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

Optimal. Leaf size=155 \[ -\frac{3}{2} \pi ^{3/2} b c^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\frac{3}{2} \pi ^{3/2} b c^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac{3}{2} \pi c^2 \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-3 \pi ^{3/2} c^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\pi ^{3/2} (-b) c^3 x-\frac{\pi ^{3/2} b c}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.302217, antiderivative size = 270, normalized size of antiderivative = 1.74, number of steps used = 11, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {5739, 5742, 5760, 4182, 2279, 2391, 8, 14} \[ -\frac{3 \pi b c^2 \sqrt{\pi c^2 x^2+\pi } \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{c^2 x^2+1}}+\frac{3 \pi b c^2 \sqrt{\pi c^2 x^2+\pi } \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{c^2 x^2+1}}+\frac{3}{2} \pi c^2 \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac{3 \pi c^2 \sqrt{\pi c^2 x^2+\pi } \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}-\frac{\pi b c^3 x \sqrt{\pi c^2 x^2+\pi }}{\sqrt{c^2 x^2+1}}-\frac{\pi b c \sqrt{\pi c^2 x^2+\pi }}{2 x \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*Pi*Sqrt[Pi + c^2*Pi*x^2])/(2*x*Sqrt[1 + c^2*x^2]) - (b*c^3*Pi*x*Sqrt[Pi + c^2*Pi*x^2])/Sqrt[1 + c^2*x^2]
+ (3*c^2*Pi*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]))/(2
*x^2) - (3*c^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] - (3*b
*c^2*Pi*Sqrt[Pi + c^2*Pi*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(2*Sqrt[1 + c^2*x^2]) + (3*b*c^2*Pi*Sqrt[Pi + c^2*P
i*x^2]*PolyLog[2, E^ArcSinh[c*x]])/(2*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 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]

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

Mathematica [A]  time = 1.71188, size = 292, normalized size = 1.88 \[ \frac{\pi ^{3/2} \left (12 b c^2 x^2 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-12 b c^2 x^2 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+8 a c^2 x^2 \sqrt{c^2 x^2+1}-4 a \sqrt{c^2 x^2+1}+12 a c^2 x^2 \log (x)-12 a c^2 x^2 \log \left (\pi \left (\sqrt{c^2 x^2+1}+1\right )\right )-8 b c^3 x^3+8 b c^2 x^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+12 b c^2 x^2 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-12 b c^2 x^2 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-b c^3 x^3 \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-b c^2 x^2 \sinh ^{-1}(c x) \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )+4 b c x \sinh ^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-4 b \sinh ^{-1}(c x) \sinh ^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )}{8 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Pi^(3/2)*(-8*b*c^3*x^3 - 4*a*Sqrt[1 + c^2*x^2] + 8*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 8*b*c^2*x^2*Sqrt[1 + c^2*x^2
]*ArcSinh[c*x] - b*c^3*x^3*Csch[ArcSinh[c*x]/2]^2 - b*c^2*x^2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 12*b*c^2*x
^2*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 12*b*c^2*x^2*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 12*a*c^2*x
^2*Log[x] - 12*a*c^2*x^2*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + 12*b*c^2*x^2*PolyLog[2, -E^(-ArcSinh[c*x])] - 12*b*
c^2*x^2*PolyLog[2, E^(-ArcSinh[c*x])] + 4*b*c*x*Sinh[ArcSinh[c*x]/2]^2 - 4*b*ArcSinh[c*x]*Sinh[ArcSinh[c*x]/2]
^2))/(8*x^2)

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Maple [A]  time = 0.371, size = 295, normalized size = 1.9 \begin{align*} -{\frac{a}{2\,\pi \,{x}^{2}} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{5}{2}}}}+{\frac{a{c}^{2}}{2} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{3}{2}}}}-{\frac{3\,a{c}^{2}{\pi }^{3/2}}{2}{\it Artanh} \left ({\sqrt{\pi }{\frac{1}{\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}}} \right ) }+{\frac{3\,a{c}^{2}\pi }{2}\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}+b{\it Arcsinh} \left ( cx \right ){\pi }^{{\frac{3}{2}}}\sqrt{{c}^{2}{x}^{2}+1}{c}^{2}-b{c}^{3}{\pi }^{{\frac{3}{2}}}x-{\frac{b{\it Arcsinh} \left ( cx \right ){\pi }^{{\frac{3}{2}}}{c}^{2}}{2}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{b{\pi }^{{\frac{3}{2}}}c}{2\,x}}-{\frac{b{\it Arcsinh} \left ( cx \right ){\pi }^{{\frac{3}{2}}}}{2\,{x}^{2}}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{3\,b{\it Arcsinh} \left ( cx \right ){\pi }^{3/2}{c}^{2}}{2}\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{3\,b{\pi }^{3/2}{c}^{2}}{2}{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }+{\frac{3\,b{\it Arcsinh} \left ( cx \right ){\pi }^{3/2}{c}^{2}}{2}\ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }+{\frac{3\,b{\pi }^{3/2}{c}^{2}}{2}{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{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^3,x)

[Out]

-1/2*a/Pi/x^2*(Pi*c^2*x^2+Pi)^(5/2)+1/2*a*c^2*(Pi*c^2*x^2+Pi)^(3/2)-3/2*a*c^2*Pi^(3/2)*arctanh(Pi^(1/2)/(Pi*c^
2*x^2+Pi)^(1/2))+3/2*a*c^2*Pi*(Pi*c^2*x^2+Pi)^(1/2)+b*arcsinh(c*x)*Pi^(3/2)*(c^2*x^2+1)^(1/2)*c^2-b*c^3*Pi^(3/
2)*x-1/2*b*Pi^(3/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^2-1/2*b*c*Pi^(3/2)/x-1/2*b*Pi^(3/2)/(c^2*x^2+1)^(1/2)/x^2
*arcsinh(c*x)-3/2*b*c^2*Pi^(3/2)*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))-3/2*b*c^2*Pi^(3/2)*polylog(2,-c*x-(c
^2*x^2+1)^(1/2))+3/2*b*c^2*Pi^(3/2)*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))+3/2*b*c^2*Pi^(3/2)*polylog(2,c*x+
(c^2*x^2+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left (3 \, \pi ^{\frac{3}{2}} c^{2} \operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) - 3 \, \pi \sqrt{\pi + \pi c^{2} x^{2}} c^{2} -{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}} c^{2} + \frac{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{5}{2}}}{\pi x^{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^{3}}\,{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^3,x, algorithm="maxima")

[Out]

-1/2*(3*pi^(3/2)*c^2*arcsinh(1/(sqrt(c^2)*abs(x))) - 3*pi*sqrt(pi + pi*c^2*x^2)*c^2 - (pi + pi*c^2*x^2)^(3/2)*
c^2 + (pi + pi*c^2*x^2)^(5/2)/(pi*x^2))*a + b*integrate((pi + pi*c^2*x^2)^(3/2)*log(c*x + sqrt(c^2*x^2 + 1))/x
^3, 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^{3}}, 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^3,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^3, 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^{3}}\, dx + \int \frac{a c^{2} \sqrt{c^{2} x^{2} + 1}}{x}\, dx + \int \frac{b \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{b c^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{x}\, 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**3,x)

[Out]

pi**(3/2)*(Integral(a*sqrt(c**2*x**2 + 1)/x**3, x) + Integral(a*c**2*sqrt(c**2*x**2 + 1)/x, x) + Integral(b*sq
rt(c**2*x**2 + 1)*asinh(c*x)/x**3, x) + Integral(b*c**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/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^{3}}\,{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^3,x, algorithm="giac")

[Out]

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