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

Optimal. Leaf size=219 \[ \frac{3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (a^2 x^2+1\right )}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac{3 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a c^3}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a c^3} \]

[Out]

(-45*Sqrt[ArcTan[a*x]])/(256*a*c^3) + (3*Sqrt[ArcTan[a*x]])/(32*a*c^3*(1 + a^2*x^2)^2) + (9*Sqrt[ArcTan[a*x]])
/(32*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(3/2))/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x]^(3/2))/(8*c^3*(1
+ a^2*x^2)) + (3*ArcTan[a*x]^(5/2))/(20*a*c^3) - (3*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*
a*c^3) - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a*c^3)

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Rubi [A]  time = 0.290959, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4900, 4892, 4930, 4904, 3312, 3304, 3352} \[ \frac{3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (a^2 x^2+1\right )}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac{3 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a c^3}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*Sqrt[ArcTan[a*x]])/(256*a*c^3) + (3*Sqrt[ArcTan[a*x]])/(32*a*c^3*(1 + a^2*x^2)^2) + (9*Sqrt[ArcTan[a*x]])
/(32*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(3/2))/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x]^(3/2))/(8*c^3*(1
+ a^2*x^2)) + (3*ArcTan[a*x]^(5/2))/(20*a*c^3) - (3*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*
a*c^3) - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a*c^3)

Rule 4900

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b*p*(d + e*x^2)^(q
+ 1)*(a + b*ArcTan[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q +
1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(
p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e
}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

Rule 4892

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

Rule 4930

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

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{\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{3}{64} \int \frac{1}{\left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{3 \int \frac{\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos ^4(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}-\frac{(9 a) \int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}\\ &=\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac{3 \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 )}{64 a c^3}-\frac{9 \int \frac{1}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{64 c}\\ &=-\frac{9 \sqrt{\tan ^{-1}(a x)}}{256 a c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{128 a c^3}-\frac{9 \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}\\ &=-\frac{9 \sqrt{\tan ^{-1}(a x)}}{256 a c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac{3 \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{256 a c^3}-\frac{3 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{64 a c^3}-\frac{9 \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}\\ &=-\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac{3 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac{3 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a c^3}-\frac{9 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{128 a c^3}\\ &=-\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac{3 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac{3 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a c^3}-\frac{9 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{64 a c^3}\\ &=-\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac{3 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac{3 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a c^3}\\ \end{align*}

Mathematica [A]  time = 0.571426, size = 123, normalized size = 0.56 \[ \frac{-15 \sqrt{2 \pi } \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )-480 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )+4 \sqrt{\tan ^{-1}(a x)} \left (8 \tan ^{-1}(a x) \left (24 \tan ^{-1}(a x)+40 \sin \left (2 \tan ^{-1}(a x)\right )+5 \sin \left (4 \tan ^{-1}(a x)\right )\right )+240 \cos \left (2 \tan ^{-1}(a x)\right )+15 \cos \left (4 \tan ^{-1}(a x)\right )\right )}{5120 a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - 480*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi
]] + 4*Sqrt[ArcTan[a*x]]*(240*Cos[2*ArcTan[a*x]] + 15*Cos[4*ArcTan[a*x]] + 8*ArcTan[a*x]*(24*ArcTan[a*x] + 40*
Sin[2*ArcTan[a*x]] + 5*Sin[4*ArcTan[a*x]])))/(5120*a*c^3)

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Maple [A]  time = 0.118, size = 132, normalized size = 0.6 \begin{align*}{\frac{1}{5120\,a{c}^{3}} \left ( 768\, \left ( \arctan \left ( ax \right ) \right ) ^{3}+1280\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\sin \left ( 2\,\arctan \left ( ax \right ) \right ) +160\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\sin \left ( 4\,\arctan \left ( ax \right ) \right ) -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 ) +60\,\cos \left ( 4\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +960\,\cos \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -480\,\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(arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x)

[Out]

1/5120/a/c^3/arctan(a*x)^(1/2)*(768*arctan(a*x)^3+1280*arctan(a*x)^2*sin(2*arctan(a*x))+160*arctan(a*x)^2*sin(
4*arctan(a*x))-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))+60*cos(4*a
rctan(a*x))*arctan(a*x)+960*cos(2*arctan(a*x))*arctan(a*x)-480*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(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(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{\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(atan(a*x)**(3/2)/(a**2*c*x**2+c)**3,x)

[Out]

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

[Out]

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