Integrand size = 23, antiderivative size = 480 \[ \int \frac {a+b \text {arctanh}(c+d x)}{x \left (e+f x^3\right )} \, dx=\frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d x}{(1-c) (1+c+d x)}\right )}{e}-\frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left (\sqrt [3]{e}+\sqrt [3]{f} x\right )}{\left (d \sqrt [3]{e}+(1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{3 e}-\frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left ((-1)^{2/3} \sqrt [3]{e}+\sqrt [3]{f} x\right )}{\left ((-1)^{2/3} d \sqrt [3]{e}+(1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{3 e}-\frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2 \sqrt [3]{-1} d \left (\sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{f} x\right )}{\left (\sqrt [3]{-1} d \sqrt [3]{e}-(1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{3 e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d x}{(1-c) (1+c+d x)}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt [3]{e}+\sqrt [3]{f} x\right )}{\left (d \sqrt [3]{e}+(1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{6 e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d \left ((-1)^{2/3} \sqrt [3]{e}+\sqrt [3]{f} x\right )}{\left ((-1)^{2/3} d \sqrt [3]{e}+(1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{6 e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt [3]{-1} d \left (\sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{f} x\right )}{\left (\sqrt [3]{-1} d \sqrt [3]{e}-(1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{6 e} \] Output:
(a+b*arctanh(d*x+c))*ln(2*d*x/(1-c)/(d*x+c+1))/e-1/3*(a+b*arctanh(d*x+c))* ln(2*d*(e^(1/3)+f^(1/3)*x)/(d*e^(1/3)+(1-c)*f^(1/3))/(d*x+c+1))/e-1/3*(a+b *arctanh(d*x+c))*ln(2*d*((-1)^(2/3)*e^(1/3)+f^(1/3)*x)/((-1)^(2/3)*d*e^(1/ 3)+(1-c)*f^(1/3))/(d*x+c+1))/e-1/3*(a+b*arctanh(d*x+c))*ln(2*(-1)^(1/3)*d* (e^(1/3)+(-1)^(2/3)*f^(1/3)*x)/((-1)^(1/3)*d*e^(1/3)-(1-c)*f^(1/3))/(d*x+c +1))/e-1/2*b*polylog(2,1-2*d*x/(1-c)/(d*x+c+1))/e+1/6*b*polylog(2,1-2*d*(e ^(1/3)+f^(1/3)*x)/(d*e^(1/3)+(1-c)*f^(1/3))/(d*x+c+1))/e+1/6*b*polylog(2,1 -2*d*((-1)^(2/3)*e^(1/3)+f^(1/3)*x)/((-1)^(2/3)*d*e^(1/3)+(1-c)*f^(1/3))/( d*x+c+1))/e+1/6*b*polylog(2,1-2*(-1)^(1/3)*d*(e^(1/3)+(-1)^(2/3)*f^(1/3)*x )/((-1)^(1/3)*d*e^(1/3)-(1-c)*f^(1/3))/(d*x+c+1))/e
Time = 16.80 (sec) , antiderivative size = 915, normalized size of antiderivative = 1.91 \[ \int \frac {a+b \text {arctanh}(c+d x)}{x \left (e+f x^3\right )} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcTanh[c + d*x])/(x*(e + f*x^3)),x]
Output:
(6*a*Log[x] - 2*a*Log[e + f*x^3] + b*(6*ArcTanh[c + d*x]*Log[x] + Log[e^(1 /3)/f^(1/3) + x]*Log[1 - c - d*x] + Log[-(((-1)^(1/3)*e^(1/3))/f^(1/3)) + x]*Log[1 - c - d*x] + Log[((-1)^(2/3)*e^(1/3))/f^(1/3) + x]*Log[1 - c - d* x] - Log[e^(1/3)/f^(1/3) + x]*Log[-((f^(1/3)*(-1 + c + d*x))/(d*e^(1/3) - (-1 + c)*f^(1/3)))] - Log[((-1)^(2/3)*e^(1/3))/f^(1/3) + x]*Log[-((f^(1/3) *(-1 + c + d*x))/((-1)^(2/3)*d*e^(1/3) - (-1 + c)*f^(1/3)))] - Log[-(((-1) ^(1/3)*e^(1/3))/f^(1/3)) + x]*Log[(f^(1/3)*(-1 + c + d*x))/((-1)^(1/3)*d*e ^(1/3) + (-1 + c)*f^(1/3))] - Log[e^(1/3)/f^(1/3) + x]*Log[1 + c + d*x] - Log[-(((-1)^(1/3)*e^(1/3))/f^(1/3)) + x]*Log[1 + c + d*x] - Log[((-1)^(2/3 )*e^(1/3))/f^(1/3) + x]*Log[1 + c + d*x] + Log[e^(1/3)/f^(1/3) + x]*Log[-( (f^(1/3)*(1 + c + d*x))/(d*e^(1/3) - (1 + c)*f^(1/3)))] + Log[((-1)^(2/3)* e^(1/3))/f^(1/3) + x]*Log[-((f^(1/3)*(1 + c + d*x))/((-1)^(2/3)*d*e^(1/3) - (1 + c)*f^(1/3)))] + Log[-(((-1)^(1/3)*e^(1/3))/f^(1/3)) + x]*Log[(f^(1/ 3)*(1 + c + d*x))/((-1)^(1/3)*d*e^(1/3) + (1 + c)*f^(1/3))] + 3*Log[x]*Log [1 + (d*x)/(-1 + c)] - 3*Log[x]*Log[1 + (d*x)/(1 + c)] - 2*ArcTanh[c + d*x ]*Log[e + f*x^3] - Log[1 - c - d*x]*Log[e + f*x^3] + Log[1 + c + d*x]*Log[ e + f*x^3] + 3*PolyLog[2, -((d*x)/(-1 + c))] - 3*PolyLog[2, -((d*x)/(1 + c ))] - PolyLog[2, (d*(e^(1/3)/f^(1/3) + x))/(1 - c + (d*e^(1/3))/f^(1/3))] - PolyLog[2, (d*(((-1)^(2/3)*e^(1/3))/f^(1/3) + x))/(1 - c + ((-1)^(2/3)*d *e^(1/3))/f^(1/3))] - PolyLog[2, (d*((-1)^(1/3)*e^(1/3) - f^(1/3)*x))/(...
Time = 1.53 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {7276, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c+d x)}{x \left (e+f x^3\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {a}{x \left (e+f x^3\right )}+\frac {b \text {arctanh}(c+d x)}{x \left (e+f x^3\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \log \left (e+f x^3\right )}{3 e}+\frac {a \log (x)}{e}-\frac {b \text {arctanh}(c+d x) \log \left (\frac {2 d \left (\sqrt [3]{e}+\sqrt [3]{f} x\right )}{(c+d x+1) \left ((1-c) \sqrt [3]{f}+d \sqrt [3]{e}\right )}\right )}{3 e}-\frac {b \text {arctanh}(c+d x) \log \left (\frac {2 d \left ((-1)^{2/3} \sqrt [3]{e}+\sqrt [3]{f} x\right )}{(c+d x+1) \left ((1-c) \sqrt [3]{f}+(-1)^{2/3} d \sqrt [3]{e}\right )}\right )}{3 e}-\frac {b \text {arctanh}(c+d x) \log \left (\frac {2 \sqrt [3]{-1} d \left (\sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{f} x\right )}{(c+d x+1) \left (\sqrt [3]{-1} d \sqrt [3]{e}-(1-c) \sqrt [3]{f}\right )}\right )}{3 e}+\frac {b \text {arctanh}(c+d x) \log \left (\frac {2 d x}{(1-c) (c+d x+1)}\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt [3]{f} x+\sqrt [3]{e}\right )}{\left (\sqrt [3]{f} (1-c)+d \sqrt [3]{e}\right ) (c+d x+1)}\right )}{6 e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt [3]{f} x+(-1)^{2/3} \sqrt [3]{e}\right )}{\left (\sqrt [3]{f} (1-c)+(-1)^{2/3} d \sqrt [3]{e}\right ) (c+d x+1)}\right )}{6 e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt [3]{-1} d \left ((-1)^{2/3} \sqrt [3]{f} x+\sqrt [3]{e}\right )}{\left (\sqrt [3]{-1} d \sqrt [3]{e}-(1-c) \sqrt [3]{f}\right ) (c+d x+1)}\right )}{6 e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d x}{(1-c) (c+d x+1)}\right )}{2 e}\) |
Input:
Int[(a + b*ArcTanh[c + d*x])/(x*(e + f*x^3)),x]
Output:
(a*Log[x])/e + (b*ArcTanh[c + d*x]*Log[(2*d*x)/((1 - c)*(1 + c + d*x))])/e - (b*ArcTanh[c + d*x]*Log[(2*d*(e^(1/3) + f^(1/3)*x))/((d*e^(1/3) + (1 - c)*f^(1/3))*(1 + c + d*x))])/(3*e) - (b*ArcTanh[c + d*x]*Log[(2*d*((-1)^(2 /3)*e^(1/3) + f^(1/3)*x))/(((-1)^(2/3)*d*e^(1/3) + (1 - c)*f^(1/3))*(1 + c + d*x))])/(3*e) - (b*ArcTanh[c + d*x]*Log[(2*(-1)^(1/3)*d*(e^(1/3) + (-1) ^(2/3)*f^(1/3)*x))/(((-1)^(1/3)*d*e^(1/3) - (1 - c)*f^(1/3))*(1 + c + d*x) )])/(3*e) - (a*Log[e + f*x^3])/(3*e) - (b*PolyLog[2, 1 - (2*d*x)/((1 - c)* (1 + c + d*x))])/(2*e) + (b*PolyLog[2, 1 - (2*d*(e^(1/3) + f^(1/3)*x))/((d *e^(1/3) + (1 - c)*f^(1/3))*(1 + c + d*x))])/(6*e) + (b*PolyLog[2, 1 - (2* d*((-1)^(2/3)*e^(1/3) + f^(1/3)*x))/(((-1)^(2/3)*d*e^(1/3) + (1 - c)*f^(1/ 3))*(1 + c + d*x))])/(6*e) + (b*PolyLog[2, 1 - (2*(-1)^(1/3)*d*(e^(1/3) + (-1)^(2/3)*f^(1/3)*x))/(((-1)^(1/3)*d*e^(1/3) - (1 - c)*f^(1/3))*(1 + c + d*x))])/(6*e)
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.31 (sec) , antiderivative size = 420, normalized size of antiderivative = 0.88
| method | result | size |
| risch | \(-\frac {b \ln \left (-d x -c +1\right ) \ln \left (-\frac {x d}{-1+c}\right )}{2 e}-\frac {b \operatorname {dilog}\left (-\frac {x d}{-1+c}\right )}{2 e}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f -6 f c +3 f \right ) \textit {\_Z} +c^{3} f -d^{3} e -3 c^{2} f +3 f c -f \right )}{\sum }\left (\ln \left (-d x -c +1\right ) \ln \left (\frac {d x +\textit {\_R1} +c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {d x +\textit {\_R1} +c -1}{\textit {\_R1}}\right )\right )\right )}{6 e}+\frac {a \ln \left (-d x \right )}{e}-\frac {a \ln \left (\left (-d x -c +1\right )^{3} f +3 \left (-d x -c +1\right )^{2} c f +3 \left (-d x -c +1\right ) c^{2} f +c^{3} f -d^{3} e -3 f \left (-d x -c +1\right )^{2}-6 \left (-d x -c +1\right ) c f -3 c^{2} f +3 \left (-d x -c +1\right ) f +3 f c -f \right )}{3 e}+\frac {b \ln \left (d x +c +1\right ) \ln \left (\frac {x d}{-1-c}\right )}{2 e}+\frac {b \operatorname {dilog}\left (\frac {x d}{-1-c}\right )}{2 e}-\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f +6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e -3 c^{2} f -3 f c -f \right )}{\sum }\left (\ln \left (d x +c +1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )\right )\right )}{6 e}\) | \(420\) |
| parts | \(\frac {a \ln \left (x \right )}{e}-\frac {a \ln \left (f \,x^{3}+e \right )}{3 e}+b \left (-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (f \left (d x +c \right )^{3}-3 c f \left (d x +c \right )^{2}+3 c^{2} f \left (d x +c \right )-c^{3} f +d^{3} e \right )}{3 e}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x \right )}{e}-\frac {d^{3} \left (\frac {-\frac {\ln \left (d x +c +1\right ) \ln \left (f \left (d x +c \right )^{3}-3 c f \left (d x +c \right )^{2}+3 c^{2} f \left (d x +c \right )-c^{3} f +d^{3} e \right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f +6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e -3 c^{2} f -3 f c -f \right )}{\sum }\left (\ln \left (d x +c +1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )\right )\right )}{2}+\frac {\ln \left (d x +c -1\right ) \ln \left (f \left (d x +c \right )^{3}-3 c f \left (d x +c \right )^{2}+3 c^{2} f \left (d x +c \right )-c^{3} f +d^{3} e \right )}{2}-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c +3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f -6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e +3 c^{2} f -3 f c +f \right )}{\sum }\left (\ln \left (d x +c -1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )\right )\right )}{2}}{d^{3} e}-\frac {3 \left (\frac {\operatorname {dilog}\left (\frac {d x +c -1}{-1+c}\right )}{2}+\frac {\ln \left (d x \right ) \ln \left (\frac {d x +c -1}{-1+c}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x +c +1}{1+c}\right )}{2}-\frac {\ln \left (d x \right ) \ln \left (\frac {d x +c +1}{1+c}\right )}{2}\right )}{d^{3} e}\right )}{3}\right )\) | \(496\) |
| derivativedivides | \(\frac {a \ln \left (-d x \right )}{e}-\frac {a \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{3 e}+b \,d^{3} \left (\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (-d x \right )}{e \,d^{3}}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{3 e \,d^{3}}-\frac {\frac {\ln \left (d x +c -1\right ) \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{2}-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c +3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f -6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e +3 c^{2} f -3 f c +f \right )}{\sum }\left (\ln \left (d x +c -1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )\right )\right )}{2}-\frac {\ln \left (d x +c +1\right ) \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f +6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e -3 c^{2} f -3 f c -f \right )}{\sum }\left (\ln \left (d x +c +1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )\right )\right )}{2}}{3 e \,d^{3}}+\frac {-\frac {\operatorname {dilog}\left (\frac {-d x -c -1}{-1-c}\right )}{2}-\frac {\ln \left (-d x \right ) \ln \left (\frac {-d x -c -1}{-1-c}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-d x -c +1}{1-c}\right )}{2}+\frac {\ln \left (-d x \right ) \ln \left (\frac {-d x -c +1}{1-c}\right )}{2}}{e \,d^{3}}\right )\) | \(565\) |
| default | \(\frac {a \ln \left (-d x \right )}{e}-\frac {a \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{3 e}+b \,d^{3} \left (\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (-d x \right )}{e \,d^{3}}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{3 e \,d^{3}}-\frac {\frac {\ln \left (d x +c -1\right ) \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{2}-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c +3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f -6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e +3 c^{2} f -3 f c +f \right )}{\sum }\left (\ln \left (d x +c -1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )\right )\right )}{2}-\frac {\ln \left (d x +c +1\right ) \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f +6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e -3 c^{2} f -3 f c -f \right )}{\sum }\left (\ln \left (d x +c +1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )\right )\right )}{2}}{3 e \,d^{3}}+\frac {-\frac {\operatorname {dilog}\left (\frac {-d x -c -1}{-1-c}\right )}{2}-\frac {\ln \left (-d x \right ) \ln \left (\frac {-d x -c -1}{-1-c}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-d x -c +1}{1-c}\right )}{2}+\frac {\ln \left (-d x \right ) \ln \left (\frac {-d x -c +1}{1-c}\right )}{2}}{e \,d^{3}}\right )\) | \(565\) |
Input:
int((a+b*arctanh(d*x+c))/x/(f*x^3+e),x,method=_RETURNVERBOSE)
Output:
-1/2*b/e*ln(-d*x-c+1)*ln(-x*d/(-1+c))-1/2*b/e*dilog(-x*d/(-1+c))+1/6*b/e*s um(ln(-d*x-c+1)*ln((d*x+_R1+c-1)/_R1)+dilog((d*x+_R1+c-1)/_R1),_R1=RootOf( f*_Z^3+(3*c*f-3*f)*_Z^2+(3*c^2*f-6*c*f+3*f)*_Z+c^3*f-d^3*e-3*c^2*f+3*f*c-f ))+a/e*ln(-d*x)-1/3*a/e*ln((-d*x-c+1)^3*f+3*(-d*x-c+1)^2*c*f+3*(-d*x-c+1)* c^2*f+c^3*f-d^3*e-3*f*(-d*x-c+1)^2-6*(-d*x-c+1)*c*f-3*c^2*f+3*(-d*x-c+1)*f +3*f*c-f)+1/2*b/e*ln(d*x+c+1)*ln(x*d/(-1-c))+1/2*b/e*dilog(x*d/(-1-c))-1/6 *b/e*sum(ln(d*x+c+1)*ln((-d*x+_R1-c-1)/_R1)+dilog((-d*x+_R1-c-1)/_R1),_R1= RootOf(f*_Z^3+(-3*c*f-3*f)*_Z^2+(3*c^2*f+6*c*f+3*f)*_Z-c^3*f+d^3*e-3*c^2*f -3*f*c-f))
\[ \int \frac {a+b \text {arctanh}(c+d x)}{x \left (e+f x^3\right )} \, dx=\int { \frac {b \operatorname {artanh}\left (d x + c\right ) + a}{{\left (f x^{3} + e\right )} x} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))/x/(f*x^3+e),x, algorithm="fricas")
Output:
integral((b*arctanh(d*x + c) + a)/(f*x^4 + e*x), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c+d x)}{x \left (e+f x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((a+b*atanh(d*x+c))/x/(f*x**3+e),x)
Output:
Timed out
\[ \int \frac {a+b \text {arctanh}(c+d x)}{x \left (e+f x^3\right )} \, dx=\int { \frac {b \operatorname {artanh}\left (d x + c\right ) + a}{{\left (f x^{3} + e\right )} x} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))/x/(f*x^3+e),x, algorithm="maxima")
Output:
-1/3*a*(log(f*x^3 + e)/e - 3*log(x)/e) + 1/2*b*integrate((log(d*x + c + 1) - log(-d*x - c + 1))/(f*x^4 + e*x), x)
\[ \int \frac {a+b \text {arctanh}(c+d x)}{x \left (e+f x^3\right )} \, dx=\int { \frac {b \operatorname {artanh}\left (d x + c\right ) + a}{{\left (f x^{3} + e\right )} x} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))/x/(f*x^3+e),x, algorithm="giac")
Output:
integrate((b*arctanh(d*x + c) + a)/((f*x^3 + e)*x), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c+d x)}{x \left (e+f x^3\right )} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c+d\,x\right )}{x\,\left (f\,x^3+e\right )} \,d x \] Input:
int((a + b*atanh(c + d*x))/(x*(e + f*x^3)),x)
Output:
int((a + b*atanh(c + d*x))/(x*(e + f*x^3)), x)
\[ \int \frac {a+b \text {arctanh}(c+d x)}{x \left (e+f x^3\right )} \, dx=\frac {3 \left (\int \frac {\mathit {atanh} \left (d x +c \right )}{f \,x^{4}+e x}d x \right ) b e -\mathrm {log}\left (e^{\frac {2}{3}}-f^{\frac {1}{3}} e^{\frac {1}{3}} x +f^{\frac {2}{3}} x^{2}\right ) a -\mathrm {log}\left (e^{\frac {1}{3}}+f^{\frac {1}{3}} x \right ) a +3 \,\mathrm {log}\left (x \right ) a}{3 e} \] Input:
int((a+b*atanh(d*x+c))/x/(f*x^3+e),x)
Output:
(3*int(atanh(c + d*x)/(e*x + f*x**4),x)*b*e - log(e**(2/3) - f**(1/3)*e**( 1/3)*x + f**(2/3)*x**2)*a - log(e**(1/3) + f**(1/3)*x)*a + 3*log(x)*a)/(3* e)