3.222 \(\int \frac{(c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)}{x^3} \, dx\)

Optimal. Leaf size=364 \[ \frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{1}{6} a^3 c^2 x \sqrt{a^2 c x^2+c}-\frac{a c^2 \sqrt{a^2 c x^2+c}}{2 x}+2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac{13}{6} a^2 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{5 a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} a^2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) \]

[Out]

-(a*c^2*Sqrt[c + a^2*c*x^2])/(2*x) - (a^3*c^2*x*Sqrt[c + a^2*c*x^2])/6 + 2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[
a*x] - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*x^2) + (a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/3 - (5*a^2*c^
3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (13*a^2*c^(5/2
)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/6 + (((5*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I
*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (((5*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]
/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]

________________________________________________________________________________________

Rubi [A]  time = 1.14402, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {4950, 4946, 4962, 264, 4958, 4954, 217, 206, 4930, 195} \[ \frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{1}{6} a^3 c^2 x \sqrt{a^2 c x^2+c}-\frac{a c^2 \sqrt{a^2 c x^2+c}}{2 x}+2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac{13}{6} a^2 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{5 a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} a^2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^3,x]

[Out]

-(a*c^2*Sqrt[c + a^2*c*x^2])/(2*x) - (a^3*c^2*x*Sqrt[c + a^2*c*x^2])/6 + 2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[
a*x] - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*x^2) + (a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/3 - (5*a^2*c^
3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (13*a^2*c^(5/2
)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/6 + (((5*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I
*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (((5*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]
/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]

Rule 4950

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Dist[
d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Dist[(c^2*d)/f^2, Int[(f*x)^(m + 2)*(d + e*
x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] &&
IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))

Rule 4946

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

Rule 4962

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

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 4958

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + c^2*
x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/(x*Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[e, c^2*d] && IGtQ[p, 0] &&  !GtQ[d, 0]

Rule 4954

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[((d + e*x^2)^
(q + 1)*(a + b*ArcTan[c*x])^p)/(2*e*(q + 1)), x] - Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTan[c
*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{x^3} \, dx &=c \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x^3} \, dx+\left (a^2 c\right ) \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x} \, dx\\ &=c^2 \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^3} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x} \, dx\right )+\left (a^4 c^2\right ) \int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^2}+\frac{1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{1}{3} \left (a^3 c^2\right ) \int \sqrt{c+a^2 c x^2} \, dx-c^3 \int \frac{\tan ^{-1}(a x)}{x^3 \sqrt{c+a^2 c x^2}} \, dx+\left (a c^3\right ) \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx+2 \left (a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\left (a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx-\left (a^3 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx\right )\\ &=-\frac{a c^2 \sqrt{c+a^2 c x^2}}{x}-\frac{1}{6} a^3 c^2 x \sqrt{c+a^2 c x^2}-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}+\frac{1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{1}{2} \left (a c^3\right ) \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{2} \left (a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{6} \left (a^3 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )+\frac{\left (a^2 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\right )\\ &=-\frac{a c^2 \sqrt{c+a^2 c x^2}}{2 x}-\frac{1}{6} a^3 c^2 x \sqrt{c+a^2 c x^2}-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}+\frac{1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+2 \left (a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{2 a^2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-a^2 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i a^2 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^2 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}\right )-\frac{1}{6} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )+\frac{\left (a^2 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{a c^2 \sqrt{c+a^2 c x^2}}{2 x}-\frac{1}{6} a^3 c^2 x \sqrt{c+a^2 c x^2}-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}+\frac{1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{a^2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{1}{6} a^2 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i a^2 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{i a^2 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}+2 \left (a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{2 a^2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-a^2 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i a^2 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^2 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 2.02946, size = 361, normalized size = 0.99 \[ \frac{a^2 c^2 \sqrt{a^2 c x^2+c} \tan \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \left (60 i \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-60 i \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )+4 a^3 x^3 \tan ^{-1}(a x) \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )-2 a^2 x^2 \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )-6 \cot ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+28 a x \tan ^{-1}(a x) \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+60 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-60 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-3 \tan ^{-1}(a x) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )-4 \sinh ^{-1}(a x) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )+3 \tan ^{-1}(a x) \csc \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \sec \left (\frac{1}{2} \tan ^{-1}(a x)\right )+48 \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \log \left (\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )-\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )-48 \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \log \left (\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )+\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )-6\right )}{24 \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^3,x]

[Out]

(a^2*c^2*Sqrt[c + a^2*c*x^2]*(-6 - 4*ArcSinh[a*x]*Cot[ArcTan[a*x]/2] - 6*Cot[ArcTan[a*x]/2]^2 - 2*a^2*x^2*Csc[
ArcTan[a*x]/2]^2 + 28*a*x*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 + 4*a^3*x^3*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 - 3*Ar
cTan[a*x]*Cot[ArcTan[a*x]/2]*Csc[ArcTan[a*x]/2]^2 + 60*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 - E^(I*ArcTan[a*x]
)] - 60*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] + 48*Cot[ArcTan[a*x]/2]*Log[Cos[ArcTan[a*x]/
2] - Sin[ArcTan[a*x]/2]] - 48*Cot[ArcTan[a*x]/2]*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + (60*I)*Cot[Arc
Tan[a*x]/2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (60*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*ArcTan[a*x])] + 3*ArcTa
n[a*x]*Csc[ArcTan[a*x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTan[a*x]/2])/(24*Sqrt[1 + a^2*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.356, size = 204, normalized size = 0.6 \begin{align*}{\frac{{c}^{2} \left ( 2\,\arctan \left ( ax \right ){x}^{4}{a}^{4}-{a}^{3}{x}^{3}+14\,\arctan \left ( ax \right ){a}^{2}{x}^{2}-3\,ax-3\,\arctan \left ( ax \right ) \right ) }{6\,{x}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{{a}^{2}{c}^{2}}{6}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 15\,\arctan \left ( ax \right ) \ln \left ( 1+{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -26\,i\arctan \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -15\,i{\it dilog} \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -15\,i{\it dilog} \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x^3,x)

[Out]

1/6*c^2*(c*(a*x-I)*(a*x+I))^(1/2)*(2*arctan(a*x)*x^4*a^4-a^3*x^3+14*arctan(a*x)*a^2*x^2-3*a*x-3*arctan(a*x))/x
^2-1/6*a^2*c^2*(c*(a*x-I)*(a*x+I))^(1/2)*(15*arctan(a*x)*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-26*I*arctan((1+I*a*
x)/(a^2*x^2+1)^(1/2))-15*I*dilog((1+I*a*x)/(a^2*x^2+1)^(1/2))-15*I*dilog(1+(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*
x^2+1)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x^3,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)*arctan(a*x)/x^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**(5/2)*atan(a*x)/x**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x^3,x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)*arctan(a*x)/x^3, x)