3.95 \(\int \frac{a+b \sinh ^{-1}(c x)}{(\pi +c^2 \pi x^2)^{3/2}} \, dx\)

Optimal. Leaf size=51 \[ \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt{\pi c^2 x^2+\pi }}-\frac{b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c} \]

[Out]

(x*(a + b*ArcSinh[c*x]))/(Pi*Sqrt[Pi + c^2*Pi*x^2]) - (b*Log[1 + c^2*x^2])/(2*c*Pi^(3/2))

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Rubi [A]  time = 0.0388463, antiderivative size = 76, normalized size of antiderivative = 1.49, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {5687, 260} \[ \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt{\pi c^2 x^2+\pi }}-\frac{b \sqrt{c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 \pi c \sqrt{\pi c^2 x^2+\pi }} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c*x])/(Pi + c^2*Pi*x^2)^(3/2),x]

[Out]

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

Rule 5687

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(x*(a + b*ArcSinh
[c*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n*Sqrt[1 + c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcSinh[
c*x])^(n - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt{\pi +c^2 \pi x^2}}-\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{x}{1+c^2 x^2} \, dx}{\pi \sqrt{\pi +c^2 \pi x^2}}\\ &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt{\pi +c^2 \pi x^2}}-\frac{b \sqrt{1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c \pi \sqrt{\pi +c^2 \pi x^2}}\\ \end{align*}

Mathematica [A]  time = 0.092849, size = 66, normalized size = 1.29 \[ \frac{2 a c x-b \sqrt{c^2 x^2+1} \log \left (c^2 x^2+1\right )+2 b c x \sinh ^{-1}(c x)}{2 \pi ^{3/2} c \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSinh[c*x])/(Pi + c^2*Pi*x^2)^(3/2),x]

[Out]

(2*a*c*x + 2*b*c*x*ArcSinh[c*x] - b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(2*c*Pi^(3/2)*Sqrt[1 + c^2*x^2])

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Maple [B]  time = 0.066, size = 132, normalized size = 2.6 \begin{align*}{\frac{ax}{\pi }{\frac{1}{\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}}}+2\,{\frac{b{\it Arcsinh} \left ( cx \right ) }{c{\pi }^{3/2}}}-{\frac{b{\it Arcsinh} \left ( cx \right ) c{x}^{2}}{{\pi }^{{\frac{3}{2}}} \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{b{\it Arcsinh} \left ( cx \right ) x}{{\pi }^{{\frac{3}{2}}}}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{b{\it Arcsinh} \left ( cx \right ) }{c{\pi }^{{\frac{3}{2}}} \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{b}{c{\pi }^{{\frac{3}{2}}}}\ln \left ( 1+ \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) } \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)^(3/2),x)

[Out]

a/Pi*x/(Pi*c^2*x^2+Pi)^(1/2)+2*b/c/Pi^(3/2)*arcsinh(c*x)-b/Pi^(3/2)*arcsinh(c*x)*c/(c^2*x^2+1)*x^2+b/Pi^(3/2)*
arcsinh(c*x)/(c^2*x^2+1)^(1/2)*x-b/Pi^(3/2)*arcsinh(c*x)/c/(c^2*x^2+1)-b/c/Pi^(3/2)*ln(1+(c*x+(c^2*x^2+1)^(1/2
))^2)

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Maxima [A]  time = 1.12228, size = 88, normalized size = 1.73 \begin{align*} -\frac{b c \sqrt{\frac{1}{\pi c^{4}}} \log \left (x^{2} + \frac{1}{c^{2}}\right )}{2 \, \pi } + \frac{b x \operatorname{arsinh}\left (c x\right )}{\pi \sqrt{\pi + \pi c^{2} x^{2}}} + \frac{a x}{\pi \sqrt{\pi + \pi c^{2} x^{2}}} \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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{\pi + \pi c^{2} x^{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{\pi ^{2} c^{4} x^{4} + 2 \, \pi ^{2} c^{2} x^{2} + \pi ^{2}}, 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)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(pi + pi*c^2*x^2)*(b*arcsinh(c*x) + a)/(pi^2*c^4*x^4 + 2*pi^2*c^2*x^2 + pi^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c^{2} x^{2} \sqrt{c^{2} x^{2} + 1} + \sqrt{c^{2} x^{2} + 1}}\, dx + \int \frac{b \operatorname{asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt{c^{2} x^{2} + 1} + \sqrt{c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac{3}{2}}} \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)**(3/2),x)

[Out]

(Integral(a/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x) + Integral(b*asinh(c*x)/(c**2*x**2*sqrt(
c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x))/pi**(3/2)

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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}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}}}\,{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)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)/(pi + pi*c^2*x^2)^(3/2), x)