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

Optimal. Leaf size=61 \[ -\frac{\sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac{\sqrt{\pi } c \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+\sqrt{\pi } b c \log (x) \]

[Out]

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

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Rubi [A]  time = 0.107924, antiderivative size = 105, normalized size of antiderivative = 1.72, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {5737, 29, 5675} \[ \frac{c \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b \sqrt{c^2 x^2+1}}-\frac{\sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac{b c \sqrt{\pi c^2 x^2+\pi } \log (x)}{\sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5737

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

Rule 29

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

Rule 5675

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

Rubi steps

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

Mathematica [A]  time = 0.163245, size = 75, normalized size = 1.23 \[ \frac{\sqrt{\pi } \left (2 \sinh ^{-1}(c x) \left (a c x-b \sqrt{c^2 x^2+1}\right )-2 a \sqrt{c^2 x^2+1}+2 b c x \log (c x)+b c x \sinh ^{-1}(c x)^2\right )}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.114, size = 155, normalized size = 2.5 \begin{align*} -{\frac{a}{\pi \,x} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{3}{2}}}}+a{c}^{2}x\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }+{a{c}^{2}\pi \ln \left ({\pi \,{c}^{2}x{\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+\sqrt{\pi \,{c}^{2}{x}^{2}+\pi } \right ){\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+{\frac{bc\sqrt{\pi } \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{2}}-bc\sqrt{\pi }{\it Arcsinh} \left ( cx \right ) -{\frac{b\sqrt{\pi }{\it Arcsinh} \left ( cx \right ) }{x}\sqrt{{c}^{2}{x}^{2}+1}}+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))*(Pi*c^2*x^2+Pi)^(1/2)/x^2,x)

[Out]

-a/Pi/x*(Pi*c^2*x^2+Pi)^(3/2)+a*c^2*x*(Pi*c^2*x^2+Pi)^(1/2)+a*c^2*Pi*ln(Pi*x*c^2/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi
)^(1/2))/(Pi*c^2)^(1/2)+1/2*b*c*Pi^(1/2)*arcsinh(c*x)^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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (\frac{\pi c^{2} \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{\pi c^{2}}} - \frac{\sqrt{\pi + \pi c^{2} x^{2}}}{x}\right )} a + b \int \frac{\sqrt{\pi + \pi c^{2} x^{2}} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{x^{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)^(1/2)/x^2,x, algorithm="maxima")

[Out]

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

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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 )}}{x^{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)^(1/2)/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 3.19936, size = 110, normalized size = 1.8 \begin{align*} - \frac{\sqrt{\pi } a c^{2} x}{\sqrt{c^{2} x^{2} + 1}} + \sqrt{\pi } a c \operatorname{asinh}{\left (c x \right )} - \frac{\sqrt{\pi } a}{x \sqrt{c^{2} x^{2} + 1}} + \sqrt{\pi } b c \log{\left (x \right )} + \frac{\sqrt{\pi } b c \operatorname{asinh}^{2}{\left (c x \right )}}{2} - \frac{\sqrt{\pi } b \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{x} \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)**(1/2)/x**2,x)

[Out]

-sqrt(pi)*a*c**2*x/sqrt(c**2*x**2 + 1) + sqrt(pi)*a*c*asinh(c*x) - sqrt(pi)*a/(x*sqrt(c**2*x**2 + 1)) + sqrt(p
i)*b*c*log(x) + sqrt(pi)*b*c*asinh(c*x)**2/2 - sqrt(pi)*b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\pi + \pi c^{2} x^{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{x^{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)^(1/2)/x^2,x, algorithm="giac")

[Out]

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