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

Optimal. Leaf size=139 \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{64 a c^3}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a c^3}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a c^3} \]

[Out]

ArcTan[a*x]^(3/2)/(4*a*c^3) - (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a*c^3) - (Sqrt[Pi]*Fre
snelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a*c^3) + (Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(4*a*c^3) + (Sqrt[Ar
cTan[a*x]]*Sin[4*ArcTan[a*x]])/(32*a*c^3)

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Rubi [A]  time = 0.144651, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4904, 3312, 3296, 3305, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{64 a c^3}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a c^3}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[a*x]^(3/2)/(4*a*c^3) - (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a*c^3) - (Sqrt[Pi]*Fre
snelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a*c^3) + (Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(4*a*c^3) + (Sqrt[Ar
cTan[a*x]]*Sin[4*ArcTan[a*x]])/(32*a*c^3)

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 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 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(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 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[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{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{x} \cos ^4(x) \, dx,x,\tan ^{-1}(a x)\right )}{a c^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{3 \sqrt{x}}{8}+\frac{1}{2} \sqrt{x} \cos (2 x)+\frac{1}{8} \sqrt{x} \cos (4 x)\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a c^3}\\ &=\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}+\frac{\operatorname{Subst}\left (\int \sqrt{x} \cos (4 x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a c^3}+\frac{\operatorname{Subst}\left (\int \sqrt{x} \cos (2 x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a c^3}\\ &=\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a c^3}-\frac{\operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a c^3}\\ &=\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a c^3}-\frac{\operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{32 a c^3}-\frac{\operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{4 a c^3}\\ &=\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}-\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{64 a c^3}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a c^3}\\ \end{align*}

Mathematica [C]  time = 0.419533, size = 192, normalized size = 1.38 \[ \frac{\frac{8 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )}{a}+\frac{8 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )}{a}+\frac{\sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )}{a}+\frac{\sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )}{a}+\frac{96 a^2 x^3 \tan ^{-1}(a x)}{\left (a^2 x^2+1\right )^2}+\frac{160 x \tan ^{-1}(a x)}{\left (a^2 x^2+1\right )^2}+\frac{64 \tan ^{-1}(a x)^2}{a}}{256 c^3 \sqrt{\tan ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((160*x*ArcTan[a*x])/(1 + a^2*x^2)^2 + (96*a^2*x^3*ArcTan[a*x])/(1 + a^2*x^2)^2 + (64*ArcTan[a*x]^2)/a + (8*Sq
rt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]])/a + (8*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*
I)*ArcTan[a*x]])/a + (Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]])/a + (Sqrt[I*ArcTan[a*x]]*Gamma[1/
2, (4*I)*ArcTan[a*x]])/a)/(256*c^3*Sqrt[ArcTan[a*x]])

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Maple [A]  time = 0.118, size = 102, normalized size = 0.7 \begin{align*}{\frac{1}{128\,a{c}^{3}} \left ( -\sqrt{2}\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +32\, \left ( \arctan \left ( ax \right ) \right ) ^{2}+32\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +4\,\sin \left ( 4\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -16\,\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \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(arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x)

[Out]

1/128/a/c^3/arctan(a*x)^(1/2)*(-2^(1/2)*arctan(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/
2))+32*arctan(a*x)^2+32*sin(2*arctan(a*x))*arctan(a*x)+4*sin(4*arctan(a*x))*arctan(a*x)-16*arctan(a*x)^(1/2)*P
i^(1/2)*FresnelS(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(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(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{\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(atan(a*x)**(1/2)/(a**2*c*x**2+c)**3,x)

[Out]

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

[Out]

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