Integrand size = 12, antiderivative size = 88 \[ \int x^3 \arctan (x) \log \left (1+x^2\right ) \, dx=-\frac {25 x}{24}+\frac {7 x^3}{72}+\frac {25 \arctan (x)}{24}+\frac {1}{4} x^2 \arctan (x)-\frac {1}{8} x^4 \arctan (x)+\frac {1}{4} x \log \left (1+x^2\right )-\frac {1}{12} x^3 \log \left (1+x^2\right )-\frac {1}{4} \arctan (x) \log \left (1+x^2\right )+\frac {1}{4} x^4 \arctan (x) \log \left (1+x^2\right ) \] Output:
-25/24*x+7/72*x^3+25/24*arctan(x)+1/4*x^2*arctan(x)-1/8*x^4*arctan(x)+1/4* x*ln(x^2+1)-1/12*x^3*ln(x^2+1)-1/4*arctan(x)*ln(x^2+1)+1/4*x^4*arctan(x)*l n(x^2+1)
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int x^3 \arctan (x) \log \left (1+x^2\right ) \, dx=\frac {1}{72} \left (x \left (-75+7 x^2-6 \left (-3+x^2\right ) \log \left (1+x^2\right )\right )+3 \arctan (x) \left (25+6 x^2-3 x^4+6 \left (-1+x^4\right ) \log \left (1+x^2\right )\right )\right ) \] Input:
Integrate[x^3*ArcTan[x]*Log[1 + x^2],x]
Output:
(x*(-75 + 7*x^2 - 6*(-3 + x^2)*Log[1 + x^2]) + 3*ArcTan[x]*(25 + 6*x^2 - 3 *x^4 + 6*(-1 + x^4)*Log[1 + x^2]))/72
Time = 0.33 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5554, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^3 \arctan (x) \log \left (x^2+1\right ) \, dx\) |
\(\Big \downarrow \) 5554 |
\(\displaystyle -\int \left (\frac {x^2 \left (2-x^2\right )}{8 \left (x^2+1\right )}-\frac {1}{4} \left (1-x^2\right ) \log \left (x^2+1\right )\right )dx-\frac {1}{8} x^4 \arctan (x)+\frac {1}{4} x^2 \arctan (x)-\frac {1}{4} \arctan (x) \log \left (x^2+1\right )+\frac {1}{4} x^4 \arctan (x) \log \left (x^2+1\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{8} x^4 \arctan (x)+\frac {1}{4} x^2 \arctan (x)-\frac {1}{4} \arctan (x) \log \left (x^2+1\right )+\frac {1}{4} x^4 \arctan (x) \log \left (x^2+1\right )+\frac {25 \arctan (x)}{24}+\frac {7 x^3}{72}+\frac {1}{4} x \log \left (x^2+1\right )-\frac {1}{12} x^3 \log \left (x^2+1\right )-\frac {25 x}{24}\) |
Input:
Int[x^3*ArcTan[x]*Log[1 + x^2],x]
Output:
(-25*x)/24 + (7*x^3)/72 + (25*ArcTan[x])/24 + (x^2*ArcTan[x])/4 - (x^4*Arc Tan[x])/8 + (x*Log[1 + x^2])/4 - (x^3*Log[1 + x^2])/12 - (ArcTan[x]*Log[1 + x^2])/4 + (x^4*ArcTan[x]*Log[1 + x^2])/4
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*( e_.))*(x_)^(m_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]) , x]}, Simp[(a + b*ArcTan[c*x]) u, x] - Simp[b*c Int[ExpandIntegrand[u/ (1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
Time = 2.50 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.81
method | result | size |
parallelrisch | \(-\frac {25 x}{24}+\frac {7 x^{3}}{72}+\frac {25 \arctan \left (x \right )}{24}+\frac {x^{2} \arctan \left (x \right )}{4}-\frac {x^{4} \arctan \left (x \right )}{8}+\frac {x \ln \left (x^{2}+1\right )}{4}-\frac {x^{3} \ln \left (x^{2}+1\right )}{12}-\frac {\arctan \left (x \right ) \ln \left (x^{2}+1\right )}{4}+\frac {x^{4} \arctan \left (x \right ) \ln \left (x^{2}+1\right )}{4}\) | \(71\) |
orering | \(\frac {\left (9 x^{8}-76 x^{6}-497 x^{4}-1570 x^{2}-1350\right ) \arctan \left (x \right ) \ln \left (x^{2}+1\right )}{12 x^{2} \left (x^{2}-3\right )}-\frac {\left (19 x^{8}-274 x^{6}-2162 x^{4}-6210 x^{2}-4725\right ) \left (3 x^{2} \arctan \left (x \right ) \ln \left (x^{2}+1\right )+\frac {x^{3} \ln \left (x^{2}+1\right )}{x^{2}+1}+\frac {2 x^{4} \arctan \left (x \right )}{x^{2}+1}\right )}{72 x^{4} \left (x^{2}-3\right )}+\frac {\left (4 x^{6}-89 x^{4}-642 x^{2}-1125\right ) \left (x^{2}+1\right ) \left (6 x \arctan \left (x \right ) \ln \left (x^{2}+1\right )+\frac {6 x^{2} \ln \left (x^{2}+1\right )}{x^{2}+1}+\frac {14 x^{3} \arctan \left (x \right )}{x^{2}+1}-\frac {2 x^{4} \ln \left (x^{2}+1\right )}{\left (x^{2}+1\right )^{2}}+\frac {4 x^{4}}{\left (x^{2}+1\right )^{2}}-\frac {4 x^{5} \arctan \left (x \right )}{\left (x^{2}+1\right )^{2}}\right )}{72 x^{3} \left (x^{2}-3\right )}-\frac {\left (x^{4}-32 x^{2}-225\right ) \left (x^{2}+1\right )^{2} \left (6 \arctan \left (x \right ) \ln \left (x^{2}+1\right )+\frac {18 \ln \left (x^{2}+1\right ) x}{x^{2}+1}+\frac {54 x^{2} \arctan \left (x \right )}{x^{2}+1}-\frac {20 x^{3} \ln \left (x^{2}+1\right )}{\left (x^{2}+1\right )^{2}}+\frac {42 x^{3}}{\left (x^{2}+1\right )^{2}}-\frac {48 x^{4} \arctan \left (x \right )}{\left (x^{2}+1\right )^{2}}+\frac {8 x^{5} \ln \left (x^{2}+1\right )}{\left (x^{2}+1\right )^{3}}-\frac {24 x^{5}}{\left (x^{2}+1\right )^{3}}+\frac {16 x^{6} \arctan \left (x \right )}{\left (x^{2}+1\right )^{3}}\right )}{144 x^{2} \left (x^{2}-3\right )}\) | \(403\) |
default | \(\text {Expression too large to display}\) | \(2834\) |
risch | \(\text {Expression too large to display}\) | \(16521\) |
Input:
int(x^3*arctan(x)*ln(x^2+1),x,method=_RETURNVERBOSE)
Output:
-25/24*x+7/72*x^3+25/24*arctan(x)+1/4*x^2*arctan(x)-1/8*x^4*arctan(x)+1/4* x*ln(x^2+1)-1/12*x^3*ln(x^2+1)-1/4*arctan(x)*ln(x^2+1)+1/4*x^4*arctan(x)*l n(x^2+1)
Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.56 \[ \int x^3 \arctan (x) \log \left (1+x^2\right ) \, dx=\frac {7}{72} \, x^{3} - \frac {1}{24} \, {\left (3 \, x^{4} - 6 \, x^{2} - 25\right )} \arctan \left (x\right ) - \frac {1}{12} \, {\left (x^{3} - 3 \, {\left (x^{4} - 1\right )} \arctan \left (x\right ) - 3 \, x\right )} \log \left (x^{2} + 1\right ) - \frac {25}{24} \, x \] Input:
integrate(x^3*arctan(x)*log(x^2+1),x, algorithm="fricas")
Output:
7/72*x^3 - 1/24*(3*x^4 - 6*x^2 - 25)*arctan(x) - 1/12*(x^3 - 3*(x^4 - 1)*a rctan(x) - 3*x)*log(x^2 + 1) - 25/24*x
Time = 0.57 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.94 \[ \int x^3 \arctan (x) \log \left (1+x^2\right ) \, dx=\frac {x^{4} \log {\left (x^{2} + 1 \right )} \operatorname {atan}{\left (x \right )}}{4} - \frac {x^{4} \operatorname {atan}{\left (x \right )}}{8} - \frac {x^{3} \log {\left (x^{2} + 1 \right )}}{12} + \frac {7 x^{3}}{72} + \frac {x^{2} \operatorname {atan}{\left (x \right )}}{4} + \frac {x \log {\left (x^{2} + 1 \right )}}{4} - \frac {25 x}{24} - \frac {\log {\left (x^{2} + 1 \right )} \operatorname {atan}{\left (x \right )}}{4} + \frac {25 \operatorname {atan}{\left (x \right )}}{24} \] Input:
integrate(x**3*atan(x)*ln(x**2+1),x)
Output:
x**4*log(x**2 + 1)*atan(x)/4 - x**4*atan(x)/8 - x**3*log(x**2 + 1)/12 + 7* x**3/72 + x**2*atan(x)/4 + x*log(x**2 + 1)/4 - 25*x/24 - log(x**2 + 1)*ata n(x)/4 + 25*atan(x)/24
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.70 \[ \int x^3 \arctan (x) \log \left (1+x^2\right ) \, dx=\frac {7}{72} \, x^{3} + \frac {1}{8} \, {\left (2 \, x^{4} \log \left (x^{2} + 1\right ) - x^{4} + 2 \, x^{2} - 2 \, \log \left (x^{2} + 1\right )\right )} \arctan \left (x\right ) - \frac {1}{12} \, {\left (x^{3} - 3 \, x\right )} \log \left (x^{2} + 1\right ) - \frac {25}{24} \, x + \frac {25}{24} \, \arctan \left (x\right ) \] Input:
integrate(x^3*arctan(x)*log(x^2+1),x, algorithm="maxima")
Output:
7/72*x^3 + 1/8*(2*x^4*log(x^2 + 1) - x^4 + 2*x^2 - 2*log(x^2 + 1))*arctan( x) - 1/12*(x^3 - 3*x)*log(x^2 + 1) - 25/24*x + 25/24*arctan(x)
Time = 0.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.41 \[ \int x^3 \arctan (x) \log \left (1+x^2\right ) \, dx=\frac {1}{8} \, \pi x^{4} \log \left (x^{2} + 1\right ) \mathrm {sgn}\left (x\right ) - \frac {1}{4} \, x^{4} \arctan \left (\frac {1}{x}\right ) \log \left (x^{2} + 1\right ) - \frac {1}{16} \, \pi x^{4} \mathrm {sgn}\left (x\right ) + \frac {1}{8} \, x^{4} \arctan \left (\frac {1}{x}\right ) - \frac {1}{12} \, x^{3} \log \left (x^{2} + 1\right ) + \frac {1}{8} \, \pi x^{2} \mathrm {sgn}\left (x\right ) + \frac {7}{72} \, x^{3} - \frac {1}{4} \, x^{2} \arctan \left (\frac {1}{x}\right ) - \frac {1}{8} \, \pi \log \left (x^{2} + 1\right ) \mathrm {sgn}\left (x\right ) + \frac {1}{4} \, x \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \arctan \left (\frac {1}{x}\right ) \log \left (x^{2} + 1\right ) - \frac {25}{24} \, \pi \mathrm {sgn}\left (x\right ) - \frac {25}{24} \, x + \frac {25}{24} \, \arctan \left (x\right ) \] Input:
integrate(x^3*arctan(x)*log(x^2+1),x, algorithm="giac")
Output:
1/8*pi*x^4*log(x^2 + 1)*sgn(x) - 1/4*x^4*arctan(1/x)*log(x^2 + 1) - 1/16*p i*x^4*sgn(x) + 1/8*x^4*arctan(1/x) - 1/12*x^3*log(x^2 + 1) + 1/8*pi*x^2*sg n(x) + 7/72*x^3 - 1/4*x^2*arctan(1/x) - 1/8*pi*log(x^2 + 1)*sgn(x) + 1/4*x *log(x^2 + 1) + 1/4*arctan(1/x)*log(x^2 + 1) - 25/24*pi*sgn(x) - 25/24*x + 25/24*arctan(x)
Time = 0.94 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int x^3 \arctan (x) \log \left (1+x^2\right ) \, dx=\frac {25\,\mathrm {atan}\left (x\right )}{24}+\frac {x^2\,\mathrm {atan}\left (x\right )}{4}+x\,\left (\frac {\ln \left (x^2+1\right )}{4}-\frac {25}{24}\right )-x^3\,\left (\frac {\ln \left (x^2+1\right )}{12}-\frac {7}{72}\right )-x^4\,\left (\frac {\mathrm {atan}\left (x\right )}{8}-\frac {\ln \left (x^2+1\right )\,\mathrm {atan}\left (x\right )}{4}\right )-\frac {\ln \left (x^2+1\right )\,\mathrm {atan}\left (x\right )}{4} \] Input:
int(x^3*log(x^2 + 1)*atan(x),x)
Output:
(25*atan(x))/24 + (x^2*atan(x))/4 + x*(log(x^2 + 1)/4 - 25/24) - x^3*(log( x^2 + 1)/12 - 7/72) - x^4*(atan(x)/8 - (log(x^2 + 1)*atan(x))/4) - (log(x^ 2 + 1)*atan(x))/4
Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int x^3 \arctan (x) \log \left (1+x^2\right ) \, dx=\frac {\mathit {atan} \left (x \right ) \mathrm {log}\left (x^{2}+1\right ) x^{4}}{4}-\frac {\mathit {atan} \left (x \right ) \mathrm {log}\left (x^{2}+1\right )}{4}-\frac {\mathit {atan} \left (x \right ) x^{4}}{8}+\frac {\mathit {atan} \left (x \right ) x^{2}}{4}+\frac {25 \mathit {atan} \left (x \right )}{24}-\frac {\mathrm {log}\left (x^{2}+1\right ) x^{3}}{12}+\frac {\mathrm {log}\left (x^{2}+1\right ) x}{4}+\frac {7 x^{3}}{72}-\frac {25 x}{24} \] Input:
int(x^3*atan(x)*log(x^2+1),x)
Output:
(18*atan(x)*log(x**2 + 1)*x**4 - 18*atan(x)*log(x**2 + 1) - 9*atan(x)*x**4 + 18*atan(x)*x**2 + 75*atan(x) - 6*log(x**2 + 1)*x**3 + 18*log(x**2 + 1)* x + 7*x**3 - 75*x)/72