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

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

[Out]

1/20*arctan(a*x)^(5/2)/a^3/c^3-1/32*arctan(a*x)^(3/2)*sin(4*arctan(a*x))/a^3/c^3+3/1024*FresnelC(2*2^(1/2)/Pi^
(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3/c^3-3/256*cos(4*arctan(a*x))*arctan(a*x)^(1/2)/a^3/c^3

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Rubi [A]  time = 0.16, antiderivative size = 108, normalized size of antiderivative = 1.00, 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 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]

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 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^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*}

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Mathematica [C]  time = 0.82, size = 353, normalized size = 3.27 \[ \frac {\frac {64 \sqrt {\tan ^{-1}(a x)} \left (64 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2+160 a x \left (a^2 x^2-1\right ) \tan ^{-1}(a x)-15 \left (a^4 x^4-6 a^2 x^2+1\right )\right )}{\left (a^2 x^2+1\right )^2}+30 \left (\sqrt {2 \pi } C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )-8 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )+12 \sqrt {\tan ^{-1}(a x)}\right )-90 \sqrt {\tan ^{-1}(a x)} \left (\frac {\Gamma \left (\frac {1}{2},-4 i \tan ^{-1}(a x)\right )}{\sqrt {-i \tan ^{-1}(a x)}}+\frac {\Gamma \left (\frac {1}{2},4 i \tan ^{-1}(a x)\right )}{\sqrt {i \tan ^{-1}(a x)}}+8\right )+\frac {15 \left (24 \tan ^{-1}(a x)-4 i \sqrt {2} \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \tan ^{-1}(a x)\right )+4 i \sqrt {2} \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \tan ^{-1}(a x)\right )-i \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \tan ^{-1}(a x)\right )+i \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \tan ^{-1}(a x)\right )\right )}{\sqrt {\tan ^{-1}(a x)}}}{81920 a^3 c^3} \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [A]  time = 0.50, size = 81, normalized size = 0.75 \[ \frac {15 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+256 \arctan \left (a x \right )^{3}-160 \arctan \left (a x \right )^{2} \sin \left (4 \arctan \left (a x \right )\right )-60 \cos \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{5120 a^{3} c^{3} \sqrt {\arctan \left (a x \right )}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]

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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