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

Optimal. Leaf size=134 \[ -\pi ^{3/2} b \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\pi ^{3/2} b \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac{1}{3} \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\pi \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-2 \pi ^{3/2} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{9} \pi ^{3/2} b c^3 x^3-\frac{4}{3} \pi ^{3/2} b c x \]

[Out]

(-4*b*c*Pi^(3/2)*x)/3 - (b*c^3*Pi^(3/2)*x^3)/9 + Pi*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + ((Pi + c^2*Pi
*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/3 - 2*Pi^(3/2)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] - b*Pi^(3/2)*Pol
yLog[2, -E^ArcSinh[c*x]] + b*Pi^(3/2)*PolyLog[2, E^ArcSinh[c*x]]

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Rubi [A]  time = 0.304125, antiderivative size = 249, normalized size of antiderivative = 1.86, number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {5744, 5742, 5760, 4182, 2279, 2391, 8} \[ -\frac{\pi b \sqrt{\pi c^2 x^2+\pi } \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}+\frac{\pi b \sqrt{\pi c^2 x^2+\pi } \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}+\frac{1}{3} \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\pi \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \pi \sqrt{\pi c^2 x^2+\pi } \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}-\frac{\pi b c^3 x^3 \sqrt{\pi c^2 x^2+\pi }}{9 \sqrt{c^2 x^2+1}}-\frac{4 \pi b c x \sqrt{\pi c^2 x^2+\pi }}{3 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*c*Pi*x*Sqrt[Pi + c^2*Pi*x^2])/(3*Sqrt[1 + c^2*x^2]) - (b*c^3*Pi*x^3*Sqrt[Pi + c^2*Pi*x^2])/(9*Sqrt[1 + c
^2*x^2]) + Pi*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/3 -
(2*Pi*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (b*Pi*Sqrt[Pi +
c^2*Pi*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (b*Pi*Sqrt[Pi + c^2*Pi*x^2]*PolyLog[2, E^ArcSinh[
c*x]])/Sqrt[1 + c^2*x^2]

Rule 5744

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 + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int
[(f*x)^m*(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 + 2*p + 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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 1])

Rule 5742

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

Rule 5760

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

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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} \, dx &=\frac{1}{3} \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\pi \int \frac{\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx-\frac{\left (b c \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c \pi x \sqrt{\pi +c^2 \pi x^2}}{3 \sqrt{1+c^2 x^2}}-\frac{b c^3 \pi x^3 \sqrt{\pi +c^2 \pi x^2}}{9 \sqrt{1+c^2 x^2}}+\pi \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{\left (\pi \sqrt{\pi +c^2 \pi x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}-\frac{\left (b c \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int 1 \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{4 b c \pi x \sqrt{\pi +c^2 \pi x^2}}{3 \sqrt{1+c^2 x^2}}-\frac{b c^3 \pi x^3 \sqrt{\pi +c^2 \pi x^2}}{9 \sqrt{1+c^2 x^2}}+\pi \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{\left (\pi \sqrt{\pi +c^2 \pi x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{4 b c \pi x \sqrt{\pi +c^2 \pi x^2}}{3 \sqrt{1+c^2 x^2}}-\frac{b c^3 \pi x^3 \sqrt{\pi +c^2 \pi x^2}}{9 \sqrt{1+c^2 x^2}}+\pi \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \pi \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b \pi \sqrt{\pi +c^2 \pi x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b \pi \sqrt{\pi +c^2 \pi x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{4 b c \pi x \sqrt{\pi +c^2 \pi x^2}}{3 \sqrt{1+c^2 x^2}}-\frac{b c^3 \pi x^3 \sqrt{\pi +c^2 \pi x^2}}{9 \sqrt{1+c^2 x^2}}+\pi \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \pi \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b \pi \sqrt{\pi +c^2 \pi x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b \pi \sqrt{\pi +c^2 \pi x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{4 b c \pi x \sqrt{\pi +c^2 \pi x^2}}{3 \sqrt{1+c^2 x^2}}-\frac{b c^3 \pi x^3 \sqrt{\pi +c^2 \pi x^2}}{9 \sqrt{1+c^2 x^2}}+\pi \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \pi \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{b \pi \sqrt{\pi +c^2 \pi x^2} \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{b \pi \sqrt{\pi +c^2 \pi x^2} \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.353417, size = 180, normalized size = 1.34 \[ \frac{1}{9} \pi ^{3/2} \left (9 b \left (\text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-c x+\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )+3 a \sqrt{c^2 x^2+1} \left (c^2 x^2+4\right )-9 a \log \left (\pi \left (\sqrt{c^2 x^2+1}+1\right )\right )+9 a \log (x)-b \left (c^3 x^3-3 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)+3 c x\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Pi^(3/2)*(3*a*Sqrt[1 + c^2*x^2]*(4 + c^2*x^2) - b*(3*c*x + c^3*x^3 - 3*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]) + 9*
a*Log[x] - 9*a*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + 9*b*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*L
og[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog
[2, E^(-ArcSinh[c*x])])))/9

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Maple [A]  time = 0.172, size = 227, normalized size = 1.7 \begin{align*}{\frac{a}{3} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{3}{2}}}}-a{\pi }^{{\frac{3}{2}}}{\it Artanh} \left ({\sqrt{\pi }{\frac{1}{\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}}} \right ) +a\pi \,\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }-{\frac{b{c}^{3}{\pi }^{{\frac{3}{2}}}{x}^{3}}{9}}+{\frac{4\,b{\it Arcsinh} \left ( cx \right ){\pi }^{3/2}}{3}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{4\,bc{\pi }^{3/2}x}{3}}+b{\pi }^{{\frac{3}{2}}}{\it Arcsinh} \left ( cx \right ) \ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) -b{\pi }^{{\frac{3}{2}}}{\it Arcsinh} \left ( cx \right ) \ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) +b{\pi }^{{\frac{3}{2}}}{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) -b{\pi }^{{\frac{3}{2}}}{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +{\frac{b{\it Arcsinh} \left ( cx \right ){\pi }^{{\frac{3}{2}}}{x}^{2}{c}^{2}}{3}\sqrt{{c}^{2}{x}^{2}+1}} \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,x)

[Out]

1/3*(Pi*c^2*x^2+Pi)^(3/2)*a-a*Pi^(3/2)*arctanh(Pi^(1/2)/(Pi*c^2*x^2+Pi)^(1/2))+a*Pi*(Pi*c^2*x^2+Pi)^(1/2)-1/9*
b*c^3*Pi^(3/2)*x^3+4/3*b*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*Pi^(3/2)-4/3*b*c*Pi^(3/2)*x+b*Pi^(3/2)*arcsinh(c*x)*ln
(1-c*x-(c^2*x^2+1)^(1/2))-b*Pi^(3/2)*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))+b*Pi^(3/2)*polylog(2,c*x+(c^2*x^
2+1)^(1/2))-b*Pi^(3/2)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+1/3*b*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*Pi^(3/2)*x^2*c^2

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

[Out]

-1/3*(3*pi^(3/2)*arcsinh(1/(sqrt(c^2)*abs(x))) - 3*pi*sqrt(pi + pi*c^2*x^2) - (pi + pi*c^2*x^2)^(3/2))*a + b*i
ntegrate((pi + pi*c^2*x^2)^(3/2)*log(c*x + sqrt(c^2*x^2 + 1))/x, 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 (\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}, 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,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, x)

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

[Out]

pi**(3/2)*(Integral(a*sqrt(c**2*x**2 + 1)/x, x) + Integral(a*c**2*x*sqrt(c**2*x**2 + 1), x) + Integral(b*sqrt(
c**2*x**2 + 1)*asinh(c*x)/x, x) + Integral(b*c**2*x*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{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{x}\,{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,x, algorithm="giac")

[Out]

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