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

Optimal. Leaf size=80 \[ \frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^3 c^2}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a^3 c^2}-\frac {x \sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (a^2 x^2+1\right )} \]

[Out]

1/3*arctan(a*x)^(3/2)/a^3/c^2+1/8*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^3/c^2-1/2*x*arctan(a*x)^(1
/2)/a^2/c^2/(a^2*x^2+1)

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Rubi [A]  time = 0.15, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4936, 4970, 4406, 12, 3305, 3351} \[ \frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^3 c^2}-\frac {x \sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a^3 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*Sqrt[ArcTan[a*x]])/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(3/2)/(3*a^3*c^2) + (Sqrt[Pi]*FresnelS[(2*Sqrt[
ArcTan[a*x]])/Sqrt[Pi]])/(8*a^3*c^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

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 4936

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^2)/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(a + b*ArcTan
[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (Dist[(b*p)/(2*c), Int[(x*(a + b*ArcTan[c*x])^(p - 1))/(d + e*x^2)^
2, x], x] - Simp[(x*(a + b*ArcTan[c*x])^p)/(2*c^2*d*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c
^2*d] && GtQ[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 \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac {x \sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a^3 c^2}+\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{4 a}\\ &=-\frac {x \sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a^3 c^2}+\frac {\operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 c^2}\\ &=-\frac {x \sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a^3 c^2}+\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 c^2}\\ &=-\frac {x \sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a^3 c^2}+\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 c^2}\\ &=-\frac {x \sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a^3 c^2}+\frac {\operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{4 a^3 c^2}\\ &=-\frac {x \sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a^3 c^2}+\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^3 c^2}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 66, normalized size = 0.82 \[ \frac {4 \sqrt {\tan ^{-1}(a x)} \left (2 \tan ^{-1}(a x)-\frac {3 a x}{a^2 x^2+1}\right )+3 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{24 a^3 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[ArcTan[a*x]]*((-3*a*x)/(1 + a^2*x^2) + 2*ArcTan[a*x]) + 3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt
[Pi]])/(24*a^3*c^2)

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

[Out]

sage0*x

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maple [A]  time = 0.51, size = 60, normalized size = 0.75 \[ \frac {3 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+8 \arctan \left (a x \right )^{2}-6 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{24 a^{3} c^{2} \sqrt {\arctan \left (a x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24/a^3/c^2*(3*arctan(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))+8*arctan(a*x)^2-6*sin(2*arct
an(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)^(1/2)/(a^2*c*x^2+c)^2,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 \[ \int \frac {x^2\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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