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

Optimal. Leaf size=108 \[ \frac{3}{2} \pi c^2 x \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac{3 \pi ^{3/2} c \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b}-\frac{1}{4} \pi ^{3/2} b c^3 x^2+\pi ^{3/2} b c \log (x) \]

[Out]

-(b*c^3*Pi^(3/2)*x^2)/4 + (3*c^2*Pi*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/2 - ((Pi + c^2*Pi*x^2)^(3/2)
*(a + b*ArcSinh[c*x]))/x + (3*c*Pi^(3/2)*(a + b*ArcSinh[c*x])^2)/(4*b) + b*c*Pi^(3/2)*Log[x]

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Rubi [A]  time = 0.167999, antiderivative size = 177, normalized size of antiderivative = 1.64, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {5739, 5682, 5675, 30, 14} \[ \frac{3}{2} \pi c^2 x \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 \pi c \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b \sqrt{c^2 x^2+1}}-\frac{\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac{\pi b c^3 x^2 \sqrt{\pi c^2 x^2+\pi }}{4 \sqrt{c^2 x^2+1}}+\frac{\pi b c \sqrt{\pi c^2 x^2+\pi } \log (x)}{\sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c^3*Pi*x^2*Sqrt[Pi + c^2*Pi*x^2])/(4*Sqrt[1 + c^2*x^2]) + (3*c^2*Pi*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh
[c*x]))/2 - ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x + (3*c*Pi*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*
x])^2)/(4*b*Sqrt[1 + c^2*x^2]) + (b*c*Pi*Sqrt[Pi + c^2*Pi*x^2]*Log[x])/Sqrt[1 + c^2*x^2]

Rule 5739

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*(a + b*ArcSinh[c*x])^n)/(f*(m + 1)), x] + (-Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x
)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], 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*ArcSinh[c*x])^(n -
1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rule 5682

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

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
]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac{\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\left (3 c^2 \pi \right ) \int \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{\left (b c \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int \frac{1+c^2 x^2}{x} \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{3}{2} c^2 \pi x \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac{\left (b c \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int \left (\frac{1}{x}+c^2 x\right ) \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (3 c^2 \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{2 \sqrt{1+c^2 x^2}}-\frac{\left (3 b c^3 \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int x \, dx}{2 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c^3 \pi x^2 \sqrt{\pi +c^2 \pi x^2}}{4 \sqrt{1+c^2 x^2}}+\frac{3}{2} c^2 \pi x \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac{3 c \pi \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b \sqrt{1+c^2 x^2}}+\frac{b c \pi \sqrt{\pi +c^2 \pi x^2} \log (x)}{\sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.28834, size = 122, normalized size = 1.13 \[ \frac{\pi ^{3/2} \left (2 \sinh ^{-1}(c x) \left (6 a c x-4 b \sqrt{c^2 x^2+1}+b c x \sinh \left (2 \sinh ^{-1}(c x)\right )\right )+4 a c^2 x^2 \sqrt{c^2 x^2+1}-8 a \sqrt{c^2 x^2+1}+8 b c x \log (c x)+6 b c x \sinh ^{-1}(c x)^2-b c x \cosh \left (2 \sinh ^{-1}(c x)\right )\right )}{8 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Pi^(3/2)*(-8*a*Sqrt[1 + c^2*x^2] + 4*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 6*b*c*x*ArcSinh[c*x]^2 - b*c*x*Cosh[2*ArcS
inh[c*x]] + 8*b*c*x*Log[c*x] + 2*ArcSinh[c*x]*(6*a*c*x - 4*b*Sqrt[1 + c^2*x^2] + b*c*x*Sinh[2*ArcSinh[c*x]])))
/(8*x)

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Maple [B]  time = 0.148, size = 222, normalized size = 2.1 \begin{align*} -{\frac{a}{\pi \,x} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{5}{2}}}}+a{c}^{2}x \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{3}{2}}}+{\frac{3\,a{c}^{2}\pi \,x}{2}\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}+{\frac{3\,a{c}^{2}{\pi }^{2}}{2}\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{3\,bc{\pi }^{3/2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{4}}+{\frac{b{\it Arcsinh} \left ( cx \right ){\pi }^{{\frac{3}{2}}}x{c}^{2}}{2}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{b{c}^{3}{\pi }^{{\frac{3}{2}}}{x}^{2}}{4}}-bc{\pi }^{{\frac{3}{2}}}{\it Arcsinh} \left ( cx \right ) -{\frac{b{\pi }^{{\frac{3}{2}}}c}{8}}-{\frac{b{\it Arcsinh} \left ( cx \right ){\pi }^{{\frac{3}{2}}}}{x}\sqrt{{c}^{2}{x}^{2}+1}}+bc{\pi }^{{\frac{3}{2}}}\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((Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x))/x^2,x)

[Out]

-a/Pi/x*(Pi*c^2*x^2+Pi)^(5/2)+a*c^2*x*(Pi*c^2*x^2+Pi)^(3/2)+3/2*a*c^2*Pi*x*(Pi*c^2*x^2+Pi)^(1/2)+3/2*a*c^2*Pi^
2*ln(Pi*x*c^2/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+3/4*b*c*Pi^(3/2)*arcsinh(c*x)^2+1/2*b*arcsi
nh(c*x)*Pi^(3/2)*(c^2*x^2+1)^(1/2)*x*c^2-1/4*b*c^3*Pi^(3/2)*x^2-b*c*Pi^(3/2)*arcsinh(c*x)-1/8*b*Pi^(3/2)*c-b*P
i^(3/2)*arcsinh(c*x)/x*(c^2*x^2+1)^(1/2)+b*c*Pi^(3/2)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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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 (\pi a c^{2} x^{2} + \pi a +{\left (\pi b c^{2} x^{2} + \pi b\right )} \operatorname{arsinh}\left (c x\right )\right )}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{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((pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x))/x^2,x, algorithm="giac")

[Out]

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