3.24 \(\int x^5 (a+b \tan ^{-1}(c x))^3 \, dx\)

Optimal. Leaf size=255 \[ -\frac {23 b^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^6}-\frac {4 b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c^4}+\frac {b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}-\frac {23 i b \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^6}-\frac {b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}-\frac {23 i b^3 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{30 c^6}-\frac {19 b^3 \tan ^{-1}(c x)}{60 c^6}+\frac {19 b^3 x}{60 c^5}-\frac {b^3 x^3}{60 c^3} \]

[Out]

19/60*b^3*x/c^5-1/60*b^3*x^3/c^3-19/60*b^3*arctan(c*x)/c^6-4/15*b^2*x^2*(a+b*arctan(c*x))/c^4+1/20*b^2*x^4*(a+
b*arctan(c*x))/c^2-23/30*I*b*(a+b*arctan(c*x))^2/c^6-1/2*b*x*(a+b*arctan(c*x))^2/c^5+1/6*b*x^3*(a+b*arctan(c*x
))^2/c^3-1/10*b*x^5*(a+b*arctan(c*x))^2/c+1/6*(a+b*arctan(c*x))^3/c^6+1/6*x^6*(a+b*arctan(c*x))^3-23/15*b^2*(a
+b*arctan(c*x))*ln(2/(1+I*c*x))/c^6-23/30*I*b^3*polylog(2,1-2/(1+I*c*x))/c^6

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Rubi [A]  time = 0.95, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {4852, 4916, 302, 203, 321, 4920, 4854, 2402, 2315, 4846, 4884} \[ -\frac {23 i b^3 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{30 c^6}+\frac {b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac {4 b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c^4}-\frac {23 b^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^6}+\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac {b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}-\frac {23 i b \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^6}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}-\frac {b^3 x^3}{60 c^3}+\frac {19 b^3 x}{60 c^5}-\frac {19 b^3 \tan ^{-1}(c x)}{60 c^6} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*ArcTan[c*x])^3,x]

[Out]

(19*b^3*x)/(60*c^5) - (b^3*x^3)/(60*c^3) - (19*b^3*ArcTan[c*x])/(60*c^6) - (4*b^2*x^2*(a + b*ArcTan[c*x]))/(15
*c^4) + (b^2*x^4*(a + b*ArcTan[c*x]))/(20*c^2) - (((23*I)/30)*b*(a + b*ArcTan[c*x])^2)/c^6 - (b*x*(a + b*ArcTa
n[c*x])^2)/(2*c^5) + (b*x^3*(a + b*ArcTan[c*x])^2)/(6*c^3) - (b*x^5*(a + b*ArcTan[c*x])^2)/(10*c) + (a + b*Arc
Tan[c*x])^3/(6*c^6) + (x^6*(a + b*ArcTan[c*x])^3)/6 - (23*b^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(15*c^6)
- (((23*I)/30)*b^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^6

Rule 203

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 4846

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

Rule 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 4854

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

Rule 4884

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

Rule 4916

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

Rule 4920

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

Rubi steps

\begin {align*} \int x^5 \left (a+b \tan ^{-1}(c x)\right )^3 \, dx &=\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {1}{2} (b c) \int \frac {x^6 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx\\ &=\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b \int x^4 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{2 c}+\frac {b \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 c}\\ &=-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {1}{5} b^2 \int \frac {x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac {b \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{2 c^3}-\frac {b \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 c^3}\\ &=\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{2 c^5}+\frac {b \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 c^5}+\frac {b^2 \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c^2}-\frac {b^2 \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c^2}-\frac {b^2 \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^2}\\ &=\frac {b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac {b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b^2 \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c^4}+\frac {b^2 \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c^4}-\frac {b^2 \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^4}+\frac {b^2 \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^4}+\frac {b^2 \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^4}-\frac {b^3 \int \frac {x^4}{1+c^2 x^2} \, dx}{20 c}\\ &=-\frac {4 b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c^4}+\frac {b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac {23 i b \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^6}-\frac {b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{5 c^5}-\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^5}-\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{c^5}+\frac {b^3 \int \frac {x^2}{1+c^2 x^2} \, dx}{10 c^3}+\frac {b^3 \int \frac {x^2}{1+c^2 x^2} \, dx}{6 c^3}-\frac {b^3 \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{20 c}\\ &=\frac {19 b^3 x}{60 c^5}-\frac {b^3 x^3}{60 c^3}-\frac {4 b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c^4}+\frac {b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac {23 i b \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^6}-\frac {b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {23 b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{15 c^6}-\frac {b^3 \int \frac {1}{1+c^2 x^2} \, dx}{20 c^5}-\frac {b^3 \int \frac {1}{1+c^2 x^2} \, dx}{10 c^5}-\frac {b^3 \int \frac {1}{1+c^2 x^2} \, dx}{6 c^5}+\frac {b^3 \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^5}+\frac {b^3 \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^5}+\frac {b^3 \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^5}\\ &=\frac {19 b^3 x}{60 c^5}-\frac {b^3 x^3}{60 c^3}-\frac {19 b^3 \tan ^{-1}(c x)}{60 c^6}-\frac {4 b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c^4}+\frac {b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac {23 i b \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^6}-\frac {b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {23 b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{15 c^6}-\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{5 c^6}-\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^6}-\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^6}\\ &=\frac {19 b^3 x}{60 c^5}-\frac {b^3 x^3}{60 c^3}-\frac {19 b^3 \tan ^{-1}(c x)}{60 c^6}-\frac {4 b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c^4}+\frac {b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac {23 i b \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^6}-\frac {b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {23 b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{15 c^6}-\frac {23 i b^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{30 c^6}\\ \end {align*}

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Mathematica [A]  time = 0.83, size = 291, normalized size = 1.14 \[ \frac {10 a^3 c^6 x^6+b \tan ^{-1}(c x) \left (30 a^2 \left (c^6 x^6+1\right )-4 a b c x \left (3 c^4 x^4-5 c^2 x^2+15\right )+b^2 \left (3 c^4 x^4-16 c^2 x^2-19\right )-92 b^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-6 a^2 b c^5 x^5+10 a^2 b c^3 x^3-30 a^2 b c x+3 a b^2 c^4 x^4-16 a b^2 c^2 x^2+46 a b^2 \log \left (c^2 x^2+1\right )+2 b^2 \tan ^{-1}(c x)^2 \left (15 a \left (c^6 x^6+1\right )+b \left (-3 c^5 x^5+5 c^3 x^3-15 c x+23 i\right )\right )-19 a b^2+10 b^3 \left (c^6 x^6+1\right ) \tan ^{-1}(c x)^3-b^3 c^3 x^3+46 i b^3 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+19 b^3 c x}{60 c^6} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-19*a*b^2 - 30*a^2*b*c*x + 19*b^3*c*x - 16*a*b^2*c^2*x^2 + 10*a^2*b*c^3*x^3 - b^3*c^3*x^3 + 3*a*b^2*c^4*x^4 -
6*a^2*b*c^5*x^5 + 10*a^3*c^6*x^6 + 2*b^2*(b*(23*I - 15*c*x + 5*c^3*x^3 - 3*c^5*x^5) + 15*a*(1 + c^6*x^6))*Arc
Tan[c*x]^2 + 10*b^3*(1 + c^6*x^6)*ArcTan[c*x]^3 + b*ArcTan[c*x]*(b^2*(-19 - 16*c^2*x^2 + 3*c^4*x^4) - 4*a*b*c*
x*(15 - 5*c^2*x^2 + 3*c^4*x^4) + 30*a^2*(1 + c^6*x^6) - 92*b^2*Log[1 + E^((2*I)*ArcTan[c*x])]) + 46*a*b^2*Log[
1 + c^2*x^2] + (46*I)*b^3*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(60*c^6)

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{5} \arctan \left (c x\right )^{3} + 3 \, a b^{2} x^{5} \arctan \left (c x\right )^{2} + 3 \, a^{2} b x^{5} \arctan \left (c x\right ) + a^{3} x^{5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^3*x^5*arctan(c*x)^3 + 3*a*b^2*x^5*arctan(c*x)^2 + 3*a^2*b*x^5*arctan(c*x) + a^3*x^5, x)

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

[Out]

sage0*x

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maple [B]  time = 0.02, size = 528, normalized size = 2.07 \[ -\frac {23 i b^{3} \ln \left (c x -i\right )^{2}}{120 c^{6}}+\frac {23 i b^{3} \ln \left (c x +i\right )^{2}}{120 c^{6}}-\frac {23 i b^{3} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{60 c^{6}}+\frac {23 i b^{3} \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{60 c^{6}}-\frac {b^{3} \arctan \left (c x \right )^{2} x^{5}}{10 c}+\frac {b^{3} \arctan \left (c x \right )^{2} x^{3}}{6 c^{3}}+\frac {b^{3} \arctan \left (c x \right ) x^{4}}{20 c^{2}}-\frac {4 b^{3} \arctan \left (c x \right ) x^{2}}{15 c^{4}}-\frac {b^{3} \arctan \left (c x \right )^{2} x}{2 c^{5}}+\frac {a \,b^{2} x^{6} \arctan \left (c x \right )^{2}}{2}+\frac {a^{2} b \,x^{6} \arctan \left (c x \right )}{2}+\frac {a^{2} b \arctan \left (c x \right )}{2 c^{6}}+\frac {a \,b^{2} \arctan \left (c x \right )^{2}}{2 c^{6}}+\frac {23 a \,b^{2} \ln \left (c^{2} x^{2}+1\right )}{30 c^{6}}+\frac {23 b^{3} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{30 c^{6}}-\frac {x \,a^{2} b}{2 c^{5}}+\frac {a \,b^{2} x^{4}}{20 c^{2}}-\frac {x^{5} a^{2} b}{10 c}+\frac {a^{2} b \,x^{3}}{6 c^{3}}-\frac {4 x^{2} a \,b^{2}}{15 c^{4}}+\frac {19 b^{3} x}{60 c^{5}}-\frac {b^{3} x^{3}}{60 c^{3}}-\frac {19 b^{3} \arctan \left (c x \right )}{60 c^{6}}+\frac {x^{6} a^{3}}{6}+\frac {b^{3} x^{6} \arctan \left (c x \right )^{3}}{6}+\frac {b^{3} \arctan \left (c x \right )^{3}}{6 c^{6}}+\frac {23 i b^{3} \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{60 c^{6}}-\frac {23 i b^{3} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{60 c^{6}}+\frac {23 i b^{3} \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{60 c^{6}}-\frac {23 i b^{3} \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{60 c^{6}}+\frac {a \,b^{2} x^{3} \arctan \left (c x \right )}{3 c^{3}}-\frac {a \,b^{2} x \arctan \left (c x \right )}{c^{5}}-\frac {a \,b^{2} x^{5} \arctan \left (c x \right )}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-23/60*I/c^6*b^3*dilog(-1/2*I*(I+c*x))-23/120*I/c^6*b^3*ln(c*x-I)^2-1/10/c*b^3*arctan(c*x)^2*x^5+1/6/c^3*b^3*a
rctan(c*x)^2*x^3+1/20/c^2*b^3*arctan(c*x)*x^4-4/15/c^4*b^3*arctan(c*x)*x^2-1/2/c^5*b^3*arctan(c*x)^2*x+1/2*a*b
^2*x^6*arctan(c*x)^2+1/2*a^2*b*x^6*arctan(c*x)+23/60*I/c^6*b^3*dilog(1/2*I*(c*x-I))+23/120*I/c^6*b^3*ln(I+c*x)
^2+1/2/c^6*a^2*b*arctan(c*x)+1/2/c^6*a*b^2*arctan(c*x)^2+23/30/c^6*a*b^2*ln(c^2*x^2+1)+23/30/c^6*b^3*arctan(c*
x)*ln(c^2*x^2+1)-1/2/c^5*x*a^2*b+1/20/c^2*a*b^2*x^4-1/10/c*x^5*a^2*b+1/6/c^3*a^2*b*x^3-4/15/c^4*x^2*a*b^2+19/6
0*b^3*x/c^5-1/60*b^3*x^3/c^3-19/60*b^3*arctan(c*x)/c^6+1/6*x^6*a^3+1/6*b^3*x^6*arctan(c*x)^3+1/6/c^6*b^3*arcta
n(c*x)^3+1/3/c^3*a*b^2*x^3*arctan(c*x)-1/c^5*a*b^2*x*arctan(c*x)-23/60*I/c^6*b^3*ln(I+c*x)*ln(c^2*x^2+1)-23/60
*I/c^6*b^3*ln(c*x-I)*ln(-1/2*I*(I+c*x))+23/60*I/c^6*b^3*ln(c*x-I)*ln(c^2*x^2+1)+23/60*I/c^6*b^3*ln(I+c*x)*ln(1
/2*I*(c*x-I))-1/5/c*a*b^2*x^5*arctan(c*x)

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maxima [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^5*(a+b*arctan(c*x))^3,x, algorithm="maxima")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*atan(c*x))^3,x)

[Out]

int(x^5*(a + b*atan(c*x))^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**5*(a + b*atan(c*x))**3, x)

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