3.669 \(\int \frac {x^2}{(c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=209 \[ \frac {x}{2 a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}-\frac {3 x}{2 a^2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}-\frac {\sqrt {a^2 x^2+1} \text {Ci}\left (\tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {9 \sqrt {a^2 x^2+1} \text {Ci}\left (3 \tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{2 a^3 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}+\frac {1}{2 a^3 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2} \]

[Out]

1/2/a^3/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^2-3/2*x/a^2/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)-1/2/a^3/c^2/arctan(a*x
)^2/(a^2*c*x^2+c)^(1/2)+1/2*x/a^2/c^2/arctan(a*x)/(a^2*c*x^2+c)^(1/2)-1/8*Ci(arctan(a*x))*(a^2*x^2+1)^(1/2)/a^
3/c^2/(a^2*c*x^2+c)^(1/2)+9/8*Ci(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/a^3/c^2/(a^2*c*x^2+c)^(1/2)

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Rubi [A]  time = 0.91, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {4964, 4902, 4942, 4905, 4904, 3302, 4968, 4971, 4970, 4406, 3312} \[ -\frac {\sqrt {a^2 x^2+1} \text {CosIntegral}\left (\tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {9 \sqrt {a^2 x^2+1} \text {CosIntegral}\left (3 \tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{2 a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}-\frac {1}{2 a^3 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}+\frac {1}{2 a^3 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]

[Out]

1/(2*a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) - 1/(2*a^3*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - (3*x)/(2*a
^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + x/(2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*C
osIntegral[ArcTan[a*x]])/(8*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (9*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(8
*a^3*c^2*Sqrt[c + a^2*c*x^2])

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 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 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 4902

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((d + e*x^2)^(q + 1)
*(a + b*ArcTan[c*x])^(p + 1))/(b*c*d*(p + 1)), x] - Dist[(2*c*(q + 1))/(b*(p + 1)), Int[x*(d + e*x^2)^q*(a + b
*ArcTan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && LtQ[p, -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 4905

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[(d^(q + 1/2)*Sqrt[1
+ c^2*x^2])/Sqrt[d + e*x^2], Int[(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, 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 4942

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

Rule 4964

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

Rule 4968

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

Rule 4971

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[(d^(q +
1/2)*Sqrt[1 + c^2*x^2])/Sqrt[d + e*x^2], Int[x^m*(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, 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}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx &=-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx}{a^2}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx}{a^2 c}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}+\frac {3 \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx}{2 a}-\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{2 a c}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-3 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx+\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{2 a^2}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{2 a^2 c}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {x^2}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{2 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{2 a^2 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos ^3(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{2 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {3 \cos (x)}{4 x}+\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\cos (x)}{4 x}-\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{2 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (9 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {9 \sqrt {1+a^2 x^2} \text {Ci}\left (3 \tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 119, normalized size = 0.57 \[ \frac {-\left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)^2 \text {Ci}\left (\tan ^{-1}(a x)\right )+9 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)^2 \text {Ci}\left (3 \tan ^{-1}(a x)\right )+4 a x \left (\left (a^2 x^2-2\right ) \tan ^{-1}(a x)-a x\right )}{8 a^3 c^2 \left (a^2 x^2+1\right ) \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]

[Out]

(4*a*x*(-(a*x) + (-2 + a^2*x^2)*ArcTan[a*x]) - (1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*CosIntegral[ArcTan[a*x]] + 9*
(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*CosIntegral[3*ArcTan[a*x]])/(8*a^3*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*Arc
Tan[a*x]^2)

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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x^{2}}{{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^2/((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*arctan(a*x)^3), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 3.33, size = 844, normalized size = 4.04 \[ -\frac {\left (9 \Ei \left (1, 3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2} x^{4} a^{4}-3 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x^{3} a^{3}+18 \Ei \left (1, 3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2} x^{2} a^{2}-i \sqrt {a^{2} x^{2}+1}\, x^{3} a^{3}+3 \sqrt {a^{2} x^{2}+1}\, x^{2} a^{2}-9 i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x^{2} a^{2}+9 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x a +9 \Ei \left (1, 3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+3 i \sqrt {a^{2} x^{2}+1}\, x a -\sqrt {a^{2} x^{2}+1}+3 i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \sqrt {a^{2} x^{2}+1}\, \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{2} c^{3} a^{3}}-\frac {\left (9 \Ei \left (1, -3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2} x^{4} a^{4}-3 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x^{3} a^{3}+18 \Ei \left (1, -3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2} x^{2} a^{2}+i \sqrt {a^{2} x^{2}+1}\, x^{3} a^{3}+3 \sqrt {a^{2} x^{2}+1}\, x^{2} a^{2}+9 i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x^{2} a^{2}+9 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x a -3 i \sqrt {a^{2} x^{2}+1}\, x a +9 \Ei \left (1, -3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-3 i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )-\sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \sqrt {a^{2} x^{2}+1}\, \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{2} c^{3} a^{3}}+\frac {\left (\Ei \left (1, i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2} x^{2} a^{2}+\sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x a +i \sqrt {a^{2} x^{2}+1}\, x a +\Ei \left (1, i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )-\sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctan \left (a x \right )^{2} c^{3} a^{3}}+\frac {\left (\Ei \left (1, -i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2} x^{2} a^{2}+\sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x a +\Ei \left (1, -i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-i \sqrt {a^{2} x^{2}+1}\, x a -\sqrt {a^{2} x^{2}+1}-i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctan \left (a x \right )^{2} c^{3} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)

[Out]

-1/16*(9*Ei(1,3*I*arctan(a*x))*arctan(a*x)^2*x^4*a^4-3*(a^2*x^2+1)^(1/2)*arctan(a*x)*x^3*a^3+18*Ei(1,3*I*arcta
n(a*x))*arctan(a*x)^2*x^2*a^2-I*(a^2*x^2+1)^(1/2)*x^3*a^3+3*(a^2*x^2+1)^(1/2)*x^2*a^2-9*I*(a^2*x^2+1)^(1/2)*ar
ctan(a*x)*x^2*a^2+9*(a^2*x^2+1)^(1/2)*arctan(a*x)*x*a+9*Ei(1,3*I*arctan(a*x))*arctan(a*x)^2+3*I*(a^2*x^2+1)^(1
/2)*x*a-(a^2*x^2+1)^(1/2)+3*I*(a^2*x^2+1)^(1/2)*arctan(a*x))/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)/(a^4*
x^4+2*a^2*x^2+1)/arctan(a*x)^2/c^3/a^3-1/16*(9*Ei(1,-3*I*arctan(a*x))*arctan(a*x)^2*x^4*a^4-3*(a^2*x^2+1)^(1/2
)*arctan(a*x)*x^3*a^3+18*Ei(1,-3*I*arctan(a*x))*arctan(a*x)^2*x^2*a^2+I*(a^2*x^2+1)^(1/2)*x^3*a^3+3*(a^2*x^2+1
)^(1/2)*x^2*a^2+9*I*(a^2*x^2+1)^(1/2)*arctan(a*x)*x^2*a^2+9*(a^2*x^2+1)^(1/2)*arctan(a*x)*x*a-3*I*(a^2*x^2+1)^
(1/2)*x*a+9*Ei(1,-3*I*arctan(a*x))*arctan(a*x)^2-3*I*(a^2*x^2+1)^(1/2)*arctan(a*x)-(a^2*x^2+1)^(1/2))/(a^2*x^2
+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)/(a^4*x^4+2*a^2*x^2+1)/arctan(a*x)^2/c^3/a^3+1/16*(Ei(1,I*arctan(a*x))*arct
an(a*x)^2*x^2*a^2+(a^2*x^2+1)^(1/2)*arctan(a*x)*x*a+I*(a^2*x^2+1)^(1/2)*x*a+Ei(1,I*arctan(a*x))*arctan(a*x)^2+
I*(a^2*x^2+1)^(1/2)*arctan(a*x)-(a^2*x^2+1)^(1/2))/(a^2*x^2+1)^(3/2)*(c*(a*x-I)*(I+a*x))^(1/2)/arctan(a*x)^2/c
^3/a^3+1/16*(Ei(1,-I*arctan(a*x))*arctan(a*x)^2*x^2*a^2+(a^2*x^2+1)^(1/2)*arctan(a*x)*x*a+Ei(1,-I*arctan(a*x))
*arctan(a*x)^2-I*(a^2*x^2+1)^(1/2)*x*a-(a^2*x^2+1)^(1/2)-I*(a^2*x^2+1)^(1/2)*arctan(a*x))/(a^2*x^2+1)^(3/2)*(c
*(a*x-I)*(I+a*x))^(1/2)/arctan(a*x)^2/c^3/a^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="maxima")

[Out]

integrate(x^2/((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(atan(a*x)^3*(c + a^2*c*x^2)^(5/2)),x)

[Out]

int(x^2/(atan(a*x)^3*(c + a^2*c*x^2)^(5/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a**2*c*x**2+c)**(5/2)/atan(a*x)**3,x)

[Out]

Integral(x**2/((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3), x)

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