3.62 \(\int \frac{\sqrt{\pi +c^2 \pi x^2} (a+b \sinh ^{-1}(c x))}{x^4} \, dx\)

Optimal. Leaf size=62 \[ -\frac{\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}+\frac{1}{3} \sqrt{\pi } b c^3 \log (x)-\frac{\sqrt{\pi } b c}{6 x^2} \]

[Out]

-(b*c*Sqrt[Pi])/(6*x^2) - ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*Pi*x^3) + (b*c^3*Sqrt[Pi]*Log[x])/
3

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Rubi [A]  time = 0.0895466, antiderivative size = 106, normalized size of antiderivative = 1.71, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {5723, 14} \[ -\frac{\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}-\frac{b c \sqrt{\pi c^2 x^2+\pi }}{6 x^2 \sqrt{c^2 x^2+1}}+\frac{b c^3 \sqrt{\pi c^2 x^2+\pi } \log (x)}{3 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x^4,x]

[Out]

-(b*c*Sqrt[Pi + c^2*Pi*x^2])/(6*x^2*Sqrt[1 + c^2*x^2]) - ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*Pi*
x^3) + (b*c^3*Sqrt[Pi + c^2*Pi*x^2]*Log[x])/(3*Sqrt[1 + c^2*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 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{\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac{\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}+\frac{\left (b c \sqrt{\pi +c^2 \pi x^2}\right ) \int \frac{1+c^2 x^2}{x^3} \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}+\frac{\left (b c \sqrt{\pi +c^2 \pi x^2}\right ) \int \left (\frac{1}{x^3}+\frac{c^2}{x}\right ) \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c \sqrt{\pi +c^2 \pi x^2}}{6 x^2 \sqrt{1+c^2 x^2}}-\frac{\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}+\frac{b c^3 \sqrt{\pi +c^2 \pi x^2} \log (x)}{3 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.129709, size = 78, normalized size = 1.26 \[ \frac{1}{3} \sqrt{\pi } b c^3 \log (x)-\frac{\sqrt{\pi } \left (2 a \left (c^2 x^2+1\right )^{3/2}+3 b c^3 x^3+2 b \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)+b c x\right )}{6 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x^4,x]

[Out]

-(Sqrt[Pi]*(b*c*x + 3*b*c^3*x^3 + 2*a*(1 + c^2*x^2)^(3/2) + 2*b*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]))/(6*x^3) + (
b*c^3*Sqrt[Pi]*Log[x])/3

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Maple [B]  time = 0.2, size = 501, normalized size = 8.1 \begin{align*} -{\frac{a}{3\,\pi \,{x}^{3}} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{3}{2}}}}-{\frac{2\,b{c}^{3}\sqrt{\pi }{\it Arcsinh} \left ( cx \right ) }{3}}+{\frac{b\sqrt{\pi }{x}^{4}{\it Arcsinh} \left ( cx \right ){c}^{7}}{3\,{c}^{4}{x}^{4}+3\,{c}^{2}{x}^{2}+1}}-{\frac{b\sqrt{\pi }{x}^{3}{\it Arcsinh} \left ( cx \right ){c}^{6}}{3\,{c}^{4}{x}^{4}+3\,{c}^{2}{x}^{2}+1}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{b\sqrt{\pi }{x}^{4}{c}^{7}}{18\,{c}^{4}{x}^{4}+18\,{c}^{2}{x}^{2}+6}}-{\frac{b\sqrt{\pi }{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ){c}^{5}}{18\,{c}^{4}{x}^{4}+18\,{c}^{2}{x}^{2}+6}}+{\frac{b\sqrt{\pi }{x}^{2}{\it Arcsinh} \left ( cx \right ){c}^{5}}{3\,{c}^{4}{x}^{4}+3\,{c}^{2}{x}^{2}+1}}-2\,{\frac{b\sqrt{\pi }x{\it Arcsinh} \left ( cx \right ) \sqrt{{c}^{2}{x}^{2}+1}{c}^{4}}{3\,{c}^{4}{x}^{4}+3\,{c}^{2}{x}^{2}+1}}-{\frac{b\sqrt{\pi } \left ({c}^{2}{x}^{2}+1 \right ){c}^{3}}{9\,{c}^{4}{x}^{4}+9\,{c}^{2}{x}^{2}+3}}+{\frac{b{c}^{3}\sqrt{\pi }{\it Arcsinh} \left ( cx \right ) }{9\,{c}^{4}{x}^{4}+9\,{c}^{2}{x}^{2}+3}}-{\frac{4\,b\sqrt{\pi }{\it Arcsinh} \left ( cx \right ){c}^{2}}{ \left ( 9\,{c}^{4}{x}^{4}+9\,{c}^{2}{x}^{2}+3 \right ) x}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{b\sqrt{\pi } \left ({c}^{2}{x}^{2}+1 \right ) c}{ \left ( 18\,{c}^{4}{x}^{4}+18\,{c}^{2}{x}^{2}+6 \right ){x}^{2}}}-{\frac{b\sqrt{\pi }{\it Arcsinh} \left ( cx \right ) }{ \left ( 9\,{c}^{4}{x}^{4}+9\,{c}^{2}{x}^{2}+3 \right ){x}^{3}}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{b{c}^{3}\sqrt{\pi }}{3}\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^4,x)

[Out]

-1/3*a/Pi/x^3*(Pi*c^2*x^2+Pi)^(3/2)-2/3*b*c^3*Pi^(1/2)*arcsinh(c*x)+b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^4*arc
sinh(c*x)*c^7-b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^6+1/6*b*Pi^(1/2)/(3*c^4*
x^4+3*c^2*x^2+1)*x^4*c^7-1/6*b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2*(c^2*x^2+1)*c^5+b*Pi^(1/2)/(3*c^4*x^4+3*c^
2*x^2+1)*x^2*arcsinh(c*x)*c^5-2*b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^4-1/3*b*
Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*(c^2*x^2+1)*c^3+1/3*b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*arcsinh(c*x)*c^3-4/3*b
*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^2-1/6*b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/
x^2*(c^2*x^2+1)*c-1/3*b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+1/3*b*c^3*Pi^(1/2)
*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)

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Maxima [B]  time = 1.19868, size = 193, normalized size = 3.11 \begin{align*} \frac{{\left (\pi ^{2} c^{4} \sqrt{\frac{1}{\pi c^{4}}} \log \left (x^{2} + \frac{1}{c^{2}}\right ) - \pi ^{\frac{3}{2}} \left (-1\right )^{2 \, \pi + 2 \, \pi c^{2} x^{2}} c^{2} \log \left (2 \, \pi c^{2} + \frac{2 \, \pi }{x^{2}}\right ) - \frac{\pi \sqrt{\pi + \pi c^{4} x^{4} + 2 \, \pi c^{2} x^{2}}}{x^{2}}\right )} b c}{6 \, \pi } - \frac{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}} b \operatorname{arsinh}\left (c x\right )}{3 \, \pi x^{3}} - \frac{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}} a}{3 \, \pi x^{3}} \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^4,x, algorithm="maxima")

[Out]

1/6*(pi^2*c^4*sqrt(1/(pi*c^4))*log(x^2 + 1/c^2) - pi^(3/2)*(-1)^(2*pi + 2*pi*c^2*x^2)*c^2*log(2*pi*c^2 + 2*pi/
x^2) - pi*sqrt(pi + pi*c^4*x^4 + 2*pi*c^2*x^2)/x^2)*b*c/pi - 1/3*(pi + pi*c^2*x^2)^(3/2)*b*arcsinh(c*x)/(pi*x^
3) - 1/3*(pi + pi*c^2*x^2)^(3/2)*a/(pi*x^3)

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Fricas [B]  time = 2.84194, size = 486, normalized size = 7.84 \begin{align*} -\frac{2 \, \sqrt{\pi + \pi c^{2} x^{2}}{\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \sqrt{\pi }{\left (b c^{5} x^{5} + b c^{3} x^{3}\right )} \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 ) + \sqrt{\pi + \pi c^{2} x^{2}}{\left (2 \, a c^{4} x^{4} + 4 \, a c^{2} x^{2} -{\left (b c x^{3} - b c x\right )} \sqrt{c^{2} x^{2} + 1} + 2 \, a\right )}}{6 \,{\left (c^{2} x^{5} + x^{3}\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^4,x, algorithm="fricas")

[Out]

-1/6*(2*sqrt(pi + pi*c^2*x^2)*(b*c^4*x^4 + 2*b*c^2*x^2 + b)*log(c*x + sqrt(c^2*x^2 + 1)) - sqrt(pi)*(b*c^5*x^5
+ b*c^3*x^3)*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)) + sqrt(pi + pi*c^2*x^2)*(2*a*c^4*x^4 + 4*a*c^2*x^2 - (b*c*x^3 - b*c*x)*sqrt(c^2*x^2
+ 1) + 2*a))/(c^2*x^5 + x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \sqrt{\pi } \left (\int \frac{a \sqrt{c^{2} x^{2} + 1}}{x^{4}}\, dx + \int \frac{b \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{x^{4}}\, dx\right ) \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**4,x)

[Out]

sqrt(pi)*(Integral(a*sqrt(c**2*x**2 + 1)/x**4, x) + Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**4, 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^{4}}\,{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^4,x, algorithm="giac")

[Out]

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