3.791 \(\int \frac{x^4 \tan ^{-1}(a x)^{3/2}}{(c+a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=230 \[ -\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a^5 c^3}+\frac{3 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^5 c^3}+\frac{3 x^4 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac{x^3 \tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}-\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 a^4 c^3 \left (a^2 x^2+1\right )}-\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a^5 c^3 \left (a^2 x^2+1\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a^5 c^3}+\frac{27 \sqrt{\tan ^{-1}(a x)}}{256 a^5 c^3} \]

[Out]

(27*Sqrt[ArcTan[a*x]])/(256*a^5*c^3) + (3*x^4*Sqrt[ArcTan[a*x]])/(32*a*c^3*(1 + a^2*x^2)^2) - (9*Sqrt[ArcTan[a
*x]])/(32*a^5*c^3*(1 + a^2*x^2)) - (x^3*ArcTan[a*x]^(3/2))/(4*a^2*c^3*(1 + a^2*x^2)^2) - (3*x*ArcTan[a*x]^(3/2
))/(8*a^4*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(5/2))/(20*a^5*c^3) - (3*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[A
rcTan[a*x]]])/(512*a^5*c^3) + (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a^5*c^3)

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Rubi [A]  time = 0.41113, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4940, 4936, 4930, 4904, 3312, 3304, 3352, 4970} \[ -\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a^5 c^3}+\frac{3 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^5 c^3}+\frac{3 x^4 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac{x^3 \tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}-\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 a^4 c^3 \left (a^2 x^2+1\right )}-\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a^5 c^3 \left (a^2 x^2+1\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a^5 c^3}+\frac{27 \sqrt{\tan ^{-1}(a x)}}{256 a^5 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]

[Out]

(27*Sqrt[ArcTan[a*x]])/(256*a^5*c^3) + (3*x^4*Sqrt[ArcTan[a*x]])/(32*a*c^3*(1 + a^2*x^2)^2) - (9*Sqrt[ArcTan[a
*x]])/(32*a^5*c^3*(1 + a^2*x^2)) - (x^3*ArcTan[a*x]^(3/2))/(4*a^2*c^3*(1 + a^2*x^2)^2) - (3*x*ArcTan[a*x]^(3/2
))/(8*a^4*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(5/2))/(20*a^5*c^3) - (3*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[A
rcTan[a*x]]])/(512*a^5*c^3) + (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a^5*c^3)

Rule 4940

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

Rule 4936

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^2)/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(a + b*ArcTan
[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (Dist[(b*p)/(2*c), Int[(x*(a + b*ArcTan[c*x])^(p - 1))/(d + e*x^2)^
2, x], x] - Simp[(x*(a + b*ArcTan[c*x])^p)/(2*c^2*d*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c
^2*d] && GtQ[p, 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 4904

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

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4970

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

Rubi steps

\begin{align*} \int \frac{x^4 \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{3 x^4 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{x^3 \tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3}{64} \int \frac{x^4}{\left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{3 \int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a^2 c}\\ &=\frac{3 x^4 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{x^3 \tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a^5 c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin ^4(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a^5 c^3}+\frac{9 \int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a^3 c}\\ &=\frac{3 x^4 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a^5 c^3}-\frac{3 \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}-\frac{\cos (2 x)}{2 \sqrt{x}}+\frac{\cos (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^5 c^3}+\frac{9 \int \frac{1}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{64 a^4 c}\\ &=-\frac{9 \sqrt{\tan ^{-1}(a x)}}{256 a^5 c^3}+\frac{3 x^4 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a^5 c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^5 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{128 a^5 c^3}+\frac{9 \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a^5 c^3}\\ &=-\frac{9 \sqrt{\tan ^{-1}(a x)}}{256 a^5 c^3}+\frac{3 x^4 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a^5 c^3}-\frac{3 \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{256 a^5 c^3}+\frac{3 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{64 a^5 c^3}+\frac{9 \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^5 c^3}\\ &=\frac{27 \sqrt{\tan ^{-1}(a x)}}{256 a^5 c^3}+\frac{3 x^4 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a^5 c^3}-\frac{3 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a^5 c^3}+\frac{3 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a^5 c^3}+\frac{9 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{128 a^5 c^3}\\ &=\frac{27 \sqrt{\tan ^{-1}(a x)}}{256 a^5 c^3}+\frac{3 x^4 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a^5 c^3}-\frac{3 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a^5 c^3}+\frac{3 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a^5 c^3}+\frac{9 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{64 a^5 c^3}\\ &=\frac{27 \sqrt{\tan ^{-1}(a x)}}{256 a^5 c^3}+\frac{3 x^4 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a^5 c^3}-\frac{3 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a^5 c^3}+\frac{3 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^5 c^3}\\ \end{align*}

Mathematica [C]  time = 0.775524, size = 355, normalized size = 1.54 \[ \frac{90 \sqrt{\tan ^{-1}(a x)} \left (\frac{\text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )}{\sqrt{-i \tan ^{-1}(a x)}}+\frac{\text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )}{\sqrt{i \tan ^{-1}(a x)}}+8\right )+\frac{225 \left (-4 i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )+4 i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )-i \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )+i \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )+24 \tan ^{-1}(a x)\right )}{\sqrt{\tan ^{-1}(a x)}}+\frac{64 \sqrt{\tan ^{-1}(a x)} \left (15 \left (17 a^4 x^4-6 a^2 x^2-15\right )+192 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2-160 a x \left (5 a^2 x^2+3\right ) \tan ^{-1}(a x)\right )}{\left (a^2 x^2+1\right )^2}-510 \left (\sqrt{2 \pi } \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )-8 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )+12 \sqrt{\tan ^{-1}(a x)}\right )}{81920 a^5 c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]

[Out]

((64*Sqrt[ArcTan[a*x]]*(15*(-15 - 6*a^2*x^2 + 17*a^4*x^4) - 160*a*x*(3 + 5*a^2*x^2)*ArcTan[a*x] + 192*(1 + a^2
*x^2)^2*ArcTan[a*x]^2))/(1 + a^2*x^2)^2 - 510*(12*Sqrt[ArcTan[a*x]] + Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[Ar
cTan[a*x]]] - 8*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]]) + 90*Sqrt[ArcTan[a*x]]*(8 + Gamma[1/2, (-4*
I)*ArcTan[a*x]]/Sqrt[(-I)*ArcTan[a*x]] + Gamma[1/2, (4*I)*ArcTan[a*x]]/Sqrt[I*ArcTan[a*x]]) + (225*(24*ArcTan[
a*x] - (4*I)*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + (4*I)*Sqrt[2]*Sqrt[I*ArcTan[a*x]]
*Gamma[1/2, (2*I)*ArcTan[a*x]] - I*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + I*Sqrt[I*ArcTan[a*x
]]*Gamma[1/2, (4*I)*ArcTan[a*x]]))/Sqrt[ArcTan[a*x]])/(81920*a^5*c^3)

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Maple [A]  time = 0.168, size = 132, normalized size = 0.6 \begin{align*}{\frac{1}{5120\,{c}^{3}{a}^{5}} \left ( 768\, \left ( \arctan \left ( ax \right ) \right ) ^{3}-1280\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\sin \left ( 2\,\arctan \left ( ax \right ) \right ) +160\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\sin \left ( 4\,\arctan \left ( ax \right ) \right ) -15\,\sqrt{2}\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +60\,\cos \left ( 4\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -960\,\cos \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +480\,\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \right ){\frac{1}{\sqrt{\arctan \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5120/a^5/c^3/arctan(a*x)^(1/2)*(768*arctan(a*x)^3-1280*arctan(a*x)^2*sin(2*arctan(a*x))+160*arctan(a*x)^2*si
n(4*arctan(a*x))-15*2^(1/2)*arctan(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))+60*cos(4
*arctan(a*x))*arctan(a*x)-960*cos(2*arctan(a*x))*arctan(a*x)+480*arctan(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*arctan(
a*x)^(1/2)/Pi^(1/2)))

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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(x^4*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{4} \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*atan(a*x)**(3/2)/(a**2*c*x**2+c)**3,x)

[Out]

Integral(x**4*atan(a*x)**(3/2)/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x)/c**3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

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