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

Optimal. Leaf size=108 \[ \frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a^3 c^3}+\frac{\tan ^{-1}(a x)^{5/2}}{20 a^3 c^3}-\frac{\tan ^{-1}(a x)^{3/2} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a^3 c^3}-\frac{3 \sqrt{\tan ^{-1}(a x)} \cos \left (4 \tan ^{-1}(a x)\right )}{256 a^3 c^3} \]

[Out]

ArcTan[a*x]^(5/2)/(20*a^3*c^3) - (3*Sqrt[ArcTan[a*x]]*Cos[4*ArcTan[a*x]])/(256*a^3*c^3) + (3*Sqrt[Pi/2]*Fresne
lC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a^3*c^3) - (ArcTan[a*x]^(3/2)*Sin[4*ArcTan[a*x]])/(32*a^3*c^3)

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Rubi [A]  time = 0.162614, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4970, 4406, 3296, 3304, 3352} \[ \frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a^3 c^3}+\frac{\tan ^{-1}(a x)^{5/2}}{20 a^3 c^3}-\frac{\tan ^{-1}(a x)^{3/2} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a^3 c^3}-\frac{3 \sqrt{\tan ^{-1}(a x)} \cos \left (4 \tan ^{-1}(a x)\right )}{256 a^3 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[a*x]^(5/2)/(20*a^3*c^3) - (3*Sqrt[ArcTan[a*x]]*Cos[4*ArcTan[a*x]])/(256*a^3*c^3) + (3*Sqrt[Pi/2]*Fresne
lC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a^3*c^3) - (ArcTan[a*x]^(3/2)*Sin[4*ArcTan[a*x]])/(32*a^3*c^3)

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])

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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]

Rubi steps

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

Mathematica [C]  time = 0.745112, size = 353, normalized size = 3.27 \[ \frac{-90 \sqrt{\tan ^{-1}(a x)} \left (\frac{\text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )}{\sqrt{-i \tan ^{-1}(a x)}}+\frac{\text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )}{\sqrt{i \tan ^{-1}(a x)}}+8\right )+\frac{15 \left (-4 i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )+4 i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )-i \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )+i \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )+24 \tan ^{-1}(a x)\right )}{\sqrt{\tan ^{-1}(a x)}}+\frac{64 \sqrt{\tan ^{-1}(a x)} \left (-15 \left (a^4 x^4-6 a^2 x^2+1\right )+64 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2+160 a x \left (a^2 x^2-1\right ) \tan ^{-1}(a x)\right )}{\left (a^2 x^2+1\right )^2}+30 \left (\sqrt{2 \pi } \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )-8 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )+12 \sqrt{\tan ^{-1}(a x)}\right )}{81920 a^3 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((64*Sqrt[ArcTan[a*x]]*(-15*(1 - 6*a^2*x^2 + a^4*x^4) + 160*a*x*(-1 + a^2*x^2)*ArcTan[a*x] + 64*(1 + a^2*x^2)^
2*ArcTan[a*x]^2))/(1 + a^2*x^2)^2 + 30*(12*Sqrt[ArcTan[a*x]] + Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*
x]]] - 8*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]]) - 90*Sqrt[ArcTan[a*x]]*(8 + Gamma[1/2, (-4*I)*ArcT
an[a*x]]/Sqrt[(-I)*ArcTan[a*x]] + Gamma[1/2, (4*I)*ArcTan[a*x]]/Sqrt[I*ArcTan[a*x]]) + (15*(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^3*c^3)

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Maple [A]  time = 0.111, size = 81, normalized size = 0.8 \begin{align*}{\frac{1}{5120\,{c}^{3}{a}^{3}} \left ( 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 ) +256\, \left ( \arctan \left ( ax \right ) \right ) ^{3}-160\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\sin \left ( 4\,\arctan \left ( ax \right ) \right ) -60\,\cos \left ( 4\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) \right ){\frac{1}{\sqrt{\arctan \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5120/a^3/c^3*(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))+256*arcta
n(a*x)^3-160*arctan(a*x)^2*sin(4*arctan(a*x))-60*cos(4*arctan(a*x))*arctan(a*x))/arctan(a*x)^(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^2*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^2*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^{2} \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**2*atan(a*x)**(3/2)/(a**2*c*x**2+c)**3,x)

[Out]

Integral(x**2*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^{2} \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^2*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

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