Integrand size = 16, antiderivative size = 246 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=-\frac {b e \arctan \left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d^2 \arctan \left (c x^3\right )}{2 e}+\frac {(d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right )}{2 e}+\frac {b e \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b e \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+\frac {\sqrt {3} b e \text {arctanh}\left (\frac {\sqrt {3} \sqrt [3]{c} x}{1+c^{2/3} x^2}\right )}{4 c^{2/3}}+\frac {b d \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b d \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}} \] Output:
-1/2*b*e*arctan(c^(1/3)*x)/c^(2/3)-1/2*b*d^2*arctan(c*x^3)/e+1/2*(e*x+d)^2 *(a+b*arctan(c*x^3))/e-1/4*b*e*arctan(-3^(1/2)+2*c^(1/3)*x)/c^(2/3)-1/4*b* e*arctan(3^(1/2)+2*c^(1/3)*x)/c^(2/3)+1/2*3^(1/2)*b*d*arctan(1/3*(1-2*c^(2 /3)*x^2)*3^(1/2))/c^(1/3)+1/4*3^(1/2)*b*e*arctanh(3^(1/2)*c^(1/3)*x/(1+c^( 2/3)*x^2))/c^(2/3)+1/2*b*d*ln(1+c^(2/3)*x^2)/c^(1/3)-1/4*b*d*ln(1-c^(2/3)* x^2+c^(4/3)*x^4)/c^(1/3)
Time = 0.07 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.26 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=a d x+\frac {1}{2} a e x^2-\frac {b e \arctan \left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+b d x \arctan \left (c x^3\right )+\frac {1}{2} b e x^2 \arctan \left (c x^3\right )+\frac {b e \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b e \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {\sqrt {3} b e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac {\sqrt {3} b e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b d \left (-2 \sqrt {3} \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )-2 \sqrt {3} \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )-2 \log \left (1+c^{2/3} x^2\right )+\log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )\right )}{4 \sqrt [3]{c}} \] Input:
Integrate[(d + e*x)*(a + b*ArcTan[c*x^3]),x]
Output:
a*d*x + (a*e*x^2)/2 - (b*e*ArcTan[c^(1/3)*x])/(2*c^(2/3)) + b*d*x*ArcTan[c *x^3] + (b*e*x^2*ArcTan[c*x^3])/2 + (b*e*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(4 *c^(2/3)) - (b*e*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(4*c^(2/3)) - (Sqrt[3]*b*e *Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) + (Sqrt[3]*b*e*Log[ 1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*d*(-2*Sqrt[3]*ArcTa n[Sqrt[3] - 2*c^(1/3)*x] - 2*Sqrt[3]*ArcTan[Sqrt[3] + 2*c^(1/3)*x] - 2*Log [1 + c^(2/3)*x^2] + Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] + Log[1 + Sqr t[3]*c^(1/3)*x + c^(2/3)*x^2]))/(4*c^(1/3))
Time = 0.78 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.21, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5395, 2370, 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.
| \(\displaystyle \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx\) |
| \(\Big \downarrow \) 5395 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right )}{2 e}-\frac {3 b c \int \frac {x^2 (d+e x)^2}{c^2 x^6+1}dx}{2 e}\) |
| \(\Big \downarrow \) 2370 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right )}{2 e}-\frac {3 b c \int \left (\frac {e^2 x^4}{c^2 x^6+1}+\frac {2 d e x^3}{c^2 x^6+1}+\frac {d^2 x^2}{c^2 x^6+1}\right )dx}{2 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right )}{2 e}-\frac {3 b c \left (-\frac {d e \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{\sqrt {3} c^{4/3}}+\frac {e^2 \arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}-\frac {e^2 \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{6 c^{5/3}}+\frac {e^2 \arctan \left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{6 c^{5/3}}+\frac {d^2 \arctan \left (c x^3\right )}{3 c}-\frac {d e \log \left (c^{2/3} x^2+1\right )}{3 c^{4/3}}+\frac {d e \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{6 c^{4/3}}+\frac {e^2 \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{4 \sqrt {3} c^{5/3}}-\frac {e^2 \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{4 \sqrt {3} c^{5/3}}\right )}{2 e}\) |
Input:
Int[(d + e*x)*(a + b*ArcTan[c*x^3]),x]
Output:
((d + e*x)^2*(a + b*ArcTan[c*x^3]))/(2*e) - (3*b*c*((e^2*ArcTan[c^(1/3)*x] )/(3*c^(5/3)) + (d^2*ArcTan[c*x^3])/(3*c) - (e^2*ArcTan[Sqrt[3] - 2*c^(1/3 )*x])/(6*c^(5/3)) + (e^2*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(6*c^(5/3)) - (d*e *ArcTan[(1 - 2*c^(2/3)*x^2)/Sqrt[3]])/(Sqrt[3]*c^(4/3)) - (d*e*Log[1 + c^( 2/3)*x^2])/(3*c^(4/3)) + (e^2*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(4 *Sqrt[3]*c^(5/3)) - (e^2*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(4*Sqrt [3]*c^(5/3)) + (d*e*Log[1 - c^(2/3)*x^2 + c^(4/3)*x^4])/(6*c^(4/3))))/(2*e )
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[ {v = Sum[(c*x)^(m + ii)*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2) )/(c^ii*(a + b*x^n))), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{ a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))*((d_) + (e_.)*(x_))^(m_.), x_Sy mbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcTan[c*x^n])/(e*(m + 1))), x] - S imp[b*c*(n/(e*(m + 1))) Int[x^(n - 1)*((d + e*x)^(m + 1)/(1 + c^2*x^(2*n) )), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
Time = 0.48 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.24
| method | result | size |
| default | \(a \left (\frac {1}{2} e \,x^{2}+d x \right )+b \left (\frac {\arctan \left (c \,x^{3}\right ) x^{2} e}{2}+\arctan \left (c \,x^{3}\right ) d x -\frac {3 c \left (\frac {c^{2} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {5}{3}} d}{6}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} e}{12}+\frac {c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {5}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, d}{3}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) e}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} e}{12}+\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d}{6}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) e}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d}{3}-\frac {\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{3}+\frac {e \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{3 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{2}\right )\) | \(305\) |
| parts | \(a \left (\frac {1}{2} e \,x^{2}+d x \right )+b \left (\frac {\arctan \left (c \,x^{3}\right ) x^{2} e}{2}+\arctan \left (c \,x^{3}\right ) d x -\frac {3 c \left (\frac {c^{2} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {5}{3}} d}{6}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} e}{12}+\frac {c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {5}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, d}{3}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) e}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} e}{12}+\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d}{6}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) e}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d}{3}-\frac {\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{3}+\frac {e \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{3 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{2}\right )\) | \(305\) |
Input:
int((e*x+d)*(a+b*arctan(c*x^3)),x,method=_RETURNVERBOSE)
Output:
a*(1/2*e*x^2+d*x)+b*(1/2*arctan(c*x^3)*x^2*e+arctan(c*x^3)*d*x-3/2*c*(1/6* c^2*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(5/3)*d+1/12*ln( x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*3^(1/2)*(1/c^2)^(5/6)*e+1/3*c^2 *(1/c^2)^(5/3)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*3^(1/2)*d+1/6/c^2/(1/c^2) ^(1/6)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*e-1/12*ln(x^2+3^(1/2)*(1/c^2)^(1/ 6)*x+(1/c^2)^(1/3))*3^(1/2)*(1/c^2)^(5/6)*e+1/6*ln(x^2+3^(1/2)*(1/c^2)^(1/ 6)*x+(1/c^2)^(1/3))*(1/c^2)^(2/3)*d+1/6/c^2/(1/c^2)^(1/6)*arctan(2*x/(1/c^ 2)^(1/6)+3^(1/2))*e-1/3*(1/c^2)^(2/3)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*3^ (1/2)*d-1/3*(1/c^2)^(2/3)*d*ln(x^2+(1/c^2)^(1/3))+1/3/c^2*e/(1/c^2)^(1/6)* arctan(x/(1/c^2)^(1/6))))
Exception generated. \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*x+d)*(a+b*arctan(c*x^3)),x, algorithm="fricas")
Output:
Exception raised: RuntimeError >> no explicit roots found
Time = 12.41 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.42 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=a d x + \frac {a e x^{2}}{2} - 3 b c d \operatorname {RootSum} {\left (216 t^{3} c^{4} + 1, \left ( t \mapsto t \log {\left (36 t^{2} c^{2} + x^{2} \right )} \right )\right )} - \frac {3 b c e \operatorname {RootSum} {\left (46656 t^{6} c^{10} + 1, \left ( t \mapsto t \log {\left (7776 t^{5} c^{8} + x \right )} \right )\right )}}{2} + b d x \operatorname {atan}{\left (c x^{3} \right )} + \frac {b e x^{2} \operatorname {atan}{\left (c x^{3} \right )}}{2} \] Input:
integrate((e*x+d)*(a+b*atan(c*x**3)),x)
Output:
a*d*x + a*e*x**2/2 - 3*b*c*d*RootSum(216*_t**3*c**4 + 1, Lambda(_t, _t*log (36*_t**2*c**2 + x**2))) - 3*b*c*e*RootSum(46656*_t**6*c**10 + 1, Lambda(_ t, _t*log(7776*_t**5*c**8 + x)))/2 + b*d*x*atan(c*x**3) + b*e*x**2*atan(c* x**3)/2
Time = 0.12 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.94 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{2} \, a e x^{2} - \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} - c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} + \frac {\log \left (c^{\frac {4}{3}} x^{4} - c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {4}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} + 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b d + \frac {1}{8} \, {\left (4 \, x^{2} \arctan \left (c x^{3}\right ) + c {\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )}\right )} b e + a d x \] Input:
integrate((e*x+d)*(a+b*arctan(c*x^3)),x, algorithm="maxima")
Output:
1/2*a*e*x^2 - 1/4*(c*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(4/3)*x^2 - c^(2/3 ))/c^(2/3))/c^(4/3) + log(c^(4/3)*x^4 - c^(2/3)*x^2 + 1)/c^(4/3) - 2*log(( c^(2/3)*x^2 + 1)/c^(2/3))/c^(4/3)) - 4*x*arctan(c*x^3))*b*d + 1/8*(4*x^2*a rctan(c*x^3) + c*(sqrt(3)*log(c^(2/3)*x^2 + sqrt(3)*c^(1/3)*x + 1)/c^(5/3) - sqrt(3)*log(c^(2/3)*x^2 - sqrt(3)*c^(1/3)*x + 1)/c^(5/3) - 4*arctan(c^( 1/3)*x)/c^(5/3) - 2*arctan((2*c^(2/3)*x + sqrt(3)*c^(1/3))/c^(1/3))/c^(5/3 ) - 2*arctan((2*c^(2/3)*x - sqrt(3)*c^(1/3))/c^(1/3))/c^(5/3)))*b*e + a*d* x
Time = 0.69 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.96 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{2} \, b e x^{2} \arctan \left (c x^{3}\right ) + \frac {1}{2} \, a e x^{2} + b d x \arctan \left (c x^{3}\right ) + a d x + \frac {b c d \log \left (x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{2 \, {\left | c \right |}^{\frac {4}{3}}} - \frac {b c e \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{2 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {{\left (2 \, \sqrt {3} b c d {\left | c \right |}^{\frac {1}{3}} - b c e\right )} \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {{\left (2 \, \sqrt {3} b c d {\left | c \right |}^{\frac {1}{3}} + b c e\right )} \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {{\left (\sqrt {3} b c e - 2 \, b c d {\left | c \right |}^{\frac {1}{3}}\right )} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {{\left (\sqrt {3} b c e + 2 \, b c d {\left | c \right |}^{\frac {1}{3}}\right )} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} \] Input:
integrate((e*x+d)*(a+b*arctan(c*x^3)),x, algorithm="giac")
Output:
1/2*b*e*x^2*arctan(c*x^3) + 1/2*a*e*x^2 + b*d*x*arctan(c*x^3) + a*d*x + 1/ 2*b*c*d*log(x^2 + 1/abs(c)^(2/3))/abs(c)^(4/3) - 1/2*b*c*e*arctan(x*abs(c) ^(1/3))/abs(c)^(5/3) + 1/4*(2*sqrt(3)*b*c*d*abs(c)^(1/3) - b*c*e)*arctan(( 2*x + sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/abs(c)^(5/3) - 1/4*(2*sqrt(3)*b* c*d*abs(c)^(1/3) + b*c*e)*arctan((2*x - sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3) )/abs(c)^(5/3) + 1/8*(sqrt(3)*b*c*e - 2*b*c*d*abs(c)^(1/3))*log(x^2 + sqrt (3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/abs(c)^(5/3) - 1/8*(sqrt(3)*b*c*e + 2 *b*c*d*abs(c)^(1/3))*log(x^2 - sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/ab s(c)^(5/3)
Time = 0.68 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.97 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\mathrm {atan}\left (c\,x^3\right )\,\left (\frac {b\,e\,x^2}{2}+b\,d\,x\right )+\left (\sum _{k=1}^6\ln \left (-\mathrm {root}\left (4096\,a^6\,c^4-1024\,a^3\,b^3\,c^3\,d^3+576\,a^2\,b^4\,c^2\,d^2\,e^2-48\,a\,b^5\,c\,d\,e^4+64\,b^6\,c^2\,d^6+b^6\,e^6,a,k\right )\,\left (\mathrm {root}\left (4096\,a^6\,c^4-1024\,a^3\,b^3\,c^3\,d^3+576\,a^2\,b^4\,c^2\,d^2\,e^2-48\,a\,b^5\,c\,d\,e^4+64\,b^6\,c^2\,d^6+b^6\,e^6,a,k\right )\,\left (\mathrm {root}\left (4096\,a^6\,c^4-1024\,a^3\,b^3\,c^3\,d^3+576\,a^2\,b^4\,c^2\,d^2\,e^2-48\,a\,b^5\,c\,d\,e^4+64\,b^6\,c^2\,d^6+b^6\,e^6,a,k\right )\,\left (-486\,b^2\,c^{10}\,e^2\,x+1944\,b^2\,c^{10}\,d\,e+\mathrm {root}\left (4096\,a^6\,c^4-1024\,a^3\,b^3\,c^3\,d^3+576\,a^2\,b^4\,c^2\,d^2\,e^2-48\,a\,b^5\,c\,d\,e^4+64\,b^6\,c^2\,d^6+b^6\,e^6,a,k\right )\,b\,c^{11}\,d\,x\,3888\right )-\frac {243\,b^3\,c^9\,e^3}{2}\right )-486\,b^4\,c^{10}\,d^4\,x\right )-\frac {243\,b^5\,c^9\,d^4\,e}{2}-\frac {243\,b^5\,c^9\,d^3\,e^2\,x}{4}\right )\,\mathrm {root}\left (4096\,a^6\,c^4-1024\,a^3\,b^3\,c^3\,d^3+576\,a^2\,b^4\,c^2\,d^2\,e^2-48\,a\,b^5\,c\,d\,e^4+64\,b^6\,c^2\,d^6+b^6\,e^6,a,k\right )\right )+a\,d\,x+\frac {a\,e\,x^2}{2} \] Input:
int((a + b*atan(c*x^3))*(d + e*x),x)
Output:
atan(c*x^3)*(b*d*x + (b*e*x^2)/2) + symsum(log(- root(4096*a^6*c^4 - 1024* a^3*b^3*c^3*d^3 + 576*a^2*b^4*c^2*d^2*e^2 - 48*a*b^5*c*d*e^4 + 64*b^6*c^2* d^6 + b^6*e^6, a, k)*(root(4096*a^6*c^4 - 1024*a^3*b^3*c^3*d^3 + 576*a^2*b ^4*c^2*d^2*e^2 - 48*a*b^5*c*d*e^4 + 64*b^6*c^2*d^6 + b^6*e^6, a, k)*(root( 4096*a^6*c^4 - 1024*a^3*b^3*c^3*d^3 + 576*a^2*b^4*c^2*d^2*e^2 - 48*a*b^5*c *d*e^4 + 64*b^6*c^2*d^6 + b^6*e^6, a, k)*(1944*b^2*c^10*d*e - 486*b^2*c^10 *e^2*x + 3888*root(4096*a^6*c^4 - 1024*a^3*b^3*c^3*d^3 + 576*a^2*b^4*c^2*d ^2*e^2 - 48*a*b^5*c*d*e^4 + 64*b^6*c^2*d^6 + b^6*e^6, a, k)*b*c^11*d*x) - (243*b^3*c^9*e^3)/2) - 486*b^4*c^10*d^4*x) - (243*b^5*c^9*d^4*e)/2 - (243* b^5*c^9*d^3*e^2*x)/4)*root(4096*a^6*c^4 - 1024*a^3*b^3*c^3*d^3 + 576*a^2*b ^4*c^2*d^2*e^2 - 48*a*b^5*c*d*e^4 + 64*b^6*c^2*d^6 + b^6*e^6, a, k), k, 1, 6) + a*d*x + (a*e*x^2)/2
Time = 0.18 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.96 \[ \int (d+e x) \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {-8 \sqrt {3}\, \mathit {atan} \left (2 c^{\frac {1}{3}} x -\sqrt {3}\right ) b c d -6 c^{\frac {2}{3}} \mathit {atan} \left (c^{\frac {1}{3}} x \right ) b e +4 \sqrt {3}\, \mathit {atan} \left (c^{\frac {1}{3}} x \right ) b c d -2 c^{\frac {2}{3}} \mathit {atan} \left (c \,x^{3}\right ) b e +8 c^{\frac {4}{3}} \mathit {atan} \left (c \,x^{3}\right ) b d x +4 c^{\frac {4}{3}} \mathit {atan} \left (c \,x^{3}\right ) b e \,x^{2}+4 \sqrt {3}\, \mathit {atan} \left (c \,x^{3}\right ) b c d -c^{\frac {2}{3}} \sqrt {3}\, \mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b e +c^{\frac {2}{3}} \sqrt {3}\, \mathrm {log}\left (c^{\frac {2}{3}} x^{2}+c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b e +8 c^{\frac {4}{3}} a d x +4 c^{\frac {4}{3}} a e \,x^{2}-2 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b c d -2 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}+c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b c d +4 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}+1\right ) b c d}{8 c^{\frac {4}{3}}} \] Input:
int((e*x+d)*(a+b*atan(c*x^3)),x)
Output:
( - 8*sqrt(3)*atan(2*c**(1/3)*x - sqrt(3))*b*c*d - 6*c**(2/3)*atan(c**(1/3 )*x)*b*e + 4*sqrt(3)*atan(c**(1/3)*x)*b*c*d - 2*c**(2/3)*atan(c*x**3)*b*e + 8*c**(1/3)*atan(c*x**3)*b*c*d*x + 4*c**(1/3)*atan(c*x**3)*b*c*e*x**2 + 4 *sqrt(3)*atan(c*x**3)*b*c*d - c**(2/3)*sqrt(3)*log(c**(2/3)*x**2 - c**(1/3 )*sqrt(3)*x + 1)*b*e + c**(2/3)*sqrt(3)*log(c**(2/3)*x**2 + c**(1/3)*sqrt( 3)*x + 1)*b*e + 8*c**(1/3)*a*c*d*x + 4*c**(1/3)*a*c*e*x**2 - 2*log(c**(2/3 )*x**2 - c**(1/3)*sqrt(3)*x + 1)*b*c*d - 2*log(c**(2/3)*x**2 + c**(1/3)*sq rt(3)*x + 1)*b*c*d + 4*log(c**(2/3)*x**2 + 1)*b*c*d)/(8*c**(1/3)*c)