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

Optimal. Leaf size=89 \[ \frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{8 a c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{4 a c^3} \]

[Out]

(3*Sqrt[ArcTan[a*x]])/(4*a*c^3) + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a*c^3) + (Sqrt[Pi]*
FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*a*c^3)

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Rubi [A]  time = 0.0955638, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4904, 3312, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{8 a c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{4 a c^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]

[Out]

(3*Sqrt[ArcTan[a*x]])/(4*a*c^3) + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a*c^3) + (Sqrt[Pi]*
FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*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 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{1}{\left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos ^4(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^3}\\ &=\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 )}{a c^3}\\ &=\frac{3 \sqrt{\tan ^{-1}(a x)}}{4 a c^3}+\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a c^3}+\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{2 a c^3}\\ &=\frac{3 \sqrt{\tan ^{-1}(a x)}}{4 a c^3}+\frac{\operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{4 a c^3}+\frac{\operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a c^3}\\ &=\frac{3 \sqrt{\tan ^{-1}(a x)}}{4 a c^3}+\frac{\sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{8 a c^3}+\frac{\sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a c^3}\\ \end{align*}

Mathematica [C]  time = 0.266006, size = 147, normalized size = 1.65 \[ \frac{-4 i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )+4 i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )-i \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )+i \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )+24 \tan ^{-1}(a x)}{32 a c^3 \sqrt{\tan ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]

[Out]

(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*A
rcTan[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]])/(32*a*c^3*Sqrt[ArcTan[a*x]])

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Maple [A]  time = 0.115, size = 68, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{16\,a{c}^{3}}{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) }+{\frac{\sqrt{\pi }}{2\,a{c}^{3}}{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) }+{\frac{3}{4\,a{c}^{3}}\sqrt{\arctan \left ( ax \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)

[Out]

1/16*FresnelC(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a/c^3+1/2*FresnelC(2*arctan(a*x)^(1/2)/Pi
^(1/2))*Pi^(1/2)/a/c^3+3/4*arctan(a*x)^(1/2)/a/c^3

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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(1/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),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(1/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),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{1}{a^{6} x^{6} \sqrt{\operatorname{atan}{\left (a x \right )}} + 3 a^{4} x^{4} \sqrt{\operatorname{atan}{\left (a x \right )}} + 3 a^{2} x^{2} \sqrt{\operatorname{atan}{\left (a x \right )}} + \sqrt{\operatorname{atan}{\left (a x \right )}}}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a**2*c*x**2+c)**3/atan(a*x)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{3} \sqrt{\arctan \left (a x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a^2*c*x^2 + c)^3*sqrt(arctan(a*x))), x)