\(\int x^4 (a+b \arctan (c x)) (d+e \log (1+c^2 x^2)) \, dx\) [1286]

Optimal result

Integrand size = 26, antiderivative size = 278 \[ \int x^4 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=-\frac {2 a e x}{5 c^4}-\frac {77 b e x^2}{300 c^3}+\frac {2 a e x^3}{15 c^2}+\frac {9 b e x^4}{200 c}-\frac {2}{25} a e x^5+\frac {2 a e \arctan (c x)}{5 c^5}-\frac {2 b e x \arctan (c x)}{5 c^4}+\frac {2 b e x^3 \arctan (c x)}{15 c^2}-\frac {2}{25} b e x^5 \arctan (c x)+\frac {b e \arctan (c x)^2}{5 c^5}+\frac {137 b e \log \left (1+c^2 x^2\right )}{300 c^5}+\frac {b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac {b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5} \]

[Out]

-2/5*a*e*x/c^4-77/300*b*e*x^2/c^3+2/15*a*e*x^3/c^2+9/200*b*e*x^4/c-2/25*a*e*x^5+2/5*a*e*arctan(c*x)/c^5-2/5*b*
e*x*arctan(c*x)/c^4+2/15*b*e*x^3*arctan(c*x)/c^2-2/25*b*e*x^5*arctan(c*x)+1/5*b*e*arctan(c*x)^2/c^5+137/300*b*
e*ln(c^2*x^2+1)/c^5+1/20*b*e*ln(c^2*x^2+1)^2/c^5+1/10*b*x^2*(d+e*ln(c^2*x^2+1))/c^3-1/20*b*x^4*(d+e*ln(c^2*x^2
+1))/c+1/5*x^5*(a+b*arctan(c*x))*(d+e*ln(c^2*x^2+1))-1/10*b*ln(c^2*x^2+1)*(d+e*ln(c^2*x^2+1))/c^5

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {4946, 272, 45, 5141, 6857, 1816, 649, 209, 266, 5036, 4930, 5004, 2525, 2437, 2338} \[ \int x^4 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {1}{5} x^5 (a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )+\frac {2 a e \arctan (c x)}{5 c^5}-\frac {2 a e x}{5 c^4}+\frac {2 a e x^3}{15 c^2}-\frac {2}{25} a e x^5+\frac {b e \arctan (c x)^2}{5 c^5}-\frac {2 b e x \arctan (c x)}{5 c^4}+\frac {2 b e x^3 \arctan (c x)}{15 c^2}-\frac {2}{25} b e x^5 \arctan (c x)-\frac {77 b e x^2}{300 c^3}-\frac {b x^4 \left (e \log \left (c^2 x^2+1\right )+d\right )}{20 c}-\frac {b \log \left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{10 c^5}+\frac {b e \log ^2\left (c^2 x^2+1\right )}{20 c^5}+\frac {137 b e \log \left (c^2 x^2+1\right )}{300 c^5}+\frac {b x^2 \left (e \log \left (c^2 x^2+1\right )+d\right )}{10 c^3}+\frac {9 b e x^4}{200 c} \]

[In]

Int[x^4*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]),x]

[Out]

(-2*a*e*x)/(5*c^4) - (77*b*e*x^2)/(300*c^3) + (2*a*e*x^3)/(15*c^2) + (9*b*e*x^4)/(200*c) - (2*a*e*x^5)/25 + (2
*a*e*ArcTan[c*x])/(5*c^5) - (2*b*e*x*ArcTan[c*x])/(5*c^4) + (2*b*e*x^3*ArcTan[c*x])/(15*c^2) - (2*b*e*x^5*ArcT
an[c*x])/25 + (b*e*ArcTan[c*x]^2)/(5*c^5) + (137*b*e*Log[1 + c^2*x^2])/(300*c^5) + (b*e*Log[1 + c^2*x^2]^2)/(2
0*c^5) + (b*x^2*(d + e*Log[1 + c^2*x^2]))/(10*c^3) - (b*x^4*(d + e*Log[1 + c^2*x^2]))/(20*c) + (x^5*(a + b*Arc
Tan[c*x])*(d + e*Log[1 + c^2*x^2]))/5 - (b*Log[1 + c^2*x^2]*(d + e*Log[1 + c^2*x^2]))/(10*c^5)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 209

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 1816

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2525

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 4930

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

Rule 4946

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

Rule 5004

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 5036

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 5141

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*(e_.))*(x_)^(m_.), x_Symbol] :> With
[{u = IntHide[x^m*(a + b*ArcTan[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - Dist[2*e*g, Int[ExpandIntegrand
[x*(u/(f + g*x^2)), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac {b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}-\left (2 c^2 e\right ) \int \left (\frac {2 b x^3-b c^2 x^5+4 a c^3 x^6+4 b c^3 x^6 \arctan (c x)}{20 c^3 \left (1+c^2 x^2\right )}-\frac {b x \log \left (1+c^2 x^2\right )}{10 c^5 \left (1+c^2 x^2\right )}\right ) \, dx \\ & = \frac {b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac {b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}+\frac {(b e) \int \frac {x \log \left (1+c^2 x^2\right )}{1+c^2 x^2} \, dx}{5 c^3}-\frac {e \int \frac {2 b x^3-b c^2 x^5+4 a c^3 x^6+4 b c^3 x^6 \arctan (c x)}{1+c^2 x^2} \, dx}{10 c} \\ & = \frac {b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac {b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}+\frac {(b e) \text {Subst}\left (\int \frac {\log \left (1+c^2 x\right )}{1+c^2 x} \, dx,x,x^2\right )}{10 c^3}-\frac {e \int \left (\frac {x^3 \left (2 b-b c^2 x^2+4 a c^3 x^3\right )}{1+c^2 x^2}+\frac {4 b c^3 x^6 \arctan (c x)}{1+c^2 x^2}\right ) \, dx}{10 c} \\ & = \frac {b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac {b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}+\frac {(b e) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+c^2 x^2\right )}{10 c^5}-\frac {e \int \frac {x^3 \left (2 b-b c^2 x^2+4 a c^3 x^3\right )}{1+c^2 x^2} \, dx}{10 c}-\frac {1}{5} \left (2 b c^2 e\right ) \int \frac {x^6 \arctan (c x)}{1+c^2 x^2} \, dx \\ & = \frac {b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac {b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}-\frac {1}{5} (2 b e) \int x^4 \arctan (c x) \, dx+\frac {1}{5} (2 b e) \int \frac {x^4 \arctan (c x)}{1+c^2 x^2} \, dx-\frac {e \int \left (\frac {4 a}{c^3}+\frac {3 b x}{c^2}-\frac {4 a x^2}{c}-b x^3+4 a c x^4-\frac {4 a+3 b c x}{c^3 \left (1+c^2 x^2\right )}\right ) \, dx}{10 c} \\ & = -\frac {2 a e x}{5 c^4}-\frac {3 b e x^2}{20 c^3}+\frac {2 a e x^3}{15 c^2}+\frac {b e x^4}{40 c}-\frac {2}{25} a e x^5-\frac {2}{25} b e x^5 \arctan (c x)+\frac {b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac {b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}+\frac {e \int \frac {4 a+3 b c x}{1+c^2 x^2} \, dx}{10 c^4}+\frac {(2 b e) \int x^2 \arctan (c x) \, dx}{5 c^2}-\frac {(2 b e) \int \frac {x^2 \arctan (c x)}{1+c^2 x^2} \, dx}{5 c^2}+\frac {1}{25} (2 b c e) \int \frac {x^5}{1+c^2 x^2} \, dx \\ & = -\frac {2 a e x}{5 c^4}-\frac {3 b e x^2}{20 c^3}+\frac {2 a e x^3}{15 c^2}+\frac {b e x^4}{40 c}-\frac {2}{25} a e x^5+\frac {2 b e x^3 \arctan (c x)}{15 c^2}-\frac {2}{25} b e x^5 \arctan (c x)+\frac {b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac {b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}+\frac {(2 a e) \int \frac {1}{1+c^2 x^2} \, dx}{5 c^4}-\frac {(2 b e) \int \arctan (c x) \, dx}{5 c^4}+\frac {(2 b e) \int \frac {\arctan (c x)}{1+c^2 x^2} \, dx}{5 c^4}+\frac {(3 b e) \int \frac {x}{1+c^2 x^2} \, dx}{10 c^3}-\frac {(2 b e) \int \frac {x^3}{1+c^2 x^2} \, dx}{15 c}+\frac {1}{25} (b c e) \text {Subst}\left (\int \frac {x^2}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -\frac {2 a e x}{5 c^4}-\frac {3 b e x^2}{20 c^3}+\frac {2 a e x^3}{15 c^2}+\frac {b e x^4}{40 c}-\frac {2}{25} a e x^5+\frac {2 a e \arctan (c x)}{5 c^5}-\frac {2 b e x \arctan (c x)}{5 c^4}+\frac {2 b e x^3 \arctan (c x)}{15 c^2}-\frac {2}{25} b e x^5 \arctan (c x)+\frac {b e \arctan (c x)^2}{5 c^5}+\frac {3 b e \log \left (1+c^2 x^2\right )}{20 c^5}+\frac {b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac {b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}+\frac {(2 b e) \int \frac {x}{1+c^2 x^2} \, dx}{5 c^3}-\frac {(b e) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{15 c}+\frac {1}{25} (b c e) \text {Subst}\left (\int \left (-\frac {1}{c^4}+\frac {x}{c^2}+\frac {1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = -\frac {2 a e x}{5 c^4}-\frac {19 b e x^2}{100 c^3}+\frac {2 a e x^3}{15 c^2}+\frac {9 b e x^4}{200 c}-\frac {2}{25} a e x^5+\frac {2 a e \arctan (c x)}{5 c^5}-\frac {2 b e x \arctan (c x)}{5 c^4}+\frac {2 b e x^3 \arctan (c x)}{15 c^2}-\frac {2}{25} b e x^5 \arctan (c x)+\frac {b e \arctan (c x)^2}{5 c^5}+\frac {39 b e \log \left (1+c^2 x^2\right )}{100 c^5}+\frac {b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac {b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}-\frac {(b e) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{15 c} \\ & = -\frac {2 a e x}{5 c^4}-\frac {77 b e x^2}{300 c^3}+\frac {2 a e x^3}{15 c^2}+\frac {9 b e x^4}{200 c}-\frac {2}{25} a e x^5+\frac {2 a e \arctan (c x)}{5 c^5}-\frac {2 b e x \arctan (c x)}{5 c^4}+\frac {2 b e x^3 \arctan (c x)}{15 c^2}-\frac {2}{25} b e x^5 \arctan (c x)+\frac {b e \arctan (c x)^2}{5 c^5}+\frac {137 b e \log \left (1+c^2 x^2\right )}{300 c^5}+\frac {b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac {b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.77 \[ \int x^4 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {c x \left (b c x \left (-30 d \left (-2+c^2 x^2\right )+e \left (-154+27 c^2 x^2\right )\right )+8 a \left (15 c^4 d x^4-2 e \left (15-5 c^2 x^2+3 c^4 x^4\right )\right )\right )+120 b e \arctan (c x)^2+\left (-60 b d+120 a c^5 e x^5+2 b e \left (137+30 c^2 x^2-15 c^4 x^4\right )\right ) \log \left (1+c^2 x^2\right )-30 b e \log ^2\left (1+c^2 x^2\right )+8 \arctan (c x) \left (30 a e+15 b c^5 d x^5-2 b c e x \left (15-5 c^2 x^2+3 c^4 x^4\right )+15 b c^5 e x^5 \log \left (1+c^2 x^2\right )\right )}{600 c^5} \]

[In]

Integrate[x^4*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]),x]

[Out]

(c*x*(b*c*x*(-30*d*(-2 + c^2*x^2) + e*(-154 + 27*c^2*x^2)) + 8*a*(15*c^4*d*x^4 - 2*e*(15 - 5*c^2*x^2 + 3*c^4*x
^4))) + 120*b*e*ArcTan[c*x]^2 + (-60*b*d + 120*a*c^5*e*x^5 + 2*b*e*(137 + 30*c^2*x^2 - 15*c^4*x^4))*Log[1 + c^
2*x^2] - 30*b*e*Log[1 + c^2*x^2]^2 + 8*ArcTan[c*x]*(30*a*e + 15*b*c^5*d*x^5 - 2*b*c*e*x*(15 - 5*c^2*x^2 + 3*c^
4*x^4) + 15*b*c^5*e*x^5*Log[1 + c^2*x^2]))/(600*c^5)

Maple [A] (verified)

Time = 2.69 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.04

method	result	size
parallelrisch	\(\frac {120 a \,c^{5} d \,x^{5}+80 x^{3} \arctan \left (c x \right ) b \,c^{3} e +154 e b -154 b \,c^{2} e \,x^{2}-48 a \,c^{5} e \,x^{5}+27 b \,c^{4} e \,x^{4}+80 a \,c^{3} e \,x^{3}+274 \ln \left (c^{2} x^{2}+1\right ) b e -48 x^{5} \arctan \left (c x \right ) b \,c^{5} e -240 x a c e -30 e b \ln \left (c^{2} x^{2}+1\right )^{2}-30 b \,c^{4} d \,x^{4}+240 e a \arctan \left (c x \right )+120 e b \arctan \left (c x \right )^{2}-60 \ln \left (c^{2} x^{2}+1\right ) b d +120 e a \ln \left (c^{2} x^{2}+1\right ) x^{5} c^{5}+60 x^{2} \ln \left (c^{2} x^{2}+1\right ) b \,c^{2} e -240 e b \arctan \left (c x \right ) x c -30 e b \ln \left (c^{2} x^{2}+1\right ) x^{4} c^{4}+120 b \arctan \left (c x \right ) x^{5} c^{5} d +120 e b \ln \left (c^{2} x^{2}+1\right ) \arctan \left (c x \right ) x^{5} c^{5}+60 c^{2} x^{2} b d -60 b d}{600 c^{5}}\)	\(289\)
default	\(\text {Expression too large to display}\)	\(4787\)
parts	\(\text {Expression too large to display}\)	\(4787\)
risch	\(\text {Expression too large to display}\)	\(23634\)

[In]

int(x^4*(a+b*arctan(c*x))*(d+e*ln(c^2*x^2+1)),x,method=_RETURNVERBOSE)

[Out]

1/600*(120*a*c^5*d*x^5+80*x^3*arctan(c*x)*b*c^3*e+154*e*b-154*b*c^2*e*x^2-48*a*c^5*e*x^5+27*b*c^4*e*x^4+80*a*c
^3*e*x^3+274*ln(c^2*x^2+1)*b*e-48*x^5*arctan(c*x)*b*c^5*e-240*x*a*c*e-30*e*b*ln(c^2*x^2+1)^2-30*b*c^4*d*x^4+24
0*e*a*arctan(c*x)+120*e*b*arctan(c*x)^2-60*ln(c^2*x^2+1)*b*d+120*e*a*ln(c^2*x^2+1)*x^5*c^5+60*x^2*ln(c^2*x^2+1
)*b*c^2*e-240*e*b*arctan(c*x)*x*c-30*e*b*ln(c^2*x^2+1)*x^4*c^4+120*b*arctan(c*x)*x^5*c^5*d+120*e*b*ln(c^2*x^2+
1)*arctan(c*x)*x^5*c^5+60*c^2*x^2*b*d-60*b*d)/c^5

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.79 \[ \int x^4 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {80 \, a c^{3} e x^{3} + 24 \, {\left (5 \, a c^{5} d - 2 \, a c^{5} e\right )} x^{5} - 3 \, {\left (10 \, b c^{4} d - 9 \, b c^{4} e\right )} x^{4} - 240 \, a c e x + 120 \, b e \arctan \left (c x\right )^{2} - 30 \, b e \log \left (c^{2} x^{2} + 1\right )^{2} + 2 \, {\left (30 \, b c^{2} d - 77 \, b c^{2} e\right )} x^{2} + 8 \, {\left (10 \, b c^{3} e x^{3} + 3 \, {\left (5 \, b c^{5} d - 2 \, b c^{5} e\right )} x^{5} - 30 \, b c e x + 30 \, a e\right )} \arctan \left (c x\right ) + 2 \, {\left (60 \, b c^{5} e x^{5} \arctan \left (c x\right ) + 60 \, a c^{5} e x^{5} - 15 \, b c^{4} e x^{4} + 30 \, b c^{2} e x^{2} - 30 \, b d + 137 \, b e\right )} \log \left (c^{2} x^{2} + 1\right )}{600 \, c^{5}} \]

[In]

integrate(x^4*(a+b*arctan(c*x))*(d+e*log(c^2*x^2+1)),x, algorithm="fricas")

[Out]

1/600*(80*a*c^3*e*x^3 + 24*(5*a*c^5*d - 2*a*c^5*e)*x^5 - 3*(10*b*c^4*d - 9*b*c^4*e)*x^4 - 240*a*c*e*x + 120*b*
e*arctan(c*x)^2 - 30*b*e*log(c^2*x^2 + 1)^2 + 2*(30*b*c^2*d - 77*b*c^2*e)*x^2 + 8*(10*b*c^3*e*x^3 + 3*(5*b*c^5
*d - 2*b*c^5*e)*x^5 - 30*b*c*e*x + 30*a*e)*arctan(c*x) + 2*(60*b*c^5*e*x^5*arctan(c*x) + 60*a*c^5*e*x^5 - 15*b
*c^4*e*x^4 + 30*b*c^2*e*x^2 - 30*b*d + 137*b*e)*log(c^2*x^2 + 1))/c^5

Sympy [A] (verification not implemented)

Time = 1.63 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.22 \[ \int x^4 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\begin {cases} \frac {a d x^{5}}{5} + \frac {a e x^{5} \log {\left (c^{2} x^{2} + 1 \right )}}{5} - \frac {2 a e x^{5}}{25} + \frac {2 a e x^{3}}{15 c^{2}} - \frac {2 a e x}{5 c^{4}} + \frac {2 a e \operatorname {atan}{\left (c x \right )}}{5 c^{5}} + \frac {b d x^{5} \operatorname {atan}{\left (c x \right )}}{5} + \frac {b e x^{5} \log {\left (c^{2} x^{2} + 1 \right )} \operatorname {atan}{\left (c x \right )}}{5} - \frac {2 b e x^{5} \operatorname {atan}{\left (c x \right )}}{25} - \frac {b d x^{4}}{20 c} - \frac {b e x^{4} \log {\left (c^{2} x^{2} + 1 \right )}}{20 c} + \frac {9 b e x^{4}}{200 c} + \frac {2 b e x^{3} \operatorname {atan}{\left (c x \right )}}{15 c^{2}} + \frac {b d x^{2}}{10 c^{3}} + \frac {b e x^{2} \log {\left (c^{2} x^{2} + 1 \right )}}{10 c^{3}} - \frac {77 b e x^{2}}{300 c^{3}} - \frac {2 b e x \operatorname {atan}{\left (c x \right )}}{5 c^{4}} - \frac {b d \log {\left (c^{2} x^{2} + 1 \right )}}{10 c^{5}} - \frac {b e \log {\left (c^{2} x^{2} + 1 \right )}^{2}}{20 c^{5}} + \frac {137 b e \log {\left (c^{2} x^{2} + 1 \right )}}{300 c^{5}} + \frac {b e \operatorname {atan}^{2}{\left (c x \right )}}{5 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d x^{5}}{5} & \text {otherwise} \end {cases} \]

[In]

integrate(x**4*(a+b*atan(c*x))*(d+e*ln(c**2*x**2+1)),x)

[Out]

Piecewise((a*d*x**5/5 + a*e*x**5*log(c**2*x**2 + 1)/5 - 2*a*e*x**5/25 + 2*a*e*x**3/(15*c**2) - 2*a*e*x/(5*c**4
) + 2*a*e*atan(c*x)/(5*c**5) + b*d*x**5*atan(c*x)/5 + b*e*x**5*log(c**2*x**2 + 1)*atan(c*x)/5 - 2*b*e*x**5*ata
n(c*x)/25 - b*d*x**4/(20*c) - b*e*x**4*log(c**2*x**2 + 1)/(20*c) + 9*b*e*x**4/(200*c) + 2*b*e*x**3*atan(c*x)/(
15*c**2) + b*d*x**2/(10*c**3) + b*e*x**2*log(c**2*x**2 + 1)/(10*c**3) - 77*b*e*x**2/(300*c**3) - 2*b*e*x*atan(
c*x)/(5*c**4) - b*d*log(c**2*x**2 + 1)/(10*c**5) - b*e*log(c**2*x**2 + 1)**2/(20*c**5) + 137*b*e*log(c**2*x**2
+ 1)/(300*c**5) + b*e*atan(c*x)**2/(5*c**5), Ne(c, 0)), (a*d*x**5/5, True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.92 \[ \int x^4 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {1}{5} \, a d x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{2} {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b e \arctan \left (c x\right ) + \frac {1}{20} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b d + \frac {1}{75} \, {\left (15 \, x^{5} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{2} {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} a e + \frac {{\left (27 \, c^{4} x^{4} - 154 \, c^{2} x^{2} - 120 \, \arctan \left (c x\right )^{2} - 2 \, {\left (15 \, c^{4} x^{4} - 30 \, c^{2} x^{2} - 137\right )} \log \left (c^{2} x^{2} + 1\right ) - 30 \, \log \left (c^{2} x^{2} + 1\right )^{2}\right )} b e}{600 \, c^{5}} \]

[In]

integrate(x^4*(a+b*arctan(c*x))*(d+e*log(c^2*x^2+1)),x, algorithm="maxima")

[Out]

1/5*a*d*x^5 + 1/75*(15*x^5*log(c^2*x^2 + 1) - 2*c^2*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))
*b*e*arctan(c*x) + 1/20*(4*x^5*arctan(c*x) - c*((c^2*x^4 - 2*x^2)/c^4 + 2*log(c^2*x^2 + 1)/c^6))*b*d + 1/75*(1
5*x^5*log(c^2*x^2 + 1) - 2*c^2*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))*a*e + 1/600*(27*c^4*
x^4 - 154*c^2*x^2 - 120*arctan(c*x)^2 - 2*(15*c^4*x^4 - 30*c^2*x^2 - 137)*log(c^2*x^2 + 1) - 30*log(c^2*x^2 +
1)^2)*b*e/c^5

Giac [F]

\[ \int x^4 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (c^{2} x^{2} + 1\right ) + d\right )} x^{4} \,d x } \]

[In]

integrate(x^4*(a+b*arctan(c*x))*(d+e*log(c^2*x^2+1)),x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 3.72 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.99 \[ \int x^4 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {a\,d\,x^5}{5}-\frac {2\,a\,e\,x^5}{25}-\frac {b\,e\,{\ln \left (c^2\,x^2+1\right )}^2}{20\,c^5}-\frac {2\,a\,e\,x}{5\,c^4}+\frac {2\,a\,e\,\mathrm {atan}\left (c\,x\right )}{5\,c^5}+\frac {b\,d\,x^5\,\mathrm {atan}\left (c\,x\right )}{5}-\frac {2\,b\,e\,x^5\,\mathrm {atan}\left (c\,x\right )}{25}-\frac {b\,d\,\ln \left (c^2\,x^2+1\right )}{10\,c^5}+\frac {137\,b\,e\,\ln \left (c^2\,x^2+1\right )}{300\,c^5}+\frac {2\,a\,e\,x^3}{15\,c^2}-\frac {b\,d\,x^4}{20\,c}+\frac {b\,d\,x^2}{10\,c^3}+\frac {9\,b\,e\,x^4}{200\,c}-\frac {77\,b\,e\,x^2}{300\,c^3}+\frac {a\,e\,x^5\,\ln \left (c^2\,x^2+1\right )}{5}+\frac {b\,e\,{\mathrm {atan}\left (c\,x\right )}^2}{5\,c^5}+\frac {2\,b\,e\,x^3\,\mathrm {atan}\left (c\,x\right )}{15\,c^2}+\frac {b\,e\,x^5\,\mathrm {atan}\left (c\,x\right )\,\ln \left (c^2\,x^2+1\right )}{5}-\frac {b\,e\,x^4\,\ln \left (c^2\,x^2+1\right )}{20\,c}+\frac {b\,e\,x^2\,\ln \left (c^2\,x^2+1\right )}{10\,c^3}-\frac {2\,b\,e\,x\,\mathrm {atan}\left (c\,x\right )}{5\,c^4} \]

[In]

int(x^4*(a + b*atan(c*x))*(d + e*log(c^2*x^2 + 1)),x)

[Out]

(a*d*x^5)/5 - (2*a*e*x^5)/25 - (b*e*log(c^2*x^2 + 1)^2)/(20*c^5) - (2*a*e*x)/(5*c^4) + (2*a*e*atan(c*x))/(5*c^
5) + (b*d*x^5*atan(c*x))/5 - (2*b*e*x^5*atan(c*x))/25 - (b*d*log(c^2*x^2 + 1))/(10*c^5) + (137*b*e*log(c^2*x^2
+ 1))/(300*c^5) + (2*a*e*x^3)/(15*c^2) - (b*d*x^4)/(20*c) + (b*d*x^2)/(10*c^3) + (9*b*e*x^4)/(200*c) - (77*b*
e*x^2)/(300*c^3) + (a*e*x^5*log(c^2*x^2 + 1))/5 + (b*e*atan(c*x)^2)/(5*c^5) + (2*b*e*x^3*atan(c*x))/(15*c^2) +
(b*e*x^5*atan(c*x)*log(c^2*x^2 + 1))/5 - (b*e*x^4*log(c^2*x^2 + 1))/(20*c) + (b*e*x^2*log(c^2*x^2 + 1))/(10*c
^3) - (2*b*e*x*atan(c*x))/(5*c^4)