^{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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