Integrand size = 18, antiderivative size = 300 \[ \int (d+e x)^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=-\frac {\sqrt {3} b d e \arctan \left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 c^{2/3}}+\frac {\sqrt {3} b d e \arctan \left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 c^{2/3}}+\frac {\sqrt {3} b d^2 \arctan \left (\frac {1+2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {b d e \text {arctanh}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+\frac {(d+e x)^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )}{3 e}-\frac {b d e \text {arctanh}\left (\frac {\sqrt [3]{c} x}{1+c^{2/3} x^2}\right )}{2 c^{2/3}}+\frac {b d^2 \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}+\frac {b \left (c d^3+e^3\right ) \log \left (1-c x^3\right )}{6 c e}-\frac {b \left (c d^3-e^3\right ) \log \left (1+c x^3\right )}{6 c e}-\frac {b d^2 \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}} \] Output:
-1/2*3^(1/2)*b*d*e*arctan(1/3*(1-2*c^(1/3)*x)*3^(1/2))/c^(2/3)+1/2*3^(1/2) *b*d*e*arctan(1/3*(1+2*c^(1/3)*x)*3^(1/2))/c^(2/3)+1/2*3^(1/2)*b*d^2*arcta n(1/3*(1+2*c^(2/3)*x^2)*3^(1/2))/c^(1/3)-b*d*e*arctanh(c^(1/3)*x)/c^(2/3)+ 1/3*(e*x+d)^3*(a+b*arctanh(c*x^3))/e-1/2*b*d*e*arctanh(c^(1/3)*x/(1+c^(2/3 )*x^2))/c^(2/3)+1/2*b*d^2*ln(1-c^(2/3)*x^2)/c^(1/3)+1/6*b*(c*d^3+e^3)*ln(- c*x^3+1)/c/e-1/6*b*(c*d^3-e^3)*ln(c*x^3+1)/c/e-1/4*b*d^2*ln(1+c^(2/3)*x^2+ c^(4/3)*x^4)/c^(1/3)
Time = 0.14 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {12 a c d^2 x+12 a c d e x^2+4 a c e^2 x^3+6 \sqrt {3} b \sqrt [3]{c} d \left (\sqrt [3]{c} d+e\right ) \arctan \left (\frac {-1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )-6 \sqrt {3} b \sqrt [3]{c} d \left (\sqrt [3]{c} d-e\right ) \arctan \left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )+4 b c x \left (3 d^2+3 d e x+e^2 x^2\right ) \text {arctanh}\left (c x^3\right )+6 b \sqrt [3]{c} d \left (\sqrt [3]{c} d+e\right ) \log \left (1-\sqrt [3]{c} x\right )+6 b \sqrt [3]{c} d \left (\sqrt [3]{c} d-e\right ) \log \left (1+\sqrt [3]{c} x\right )-3 b \sqrt [3]{c} d \left (\sqrt [3]{c} d-e\right ) \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )-3 b \sqrt [3]{c} d \left (\sqrt [3]{c} d+e\right ) \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )+2 b e^2 \log \left (1-c^2 x^6\right )}{12 c} \] Input:
Integrate[(d + e*x)^2*(a + b*ArcTanh[c*x^3]),x]
Output:
(12*a*c*d^2*x + 12*a*c*d*e*x^2 + 4*a*c*e^2*x^3 + 6*Sqrt[3]*b*c^(1/3)*d*(c^ (1/3)*d + e)*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]] - 6*Sqrt[3]*b*c^(1/3)*d*(c ^(1/3)*d - e)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]] + 4*b*c*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTanh[c*x^3] + 6*b*c^(1/3)*d*(c^(1/3)*d + e)*Log[1 - c^(1/3) *x] + 6*b*c^(1/3)*d*(c^(1/3)*d - e)*Log[1 + c^(1/3)*x] - 3*b*c^(1/3)*d*(c^ (1/3)*d - e)*Log[1 - c^(1/3)*x + c^(2/3)*x^2] - 3*b*c^(1/3)*d*(c^(1/3)*d + e)*Log[1 + c^(1/3)*x + c^(2/3)*x^2] + 2*b*e^2*Log[1 - c^2*x^6])/(12*c)
Time = 0.75 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6486, 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.
| \(\displaystyle \int (d+e x)^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6486 |
| \(\displaystyle \frac {(d+e x)^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )}{3 e}-\frac {b c \int \frac {x^2 (d+e x)^3}{1-c^2 x^6}dx}{e}\) |
| \(\Big \downarrow \) 2370 |
| \(\displaystyle \frac {(d+e x)^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )}{3 e}-\frac {b c \int \left (\frac {3 d e^2 x^4}{1-c^2 x^6}+\frac {3 d^2 e x^3}{1-c^2 x^6}+\frac {\left (d^3+e^3 x^3\right ) x^2}{1-c^2 x^6}\right )dx}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )}{3 e}-\frac {b c \left (-\frac {\sqrt {3} d^2 e \arctan \left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right )}{2 c^{4/3}}+\frac {\sqrt {3} d e^2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 c^{5/3}}-\frac {\sqrt {3} d e^2 \arctan \left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 c^{5/3}}+\frac {d e^2 \text {arctanh}\left (\sqrt [3]{c} x\right )}{c^{5/3}}+\frac {d^3 \text {arctanh}\left (c x^3\right )}{3 c}-\frac {d^2 e \log \left (1-c^{2/3} x^2\right )}{2 c^{4/3}}+\frac {d^2 e \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )}{4 c^{4/3}}-\frac {d e^2 \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{4 c^{5/3}}+\frac {d e^2 \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{4 c^{5/3}}-\frac {e^3 \log \left (1-c^2 x^6\right )}{6 c^2}\right )}{e}\) |
Input:
Int[(d + e*x)^2*(a + b*ArcTanh[c*x^3]),x]
Output:
((d + e*x)^3*(a + b*ArcTanh[c*x^3]))/(3*e) - (b*c*((Sqrt[3]*d*e^2*ArcTan[1 /Sqrt[3] - (2*c^(1/3)*x)/Sqrt[3]])/(2*c^(5/3)) - (Sqrt[3]*d*e^2*ArcTan[1/S qrt[3] + (2*c^(1/3)*x)/Sqrt[3]])/(2*c^(5/3)) - (Sqrt[3]*d^2*e*ArcTan[(1 + 2*c^(2/3)*x^2)/Sqrt[3]])/(2*c^(4/3)) + (d*e^2*ArcTanh[c^(1/3)*x])/c^(5/3) + (d^3*ArcTanh[c*x^3])/(3*c) - (d^2*e*Log[1 - c^(2/3)*x^2])/(2*c^(4/3)) - (d*e^2*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(4*c^(5/3)) + (d*e^2*Log[1 + c^(1 /3)*x + c^(2/3)*x^2])/(4*c^(5/3)) + (d^2*e*Log[1 + c^(2/3)*x^2 + c^(4/3)*x ^4])/(4*c^(4/3)) - (e^3*Log[1 - c^2*x^6])/(6*c^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_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_ Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcTanh[c*x^n])/(e*(m + 1))), x] - Simp[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 = 0.37 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.56
| method | result | size |
| default | \(\frac {a \left (e x +d \right )^{3}}{3 e}+b \left (\frac {e^{2} \operatorname {arctanh}\left (c \,x^{3}\right ) x^{3}}{3}+e \,\operatorname {arctanh}\left (c \,x^{3}\right ) x^{2} d +\operatorname {arctanh}\left (c \,x^{3}\right ) x \,d^{2}+\frac {\operatorname {arctanh}\left (c \,x^{3}\right ) d^{3}}{3 e}-\frac {c \left (\frac {-3 d^{2} e \left (\frac {\ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}\right )-3 d \,e^{2} \left (\frac {\ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\right )+\frac {\left (-c \,d^{3}-e^{3}\right ) \ln \left (c \,x^{3}-1\right )}{3 c}}{2 c}+\frac {-3 d^{2} e \left (\frac {\ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}\right )-3 d \,e^{2} \left (-\frac {\ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\right )+\frac {\left (c \,d^{3}-e^{3}\right ) \ln \left (c \,x^{3}+1\right )}{3 c}}{2 c}\right )}{e}\right )\) | \(467\) |
| parts | \(\frac {a \left (e x +d \right )^{3}}{3 e}+b \left (\frac {e^{2} \operatorname {arctanh}\left (c \,x^{3}\right ) x^{3}}{3}+e \,\operatorname {arctanh}\left (c \,x^{3}\right ) x^{2} d +\operatorname {arctanh}\left (c \,x^{3}\right ) x \,d^{2}+\frac {\operatorname {arctanh}\left (c \,x^{3}\right ) d^{3}}{3 e}-\frac {c \left (\frac {-3 d^{2} e \left (\frac {\ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}\right )-3 d \,e^{2} \left (\frac {\ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\right )+\frac {\left (-c \,d^{3}-e^{3}\right ) \ln \left (c \,x^{3}-1\right )}{3 c}}{2 c}+\frac {-3 d^{2} e \left (\frac {\ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}\right )-3 d \,e^{2} \left (-\frac {\ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\right )+\frac {\left (c \,d^{3}-e^{3}\right ) \ln \left (c \,x^{3}+1\right )}{3 c}}{2 c}\right )}{e}\right )\) | \(467\) |
| risch | \(-\frac {b \,e^{2}}{6 c}+\frac {b \,d^{2} \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \,d^{2} \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \,d^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b e d \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b e d \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b e d \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b e d \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b e d \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b e d \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (c \,x^{3}+1\right ) d^{3}}{6 e}+\frac {b \,e^{2} \ln \left (c \,x^{3}+1\right )}{6 c}-\frac {b \,e^{2} \ln \left (-c \,x^{3}+1\right ) x^{3}}{6}+\frac {b \,e^{2} \ln \left (-c \,x^{3}+1\right )}{6 c}+\frac {\left (e x +d \right )^{3} b \ln \left (c \,x^{3}+1\right )}{6 e}+\frac {e^{2} a \,x^{3}}{3}+e a d \,x^{2}+a \,d^{2} x -\frac {b \,d^{2} x \ln \left (-c \,x^{3}+1\right )}{2}+\frac {b \,d^{2} \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \,d^{2} \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b d e \ln \left (-c \,x^{3}+1\right ) x^{2}}{2}+\frac {b \,d^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}\) | \(501\) |
Input:
int((e*x+d)^2*(a+b*arctanh(c*x^3)),x,method=_RETURNVERBOSE)
Output:
1/3*a*(e*x+d)^3/e+b*(1/3*e^2*arctanh(c*x^3)*x^3+e*arctanh(c*x^3)*x^2*d+arc tanh(c*x^3)*x*d^2+1/3/e*arctanh(c*x^3)*d^3-1/e*c*(1/2*(-3*d^2*e*(1/3/c/(1/ c)^(2/3)*ln(x-(1/c)^(1/3))-1/6/c/(1/c)^(2/3)*ln(x^2+(1/c)^(1/3)*x+(1/c)^(2 /3))-1/3/c/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x+1)))-3* d*e^2*(1/3/c/(1/c)^(1/3)*ln(x-(1/c)^(1/3))-1/6/c/(1/c)^(1/3)*ln(x^2+(1/c)^ (1/3)*x+(1/c)^(2/3))+1/3*3^(1/2)/c/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c) ^(1/3)*x+1)))+1/3*(-c*d^3-e^3)/c*ln(c*x^3-1))/c+1/2*(-3*d^2*e*(1/3/c/(1/c) ^(2/3)*ln(x+(1/c)^(1/3))-1/6/c/(1/c)^(2/3)*ln(x^2-(1/c)^(1/3)*x+(1/c)^(2/3 ))+1/3/c/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x-1)))-3*d* e^2*(-1/3/c/(1/c)^(1/3)*ln(x+(1/c)^(1/3))+1/6/c/(1/c)^(1/3)*ln(x^2-(1/c)^( 1/3)*x+(1/c)^(2/3))+1/3*3^(1/2)/c/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^ (1/3)*x-1)))+1/3*(c*d^3-e^3)/c*ln(c*x^3+1))/c))
Result contains complex when optimal does not.
Time = 0.99 (sec) , antiderivative size = 9282, normalized size of antiderivative = 30.94 \[ \int (d+e x)^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^2*(a+b*arctanh(c*x^3)),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int (d+e x)^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**2*(a+b*atanh(c*x**3)),x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.98 \[ \int (d+e x)^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} - \frac {\log \left (c^{\frac {4}{3}} x^{4} + c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {4}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} - 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}}\right )} + 4 \, x \operatorname {artanh}\left (c x^{3}\right )\right )} b d^{2} + \frac {1}{4} \, {\left (4 \, x^{2} \operatorname {artanh}\left (c x^{3}\right ) + c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {\log \left (c^{\frac {2}{3}} x^{2} + c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} + \frac {\log \left (c^{\frac {2}{3}} x^{2} - c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} x + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} x - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )}\right )} b d e + a d^{2} x + \frac {{\left (2 \, c x^{3} \operatorname {artanh}\left (c x^{3}\right ) + \log \left (-c^{2} x^{6} + 1\right )\right )} b e^{2}}{6 \, c} \] Input:
integrate((e*x+d)^2*(a+b*arctanh(c*x^3)),x, algorithm="maxima")
Output:
1/3*a*e^2*x^3 + a*d*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*arctanh(c*x^3))*b*d ^2 + 1/4*(4*x^2*arctanh(c*x^3) + c*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3 )*x + c^(1/3))/c^(1/3))/c^(5/3) + 2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3)* x - c^(1/3))/c^(1/3))/c^(5/3) - log(c^(2/3)*x^2 + c^(1/3)*x + 1)/c^(5/3) + log(c^(2/3)*x^2 - c^(1/3)*x + 1)/c^(5/3) - 2*log((c^(1/3)*x + 1)/c^(1/3)) /c^(5/3) + 2*log((c^(1/3)*x - 1)/c^(1/3))/c^(5/3)))*b*d*e + a*d^2*x + 1/6* (2*c*x^3*arctanh(c*x^3) + log(-c^2*x^6 + 1))*b*e^2/c
Time = 11.46 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.15 \[ \int (d+e x)^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + a d^{2} x + \frac {1}{6} \, {\left (b e^{2} x^{3} + 3 \, b d e x^{2} + 3 \, b d^{2} x\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) - \frac {\sqrt {3} {\left (b c d^{2} {\left | c \right |}^{\frac {2}{3}} - b c d e {\left | c \right |}^{\frac {1}{3}}\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{2 \, c^{2}} + \frac {\sqrt {3} {\left (b c d^{2} {\left | c \right |}^{\frac {2}{3}} + b c d e {\left | c \right |}^{\frac {1}{3}}\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{2 \, c^{2}} - \frac {{\left (3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} + 3 \, b c d e {\left | c \right |}^{\frac {1}{3}} - 2 \, b c e^{2}\right )} \log \left (x^{2} + \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{12 \, c^{2}} - \frac {{\left (3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} - 3 \, b c d e {\left | c \right |}^{\frac {1}{3}} - 2 \, b c e^{2}\right )} \log \left (x^{2} - \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{12 \, c^{2}} + \frac {{\left (3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} - 3 \, b c d e {\left | c \right |}^{\frac {1}{3}} + b c e^{2}\right )} \log \left ({\left | x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{6 \, c^{2}} + \frac {{\left (3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} + 3 \, b c d e {\left | c \right |}^{\frac {1}{3}} + b c e^{2}\right )} \log \left ({\left | x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{6 \, c^{2}} \] Input:
integrate((e*x+d)^2*(a+b*arctanh(c*x^3)),x, algorithm="giac")
Output:
1/3*a*e^2*x^3 + a*d*e*x^2 + a*d^2*x + 1/6*(b*e^2*x^3 + 3*b*d*e*x^2 + 3*b*d ^2*x)*log(-(c*x^3 + 1)/(c*x^3 - 1)) - 1/2*sqrt(3)*(b*c*d^2*abs(c)^(2/3) - b*c*d*e*abs(c)^(1/3))*arctan(1/3*sqrt(3)*(2*x + 1/abs(c)^(1/3))*abs(c)^(1/ 3))/c^2 + 1/2*sqrt(3)*(b*c*d^2*abs(c)^(2/3) + b*c*d*e*abs(c)^(1/3))*arctan (1/3*sqrt(3)*(2*x - 1/abs(c)^(1/3))*abs(c)^(1/3))/c^2 - 1/12*(3*b*c*d^2*ab s(c)^(2/3) + 3*b*c*d*e*abs(c)^(1/3) - 2*b*c*e^2)*log(x^2 + x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^2 - 1/12*(3*b*c*d^2*abs(c)^(2/3) - 3*b*c*d*e*abs(c)^(1 /3) - 2*b*c*e^2)*log(x^2 - x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^2 + 1/6*(3*b *c*d^2*abs(c)^(2/3) - 3*b*c*d*e*abs(c)^(1/3) + b*c*e^2)*log(abs(x + 1/abs( c)^(1/3)))/c^2 + 1/6*(3*b*c*d^2*abs(c)^(2/3) + 3*b*c*d*e*abs(c)^(1/3) + b* c*e^2)*log(abs(x - 1/abs(c)^(1/3)))/c^2
Time = 3.63 (sec) , antiderivative size = 1081, normalized size of antiderivative = 3.60 \[ \int (d+e x)^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx =\text {Too large to display} \] Input:
int((a + b*atanh(c*x^3))*(d + e*x)^2,x)
Output:
symsum(log(x*(6*b^5*c^7*d^2*e^8 + 162*b^5*c^9*d^8*e^2) + root(216*c^3*z^3 - 108*b*c^2*e^2*z^2 - 162*b^2*c^2*d^3*e*z + 18*b^2*c*e^4*z - 27*b^3*c^2*d^ 6 - b^3*e^6, z, k)*(x*(486*b^4*c^10*d^8 - 90*b^4*c^8*d^2*e^6) + root(216*c ^3*z^3 - 108*b*c^2*e^2*z^2 - 162*b^2*c^2*d^3*e*z + 18*b^2*c*e^4*z - 27*b^3 *c^2*d^6 - b^3*e^6, z, k)*(root(216*c^3*z^3 - 108*b*c^2*e^2*z^2 - 162*b^2* c^2*d^3*e*z + 18*b^2*c*e^4*z - 27*b^3*c^2*d^6 - b^3*e^6, z, k)*(3888*b^2*c ^10*d^3*e - 3888*root(216*c^3*z^3 - 108*b*c^2*e^2*z^2 - 162*b^2*c^2*d^3*e* z + 18*b^2*c*e^4*z - 27*b^3*c^2*d^6 - b^3*e^6, z, k)*b*c^11*d^2*x + 648*b^ 2*c^10*d^2*e^2*x) - 972*b^3*c^9*d^3*e^3 + 324*b^3*c^9*d^2*e^4*x)) + 243*b^ 5*c^9*d^9*e + 9*b^5*c^7*d^3*e^7)*root(216*c^3*z^3 - 108*b*c^2*e^2*z^2 - 16 2*b^2*c^2*d^3*e*z + 18*b^2*c*e^4*z - 27*b^3*c^2*d^6 - b^3*e^6, z, k), k, 1 , 3) + symsum(log(x*(6*b^5*c^7*d^2*e^8 + 162*b^5*c^9*d^8*e^2) + root(216*c ^3*z^3 - 108*b*c^2*e^2*z^2 + 162*b^2*c^2*d^3*e*z + 18*b^2*c*e^4*z - 27*b^3 *c^2*d^6 - b^3*e^6, z, k)*(x*(486*b^4*c^10*d^8 - 90*b^4*c^8*d^2*e^6) + roo t(216*c^3*z^3 - 108*b*c^2*e^2*z^2 + 162*b^2*c^2*d^3*e*z + 18*b^2*c*e^4*z - 27*b^3*c^2*d^6 - b^3*e^6, z, k)*(root(216*c^3*z^3 - 108*b*c^2*e^2*z^2 + 1 62*b^2*c^2*d^3*e*z + 18*b^2*c*e^4*z - 27*b^3*c^2*d^6 - b^3*e^6, z, k)*(388 8*b^2*c^10*d^3*e - 3888*root(216*c^3*z^3 - 108*b*c^2*e^2*z^2 + 162*b^2*c^2 *d^3*e*z + 18*b^2*c*e^4*z - 27*b^3*c^2*d^6 - b^3*e^6, z, k)*b*c^11*d^2*x + 648*b^2*c^10*d^2*e^2*x) - 972*b^3*c^9*d^3*e^3 + 324*b^3*c^9*d^2*e^4*x)...
Time = 0.18 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.23 \[ \int (d+e x)^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {6 c^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {2 c^{\frac {1}{3}} x -1}{\sqrt {3}}\right ) b \,d^{2}+6 \sqrt {3}\, \mathit {atan} \left (\frac {2 c^{\frac {1}{3}} x -1}{\sqrt {3}}\right ) b c d e -6 c^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {2 c^{\frac {1}{3}} x +1}{\sqrt {3}}\right ) b \,d^{2}+6 \sqrt {3}\, \mathit {atan} \left (\frac {2 c^{\frac {1}{3}} x +1}{\sqrt {3}}\right ) b c d e +12 c^{\frac {5}{3}} \mathit {atanh} \left (c \,x^{3}\right ) b \,d^{2} x +12 c^{\frac {5}{3}} \mathit {atanh} \left (c \,x^{3}\right ) b d e \,x^{2}+4 c^{\frac {5}{3}} \mathit {atanh} \left (c \,x^{3}\right ) b \,e^{2} x^{3}-4 c^{\frac {2}{3}} \mathit {atanh} \left (c \,x^{3}\right ) b \,e^{2}+6 c^{\frac {4}{3}} \mathit {atanh} \left (c \,x^{3}\right ) b \,d^{2}+6 \mathit {atanh} \left (c \,x^{3}\right ) b c d e +4 c^{\frac {2}{3}} \mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} x +1\right ) b \,e^{2}+4 c^{\frac {2}{3}} \mathrm {log}\left (c^{\frac {2}{3}} x +c^{\frac {1}{3}}\right ) b \,e^{2}+12 c^{\frac {5}{3}} a \,d^{2} x +12 c^{\frac {5}{3}} a d e \,x^{2}+4 c^{\frac {5}{3}} a \,e^{2} x^{3}-6 c^{\frac {4}{3}} \mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} x +1\right ) b \,d^{2}+3 c^{\frac {4}{3}} \mathrm {log}\left (c^{\frac {2}{3}} x +c^{\frac {1}{3}}\right ) b \,d^{2}+9 c^{\frac {4}{3}} \mathrm {log}\left (c^{\frac {2}{3}} x -c^{\frac {1}{3}}\right ) b \,d^{2}-9 \,\mathrm {log}\left (c^{\frac {2}{3}} x +c^{\frac {1}{3}}\right ) b c d e +9 \,\mathrm {log}\left (c^{\frac {2}{3}} x -c^{\frac {1}{3}}\right ) b c d e}{12 c^{\frac {5}{3}}} \] Input:
int((e*x+d)^2*(a+b*atanh(c*x^3)),x)
Output:
(6*c**(1/3)*sqrt(3)*atan((2*c**(1/3)*x - 1)/sqrt(3))*b*c*d**2 + 6*sqrt(3)* atan((2*c**(1/3)*x - 1)/sqrt(3))*b*c*d*e - 6*c**(1/3)*sqrt(3)*atan((2*c**( 1/3)*x + 1)/sqrt(3))*b*c*d**2 + 6*sqrt(3)*atan((2*c**(1/3)*x + 1)/sqrt(3)) *b*c*d*e + 12*c**(2/3)*atanh(c*x**3)*b*c*d**2*x + 12*c**(2/3)*atanh(c*x**3 )*b*c*d*e*x**2 + 4*c**(2/3)*atanh(c*x**3)*b*c*e**2*x**3 - 4*c**(2/3)*atanh (c*x**3)*b*e**2 + 6*c**(1/3)*atanh(c*x**3)*b*c*d**2 + 6*atanh(c*x**3)*b*c* d*e + 4*c**(2/3)*log(c**(2/3)*x**2 - c**(1/3)*x + 1)*b*e**2 + 4*c**(2/3)*l og(c**(2/3)*x + c**(1/3))*b*e**2 + 12*c**(2/3)*a*c*d**2*x + 12*c**(2/3)*a* c*d*e*x**2 + 4*c**(2/3)*a*c*e**2*x**3 - 6*c**(1/3)*log(c**(2/3)*x**2 - c** (1/3)*x + 1)*b*c*d**2 + 3*c**(1/3)*log(c**(2/3)*x + c**(1/3))*b*c*d**2 + 9 *c**(1/3)*log(c**(2/3)*x - c**(1/3))*b*c*d**2 - 9*log(c**(2/3)*x + c**(1/3 ))*b*c*d*e + 9*log(c**(2/3)*x - c**(1/3))*b*c*d*e)/(12*c**(2/3)*c)