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

Optimal. Leaf size=118 \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{64 a^4 c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{16 a^4 c^3}+\frac{x^4 \sqrt{\tan ^{-1}(a x)}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a^4 c^3} \]

[Out]

(-3*Sqrt[ArcTan[a*x]])/(32*a^4*c^3) + (x^4*Sqrt[ArcTan[a*x]])/(4*c^3*(1 + a^2*x^2)^2) - (Sqrt[Pi/2]*FresnelC[2
*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a^4*c^3) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(16*a^4*c^3
)

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Rubi [A]  time = 0.214544, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4944, 4970, 3312, 3304, 3352} \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{64 a^4 c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{16 a^4 c^3}+\frac{x^4 \sqrt{\tan ^{-1}(a x)}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a^4 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[ArcTan[a*x]])/(32*a^4*c^3) + (x^4*Sqrt[ArcTan[a*x]])/(4*c^3*(1 + a^2*x^2)^2) - (Sqrt[Pi/2]*FresnelC[2
*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a^4*c^3) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(16*a^4*c^3
)

Rule 4944

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(d*f*(m + 1)), x] - Dist[(b*c*p)/(f*(m + 1)), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

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 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]

Rubi steps

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

Mathematica [C]  time = 0.649943, size = 230, normalized size = 1.95 \[ \frac{\frac{-12 i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )+12 i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )+3 i \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )-3 i \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )+\frac{64 \left (5 a^4 x^4-6 a^2 x^2-3\right ) \tan ^{-1}(a x)}{\left (a^2 x^2+1\right )^2}}{\sqrt{\tan ^{-1}(a x)}}-10 \sqrt{2 \pi } \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )+80 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2048 a^4 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-10*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + 80*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]
] + ((64*(-3 - 6*a^2*x^2 + 5*a^4*x^4)*ArcTan[a*x])/(1 + a^2*x^2)^2 - (12*I)*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gam
ma[1/2, (-2*I)*ArcTan[a*x]] + (12*I)*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + (3*I)*Sqrt[(-
I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - (3*I)*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/Sqrt
[ArcTan[a*x]])/(2048*a^4*c^3)

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Maple [A]  time = 0.114, size = 93, normalized size = 0.8 \begin{align*} -{\frac{1}{128\,{c}^{3}{a}^{4}} \left ( \sqrt{2}\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +16\,\cos \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -4\,\cos \left ( 4\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -8\,\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^3*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x)

[Out]

-1/128/a^4/c^3/arctan(a*x)^(1/2)*(2^(1/2)*arctan(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(
1/2))+16*cos(2*arctan(a*x))*arctan(a*x)-4*cos(4*arctan(a*x))*arctan(a*x)-8*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^3*arctan(a*x)^(1/2)/(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^3*arctan(a*x)^(1/2)/(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^{3} \sqrt{\operatorname{atan}{\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**3*atan(a*x)**(1/2)/(a**2*c*x**2+c)**3,x)

[Out]

Integral(x**3*sqrt(atan(a*x))/(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^{3} \sqrt{\arctan \left (a x\right )}}{{\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^3*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

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