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

Optimal. Leaf size=798 \[ \frac {1}{9} a^4 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^8-\frac {1}{24} a^3 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x^7+\frac {19}{63} a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^6+\frac {1}{84} a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) x^6-\frac {103 a c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x^5}{1008}-\frac {1}{504} a c^2 \sqrt {a^2 c x^2+c} x^5+\frac {5}{21} c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^4+\frac {67 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) x^4}{2520}-\frac {205 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x^3}{4032 a}-\frac {c^2 \sqrt {a^2 c x^2+c} x^3}{240 a}+\frac {c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^2}{63 a^2}-\frac {47 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) x^2}{30240 a^2}+\frac {47 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x}{896 a^3}+\frac {85 c^2 \sqrt {a^2 c x^2+c} x}{12096 a^3}-\frac {2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3}{63 a^4}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{1344 a^4 \sqrt {a^2 c x^2+c}}-\frac {6157 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{60480 a^4}+\frac {1433 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{15120 a^4}+\frac {115 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}}+\frac {115 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}} \]

[Out]

1433/15120*c^(5/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^4+115/1344*I*c^3*arctan(a*x)*polylog(2,-I*(1+I*a
*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-115/1344*I*c^3*arctan(a*x)*polylog(2,I*(1+I*a
*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-115/1344*I*c^3*arctan((1+I*a*x)/(a^2*x^2+1)^(
1/2))*arctan(a*x)^2*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-115/1344*c^3*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^
(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)+115/1344*c^3*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^
2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)+85/12096*c^2*x*(a^2*c*x^2+c)^(1/2)/a^3-1/240*c^2*x^3*(a^2*c*x^2+c)^(1/2)/a-
1/504*a*c^2*x^5*(a^2*c*x^2+c)^(1/2)-6157/60480*c^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^4-47/30240*c^2*x^2*arctan
(a*x)*(a^2*c*x^2+c)^(1/2)/a^2+67/2520*c^2*x^4*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+1/84*a^2*c^2*x^6*arctan(a*x)*(a^
2*c*x^2+c)^(1/2)+47/896*c^2*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^3-205/4032*c^2*x^3*arctan(a*x)^2*(a^2*c*x^2+
c)^(1/2)/a-103/1008*a*c^2*x^5*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)-1/24*a^3*c^2*x^7*arctan(a*x)^2*(a^2*c*x^2+c)^(
1/2)-2/63*c^2*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^4+1/63*c^2*x^2*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^2+5/21*c^
2*x^4*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)+19/63*a^2*c^2*x^6*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)+1/9*a^4*c^2*x^8*ar
ctan(a*x)^3*(a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 19.66, antiderivative size = 798, normalized size of antiderivative = 1.00, number of steps used = 547, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321} \[ \frac {1}{9} a^4 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^8-\frac {1}{24} a^3 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x^7+\frac {19}{63} a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^6+\frac {1}{84} a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) x^6-\frac {103 a c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x^5}{1008}-\frac {1}{504} a c^2 \sqrt {a^2 c x^2+c} x^5+\frac {5}{21} c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^4+\frac {67 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) x^4}{2520}-\frac {205 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x^3}{4032 a}-\frac {c^2 \sqrt {a^2 c x^2+c} x^3}{240 a}+\frac {c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^2}{63 a^2}-\frac {47 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) x^2}{30240 a^2}+\frac {47 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x}{896 a^3}+\frac {85 c^2 \sqrt {a^2 c x^2+c} x}{12096 a^3}-\frac {2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3}{63 a^4}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{1344 a^4 \sqrt {a^2 c x^2+c}}-\frac {6157 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{60480 a^4}+\frac {1433 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{15120 a^4}+\frac {115 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}}+\frac {115 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(85*c^2*x*Sqrt[c + a^2*c*x^2])/(12096*a^3) - (c^2*x^3*Sqrt[c + a^2*c*x^2])/(240*a) - (a*c^2*x^5*Sqrt[c + a^2*c
*x^2])/504 - (6157*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(60480*a^4) - (47*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a
*x])/(30240*a^2) + (67*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2520 + (a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan
[a*x])/84 + (47*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(896*a^3) - (205*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a
*x]^2)/(4032*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/1008 - (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2]*Ar
cTan[a*x]^2)/24 - (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^4*Sqrt[c +
a^2*c*x^2]) - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(63*a^4) + (c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3
)/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/21 + (19*a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x
]^3)/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/9 + (1433*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2
*c*x^2]])/(15120*a^4) + (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/
(a^4*Sqrt[c + a^2*c*x^2]) - (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])
/(a^4*Sqrt[c + a^2*c*x^2]) - (115*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c +
a^2*c*x^2]) + (115*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c + a^2*c*x^2])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4888

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

Rule 4890

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*
d] && IGtQ[p, 0] &&  !GtQ[d, 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 4950

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

Rule 4952

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

Rule 6589

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

Rubi steps

\begin {align*} \int x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3 \, dx &=c \int x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx+\left (a^2 c\right ) \int x^5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 7.27, size = 850, normalized size = 1.07 \[ \frac {c^2 \sqrt {a^2 c x^2+c} \left (-\left (\left (1536 \left (711 \cos \left (2 \tan ^{-1}(a x)\right )-126 \cos \left (4 \tan ^{-1}(a x)\right )+105 \cos \left (6 \tan ^{-1}(a x)\right )-178\right ) \tan ^{-1}(a x)^3+3 \left (13074 \sin \left (2 \tan ^{-1}(a x)\right )-26742 \sin \left (4 \tan ^{-1}(a x)\right )+6362 \sin \left (6 \tan ^{-1}(a x)\right )-5469 \sin \left (8 \tan ^{-1}(a x)\right )\right ) \tan ^{-1}(a x)^2+\frac {8 \left (153529 \cos \left (2 \tan ^{-1}(a x)\right )+59266 \cos \left (4 \tan ^{-1}(a x)\right )+16407 \cos \left (6 \tan ^{-1}(a x)\right )+87630\right ) \tan ^{-1}(a x)}{a^2 x^2+1}+74932 \sin \left (2 \tan ^{-1}(a x)\right )+77908 \sin \left (4 \tan ^{-1}(a x)\right )+36612 \sin \left (6 \tan ^{-1}(a x)\right )+7238 \sin \left (8 \tan ^{-1}(a x)\right )\right ) \left (a^2 x^2+1\right )^{9/2}\right )-16128 \left (32 \left (5 \cos \left (2 \tan ^{-1}(a x)\right )-1\right ) \tan ^{-1}(a x)^3+\left (6 \sin \left (2 \tan ^{-1}(a x)\right )-33 \sin \left (4 \tan ^{-1}(a x)\right )\right ) \tan ^{-1}(a x)^2+6 \left (36 \cos \left (2 \tan ^{-1}(a x)\right )+11 \cos \left (4 \tan ^{-1}(a x)\right )+25\right ) \tan ^{-1}(a x)+\frac {48 a x}{\left (a^2 x^2+1\right )^2}\right ) \left (a^2 x^2+1\right )^{5/2}+774144 \left (-11 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2+11 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-11 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+10 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )-11 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )+11 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )\right )+256 \left (-16407 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2+16407 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-16407 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+12788 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )-16407 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )+16407 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )\right )+576 \left (\left (64 \left (-28 \cos \left (2 \tan ^{-1}(a x)\right )+35 \cos \left (4 \tan ^{-1}(a x)\right )+57\right ) \tan ^{-1}(a x)^3-3 \left (211 \sin \left (2 \tan ^{-1}(a x)\right )-60 \sin \left (4 \tan ^{-1}(a x)\right )+103 \sin \left (6 \tan ^{-1}(a x)\right )\right ) \tan ^{-1}(a x)^2+\frac {8 \left (764 \cos \left (2 \tan ^{-1}(a x)\right )+309 \cos \left (4 \tan ^{-1}(a x)\right )+647\right ) \tan ^{-1}(a x)}{a^2 x^2+1}+4 \left (101 \sin \left (2 \tan ^{-1}(a x)\right )+88 \sin \left (4 \tan ^{-1}(a x)\right )+25 \sin \left (6 \tan ^{-1}(a x)\right )\right )\right ) \left (a^2 x^2+1\right )^{7/2}+64 \left (309 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-309 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+309 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-259 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )+309 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-309 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )\right )\right )\right )}{15482880 a^4 \sqrt {a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*(774144*((-11*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*ArcTanh[(a*x)/Sqrt[1 +
a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTa
n[a*x])] - 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 11*PolyLog[3, I*E^(I*ArcTan[a*x])]) + 256*((-16407*I)*ArcTa
n[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 12788*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (16407*I)*ArcTan[a*x]*PolyLog[2,
(-I)*E^(I*ArcTan[a*x])] - (16407*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 16407*PolyLog[3, (-I)*E^(I*
ArcTan[a*x])] + 16407*PolyLog[3, I*E^(I*ArcTan[a*x])]) - 16128*(1 + a^2*x^2)^(5/2)*((48*a*x)/(1 + a^2*x^2)^2 +
32*ArcTan[a*x]^3*(-1 + 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcTan[a*x]] + 11*Cos[4*ArcTan[a*
x]]) + ArcTan[a*x]^2*(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*ArcTan[a*x]])) + 576*(64*((309*I)*ArcTan[E^(I*ArcTan[a*x
])]*ArcTan[a*x]^2 - 259*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - (309*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x
])] + (309*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 309*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 309*PolyL
og[3, I*E^(I*ArcTan[a*x])]) + (1 + a^2*x^2)^(7/2)*(64*ArcTan[a*x]^3*(57 - 28*Cos[2*ArcTan[a*x]] + 35*Cos[4*Arc
Tan[a*x]]) + (8*ArcTan[a*x]*(647 + 764*Cos[2*ArcTan[a*x]] + 309*Cos[4*ArcTan[a*x]]))/(1 + a^2*x^2) + 4*(101*Si
n[2*ArcTan[a*x]] + 88*Sin[4*ArcTan[a*x]] + 25*Sin[6*ArcTan[a*x]]) - 3*ArcTan[a*x]^2*(211*Sin[2*ArcTan[a*x]] -
60*Sin[4*ArcTan[a*x]] + 103*Sin[6*ArcTan[a*x]]))) - (1 + a^2*x^2)^(9/2)*(1536*ArcTan[a*x]^3*(-178 + 711*Cos[2*
ArcTan[a*x]] - 126*Cos[4*ArcTan[a*x]] + 105*Cos[6*ArcTan[a*x]]) + (8*ArcTan[a*x]*(87630 + 153529*Cos[2*ArcTan[
a*x]] + 59266*Cos[4*ArcTan[a*x]] + 16407*Cos[6*ArcTan[a*x]]))/(1 + a^2*x^2) + 74932*Sin[2*ArcTan[a*x]] + 77908
*Sin[4*ArcTan[a*x]] + 36612*Sin[6*ArcTan[a*x]] + 3*ArcTan[a*x]^2*(13074*Sin[2*ArcTan[a*x]] - 26742*Sin[4*ArcTa
n[a*x]] + 6362*Sin[6*ArcTan[a*x]] - 5469*Sin[8*ArcTan[a*x]]) + 7238*Sin[8*ArcTan[a*x]])))/(15482880*a^4*Sqrt[1
+ a^2*x^2])

________________________________________________________________________________________

fricas [F]  time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{7} + 2 \, a^{2} c^{2} x^{5} + c^{2} x^{3}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^4*c^2*x^7 + 2*a^2*c^2*x^5 + c^2*x^3)*sqrt(a^2*c*x^2 + c)*arctan(a*x)^3, x)

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 2.85, size = 525, normalized size = 0.66 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (13440 \arctan \left (a x \right )^{3} x^{8} a^{8}-5040 \arctan \left (a x \right )^{2} x^{7} a^{7}+36480 \arctan \left (a x \right )^{3} x^{6} a^{6}+1440 \arctan \left (a x \right ) x^{6} a^{6}-12360 \arctan \left (a x \right )^{2} x^{5} a^{5}+28800 \arctan \left (a x \right )^{3} x^{4} a^{4}-240 x^{5} a^{5}+3216 \arctan \left (a x \right ) x^{4} a^{4}-6150 \arctan \left (a x \right )^{2} x^{3} a^{3}+1920 \arctan \left (a x \right )^{3} x^{2} a^{2}-504 a^{3} x^{3}-188 \arctan \left (a x \right ) a^{2} x^{2}+6345 \arctan \left (a x \right )^{2} x a -3840 \arctan \left (a x \right )^{3}+850 a x -12314 \arctan \left (a x \right )\right )}{120960 a^{4}}+\frac {115 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (-i \arctan \left (a x \right )^{3}+3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8064 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {115 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (-i \arctan \left (a x \right )^{3}+3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8064 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {1433 i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{7560 a^{4} \sqrt {a^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120960*c^2/a^4*(c*(a*x-I)*(I+a*x))^(1/2)*(13440*arctan(a*x)^3*x^8*a^8-5040*arctan(a*x)^2*x^7*a^7+36480*arcta
n(a*x)^3*x^6*a^6+1440*arctan(a*x)*x^6*a^6-12360*arctan(a*x)^2*x^5*a^5+28800*arctan(a*x)^3*x^4*a^4-240*x^5*a^5+
3216*arctan(a*x)*x^4*a^4-6150*arctan(a*x)^2*x^3*a^3+1920*arctan(a*x)^3*x^2*a^2-504*a^3*x^3-188*arctan(a*x)*a^2
*x^2+6345*arctan(a*x)^2*x*a-3840*arctan(a*x)^3+850*a*x-12314*arctan(a*x))+115/8064*c^2*(c*(a*x-I)*(I+a*x))^(1/
2)*(-I*arctan(a*x)^3+3*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*arctan(a*x)*polylog(2,I*(1+I*a*x)
/(a^2*x^2+1)^(1/2))+6*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1/2)-115/8064*c^2*(c*(a*x-I)*
(I+a*x))^(1/2)*(-I*arctan(a*x)^3+3*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*arctan(a*x)*polylog(2
,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1/2)-1433/7560*
I*c^2/a^4*(c*(a*x-I)*(I+a*x))^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*x^2+1)^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________