Integrand size = 20, antiderivative size = 679 \[ \int \frac {a+b \text {arctanh}(c+d x)}{e+f x^3} \, dx=-\frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1+c+d x}\right )}{3 e^{2/3} \sqrt [3]{f}}+\frac {\sqrt [3]{-1} (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1+c+d x}\right )}{3 e^{2/3} \sqrt [3]{f}}-\frac {(-1)^{2/3} (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1+c+d x}\right )}{3 e^{2/3} \sqrt [3]{f}}+\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^{2/3} \sqrt [3]{f}}+\frac {(-1)^{2/3} (a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left (\sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{f} x\right )}{\left (d \sqrt [3]{e}-\sqrt [3]{-1} (1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{3 e^{2/3} \sqrt [3]{f}}-\frac {\sqrt [3]{-1} (a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left (\sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{f} x\right )}{\left (d \sqrt [3]{e}+(-1)^{2/3} (1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{3 e^{2/3} \sqrt [3]{f}}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{6 e^{2/3} \sqrt [3]{f}}-\frac {\sqrt [3]{-1} b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{6 e^{2/3} \sqrt [3]{f}}+\frac {(-1)^{2/3} b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{6 e^{2/3} \sqrt [3]{f}}-\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^{2/3} \sqrt [3]{f}}-\frac {(-1)^{2/3} b \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{f} x\right )}{\left (d \sqrt [3]{e}-\sqrt [3]{-1} (1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{6 e^{2/3} \sqrt [3]{f}}+\frac {\sqrt [3]{-1} b \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{f} x\right )}{\left (d \sqrt [3]{e}+(-1)^{2/3} (1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{6 e^{2/3} \sqrt [3]{f}} \] Output:
-1/3*(a+b*arctanh(d*x+c))*ln(2/(d*x+c+1))/e^(2/3)/f^(1/3)+1/3*(-1)^(1/3)*( a+b*arctanh(d*x+c))*ln(2/(d*x+c+1))/e^(2/3)/f^(1/3)-1/3*(-1)^(2/3)*(a+b*ar ctanh(d*x+c))*ln(2/(d*x+c+1))/e^(2/3)/f^(1/3)+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^(2/3)/f^(1/ 3)+1/3*(-1)^(2/3)*(a+b*arctanh(d*x+c))*ln(2*d*(e^(1/3)-(-1)^(1/3)*f^(1/3)* x)/(d*e^(1/3)-(-1)^(1/3)*(1-c)*f^(1/3))/(d*x+c+1))/e^(2/3)/f^(1/3)-1/3*(-1 )^(1/3)*(a+b*arctanh(d*x+c))*ln(2*d*(e^(1/3)+(-1)^(2/3)*f^(1/3)*x)/(d*e^(1 /3)+(-1)^(2/3)*(1-c)*f^(1/3))/(d*x+c+1))/e^(2/3)/f^(1/3)+1/6*b*polylog(2,1 -2/(d*x+c+1))/e^(2/3)/f^(1/3)-1/6*(-1)^(1/3)*b*polylog(2,1-2/(d*x+c+1))/e^ (2/3)/f^(1/3)+1/6*(-1)^(2/3)*b*polylog(2,1-2/(d*x+c+1))/e^(2/3)/f^(1/3)-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^(2/3)/f^(1/3)-1/6*(-1)^(2/3)*b*polylog(2,1-2*d*(e^(1/3)-(-1)^(1/3)*f^ (1/3)*x)/(d*e^(1/3)-(-1)^(1/3)*(1-c)*f^(1/3))/(d*x+c+1))/e^(2/3)/f^(1/3)+1 /6*(-1)^(1/3)*b*polylog(2,1-2*d*(e^(1/3)+(-1)^(2/3)*f^(1/3)*x)/(d*e^(1/3)+ (-1)^(2/3)*(1-c)*f^(1/3))/(d*x+c+1))/e^(2/3)/f^(1/3)
Time = 0.54 (sec) , antiderivative size = 714, normalized size of antiderivative = 1.05 \[ \int \frac {a+b \text {arctanh}(c+d x)}{e+f x^3} \, dx=-\frac {2 \sqrt {3} a \arctan \left (\frac {1-\frac {2 \sqrt [3]{f} x}{\sqrt [3]{e}}}{\sqrt {3}}\right )-2 a \log \left (\sqrt [3]{e}+\sqrt [3]{f} x\right )+b \log (1-c-d x) \log \left (\frac {d \left (\sqrt [3]{e}+\sqrt [3]{f} x\right )}{d \sqrt [3]{e}-(-1+c) \sqrt [3]{f}}\right )-b \log (1+c+d x) \log \left (\frac {d \left (\sqrt [3]{e}+\sqrt [3]{f} x\right )}{d \sqrt [3]{e}-(1+c) \sqrt [3]{f}}\right )+(-1)^{2/3} b \log (1-c-d x) \log \left (\frac {d \left (\sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{f} x\right )}{d \sqrt [3]{e}+\sqrt [3]{-1} (-1+c) \sqrt [3]{f}}\right )-(-1)^{2/3} b \log (1+c+d x) \log \left (\frac {d \left (\sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{f} x\right )}{d \sqrt [3]{e}+\sqrt [3]{-1} (1+c) \sqrt [3]{f}}\right )-\sqrt [3]{-1} b \log (1-c-d x) \log \left (\frac {d \left (\sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{f} x\right )}{d \sqrt [3]{e}-(-1)^{2/3} (-1+c) \sqrt [3]{f}}\right )+\sqrt [3]{-1} b \log (1+c+d x) \log \left (\frac {d \left (\sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{f} x\right )}{d \sqrt [3]{e}-(-1)^{2/3} (1+c) \sqrt [3]{f}}\right )+a \log \left (e^{2/3}-\sqrt [3]{e} \sqrt [3]{f} x+f^{2/3} x^2\right )+b \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{f} (-1+c+d x)}{d \sqrt [3]{e}-(-1+c) \sqrt [3]{f}}\right )+(-1)^{2/3} b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{f} (-1+c+d x)}{d \sqrt [3]{e}+\sqrt [3]{-1} (-1+c) \sqrt [3]{f}}\right )-\sqrt [3]{-1} b \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{f} (-1+c+d x)}{-d \sqrt [3]{e}+(-1)^{2/3} (-1+c) \sqrt [3]{f}}\right )-b \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{f} (1+c+d x)}{d \sqrt [3]{e}-(1+c) \sqrt [3]{f}}\right )-(-1)^{2/3} b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{f} (1+c+d x)}{d \sqrt [3]{e}+\sqrt [3]{-1} (1+c) \sqrt [3]{f}}\right )+\sqrt [3]{-1} b \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{f} (1+c+d x)}{-d \sqrt [3]{e}+(-1)^{2/3} (1+c) \sqrt [3]{f}}\right )}{6 e^{2/3} \sqrt [3]{f}} \] Input:
Integrate[(a + b*ArcTanh[c + d*x])/(e + f*x^3),x]
Output:
-1/6*(2*Sqrt[3]*a*ArcTan[(1 - (2*f^(1/3)*x)/e^(1/3))/Sqrt[3]] - 2*a*Log[e^ (1/3) + f^(1/3)*x] + b*Log[1 - c - d*x]*Log[(d*(e^(1/3) + f^(1/3)*x))/(d*e ^(1/3) - (-1 + c)*f^(1/3))] - b*Log[1 + c + d*x]*Log[(d*(e^(1/3) + f^(1/3) *x))/(d*e^(1/3) - (1 + c)*f^(1/3))] + (-1)^(2/3)*b*Log[1 - c - d*x]*Log[(d *(e^(1/3) - (-1)^(1/3)*f^(1/3)*x))/(d*e^(1/3) + (-1)^(1/3)*(-1 + c)*f^(1/3 ))] - (-1)^(2/3)*b*Log[1 + c + d*x]*Log[(d*(e^(1/3) - (-1)^(1/3)*f^(1/3)*x ))/(d*e^(1/3) + (-1)^(1/3)*(1 + c)*f^(1/3))] - (-1)^(1/3)*b*Log[1 - c - d* x]*Log[(d*(e^(1/3) + (-1)^(2/3)*f^(1/3)*x))/(d*e^(1/3) - (-1)^(2/3)*(-1 + c)*f^(1/3))] + (-1)^(1/3)*b*Log[1 + c + d*x]*Log[(d*(e^(1/3) + (-1)^(2/3)* f^(1/3)*x))/(d*e^(1/3) - (-1)^(2/3)*(1 + c)*f^(1/3))] + a*Log[e^(2/3) - e^ (1/3)*f^(1/3)*x + f^(2/3)*x^2] + b*PolyLog[2, -((f^(1/3)*(-1 + c + d*x))/( d*e^(1/3) - (-1 + c)*f^(1/3)))] + (-1)^(2/3)*b*PolyLog[2, ((-1)^(1/3)*f^(1 /3)*(-1 + c + d*x))/(d*e^(1/3) + (-1)^(1/3)*(-1 + c)*f^(1/3))] - (-1)^(1/3 )*b*PolyLog[2, ((-1)^(2/3)*f^(1/3)*(-1 + c + d*x))/(-(d*e^(1/3)) + (-1)^(2 /3)*(-1 + c)*f^(1/3))] - b*PolyLog[2, -((f^(1/3)*(1 + c + d*x))/(d*e^(1/3) - (1 + c)*f^(1/3)))] - (-1)^(2/3)*b*PolyLog[2, ((-1)^(1/3)*f^(1/3)*(1 + c + d*x))/(d*e^(1/3) + (-1)^(1/3)*(1 + c)*f^(1/3))] + (-1)^(1/3)*b*PolyLog[ 2, ((-1)^(2/3)*f^(1/3)*(1 + c + d*x))/(-(d*e^(1/3)) + (-1)^(2/3)*(1 + c)*f ^(1/3))])/(e^(2/3)*f^(1/3))
Time = 1.55 (sec) , antiderivative size = 919, normalized size of antiderivative = 1.35, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c+d x)}{e+f x^3} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {a}{e+f x^3}+\frac {b \text {arctanh}(c+d x)}{e+f x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \arctan \left (\frac {\sqrt [3]{e}-2 \sqrt [3]{f} x}{\sqrt {3} \sqrt [3]{e}}\right )}{\sqrt {3} e^{2/3} \sqrt [3]{f}}+\frac {a \log \left (\sqrt [3]{f} x+\sqrt [3]{e}\right )}{3 e^{2/3} \sqrt [3]{f}}-\frac {b \log (-c-d x+1) \log \left (\frac {d \left (\sqrt [3]{f} x+\sqrt [3]{e}\right )}{\sqrt [3]{f} (1-c)+d \sqrt [3]{e}}\right )}{6 e^{2/3} \sqrt [3]{f}}+\frac {b \log (c+d x+1) \log \left (\frac {d \left (\sqrt [3]{f} x+\sqrt [3]{e}\right )}{d \sqrt [3]{e}-(c+1) \sqrt [3]{f}}\right )}{6 e^{2/3} \sqrt [3]{f}}-\frac {(-1)^{2/3} b \log (-c-d x+1) \log \left (\frac {d \left (\sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{f} x\right )}{d \sqrt [3]{e}-\sqrt [3]{-1} (1-c) \sqrt [3]{f}}\right )}{6 e^{2/3} \sqrt [3]{f}}+\frac {(-1)^{2/3} b \log (c+d x+1) \log \left (\frac {d \left (\sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{f} x\right )}{\sqrt [3]{-1} \sqrt [3]{f} (c+1)+d \sqrt [3]{e}}\right )}{6 e^{2/3} \sqrt [3]{f}}+\frac {\sqrt [3]{-1} b \log (-c-d x+1) \log \left (\frac {d \left ((-1)^{2/3} \sqrt [3]{f} x+\sqrt [3]{e}\right )}{(-1)^{2/3} \sqrt [3]{f} (1-c)+d \sqrt [3]{e}}\right )}{6 e^{2/3} \sqrt [3]{f}}-\frac {\sqrt [3]{-1} b \log (c+d x+1) \log \left (\frac {d \left ((-1)^{2/3} \sqrt [3]{f} x+\sqrt [3]{e}\right )}{d \sqrt [3]{e}-(-1)^{2/3} (c+1) \sqrt [3]{f}}\right )}{6 e^{2/3} \sqrt [3]{f}}-\frac {a \log \left (f^{2/3} x^2-\sqrt [3]{e} \sqrt [3]{f} x+e^{2/3}\right )}{6 e^{2/3} \sqrt [3]{f}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{f} (-c-d x+1)}{\sqrt [3]{f} (1-c)+d \sqrt [3]{e}}\right )}{6 e^{2/3} \sqrt [3]{f}}-\frac {(-1)^{2/3} b \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{-1} \sqrt [3]{f} (-c-d x+1)}{d \sqrt [3]{e}-\sqrt [3]{-1} (1-c) \sqrt [3]{f}}\right )}{6 e^{2/3} \sqrt [3]{f}}+\frac {\sqrt [3]{-1} b \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{f} (-c-d x+1)}{(-1)^{2/3} \sqrt [3]{f} (1-c)+d \sqrt [3]{e}}\right )}{6 e^{2/3} \sqrt [3]{f}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{f} (c+d x+1)}{d \sqrt [3]{e}-(c+1) \sqrt [3]{f}}\right )}{6 e^{2/3} \sqrt [3]{f}}+\frac {(-1)^{2/3} b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{f} (c+d x+1)}{\sqrt [3]{-1} \sqrt [3]{f} (c+1)+d \sqrt [3]{e}}\right )}{6 e^{2/3} \sqrt [3]{f}}-\frac {\sqrt [3]{-1} b \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{f} \left ((-1)^{2/3} c+(-1)^{2/3} d x+(-1)^{2/3}\right )}{d \sqrt [3]{e}-(-1)^{2/3} (c+1) \sqrt [3]{f}}\right )}{6 e^{2/3} \sqrt [3]{f}}\) |
Input:
Int[(a + b*ArcTanh[c + d*x])/(e + f*x^3),x]
Output:
-((a*ArcTan[(e^(1/3) - 2*f^(1/3)*x)/(Sqrt[3]*e^(1/3))])/(Sqrt[3]*e^(2/3)*f ^(1/3))) + (a*Log[e^(1/3) + f^(1/3)*x])/(3*e^(2/3)*f^(1/3)) - (b*Log[1 - c - d*x]*Log[(d*(e^(1/3) + f^(1/3)*x))/(d*e^(1/3) + (1 - c)*f^(1/3))])/(6*e ^(2/3)*f^(1/3)) + (b*Log[1 + c + d*x]*Log[(d*(e^(1/3) + f^(1/3)*x))/(d*e^( 1/3) - (1 + c)*f^(1/3))])/(6*e^(2/3)*f^(1/3)) - ((-1)^(2/3)*b*Log[1 - c - d*x]*Log[(d*(e^(1/3) - (-1)^(1/3)*f^(1/3)*x))/(d*e^(1/3) - (-1)^(1/3)*(1 - c)*f^(1/3))])/(6*e^(2/3)*f^(1/3)) + ((-1)^(2/3)*b*Log[1 + c + d*x]*Log[(d *(e^(1/3) - (-1)^(1/3)*f^(1/3)*x))/(d*e^(1/3) + (-1)^(1/3)*(1 + c)*f^(1/3) )])/(6*e^(2/3)*f^(1/3)) + ((-1)^(1/3)*b*Log[1 - c - d*x]*Log[(d*(e^(1/3) + (-1)^(2/3)*f^(1/3)*x))/(d*e^(1/3) + (-1)^(2/3)*(1 - c)*f^(1/3))])/(6*e^(2 /3)*f^(1/3)) - ((-1)^(1/3)*b*Log[1 + c + d*x]*Log[(d*(e^(1/3) + (-1)^(2/3) *f^(1/3)*x))/(d*e^(1/3) - (-1)^(2/3)*(1 + c)*f^(1/3))])/(6*e^(2/3)*f^(1/3) ) - (a*Log[e^(2/3) - e^(1/3)*f^(1/3)*x + f^(2/3)*x^2])/(6*e^(2/3)*f^(1/3)) - (b*PolyLog[2, (f^(1/3)*(1 - c - d*x))/(d*e^(1/3) + (1 - c)*f^(1/3))])/( 6*e^(2/3)*f^(1/3)) - ((-1)^(2/3)*b*PolyLog[2, -(((-1)^(1/3)*f^(1/3)*(1 - c - d*x))/(d*e^(1/3) - (-1)^(1/3)*(1 - c)*f^(1/3)))])/(6*e^(2/3)*f^(1/3)) + ((-1)^(1/3)*b*PolyLog[2, ((-1)^(2/3)*f^(1/3)*(1 - c - d*x))/(d*e^(1/3) + (-1)^(2/3)*(1 - c)*f^(1/3))])/(6*e^(2/3)*f^(1/3)) + (b*PolyLog[2, -((f^(1/ 3)*(1 + c + d*x))/(d*e^(1/3) - (1 + c)*f^(1/3)))])/(6*e^(2/3)*f^(1/3)) + ( (-1)^(2/3)*b*PolyLog[2, ((-1)^(1/3)*f^(1/3)*(1 + c + d*x))/(d*e^(1/3) +...
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 = 0.32 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.54
| method | result | size |
| risch | \(-\frac {d^{2} b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f -6 f c +3 f \right ) \textit {\_Z} +c^{3} f -d^{3} e -3 c^{2} f +3 f c -f \right )}{\sum }\frac {\ln \left (-d x -c +1\right ) \ln \left (\frac {d x +\textit {\_R1} +c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {d x +\textit {\_R1} +c -1}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}+2 \textit {\_R1} c +c^{2}-2 \textit {\_R1} -2 c +1}\right )}{6 f}+\frac {d^{2} a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f -6 f c +3 f \right ) \textit {\_Z} +c^{3} f -d^{3} e -3 c^{2} f +3 f c -f \right )}{\sum }\frac {\ln \left (-d x -\textit {\_R} -c +1\right )}{\textit {\_R}^{2}+2 \textit {\_R} c +c^{2}-2 \textit {\_R} -2 c +1}\right )}{3 f}+\frac {b \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f +6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e -3 c^{2} f -3 f c -f \right )}{\sum }\frac {\ln \left (d x +c +1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}-2 \textit {\_R1} +2 c +1}\right )}{6 f}\) | \(366\) |
| derivativedivides | \(\frac {-\frac {a \,d^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}-3 f c \,\textit {\_Z}^{2}+3 c^{2} f \textit {\_Z} -c^{3} f +d^{3} e \right )}{\sum }\frac {\ln \left (d x -\textit {\_R} +c \right )}{-\textit {\_R}^{2}+2 \textit {\_R} c -c^{2}}\right )}{3 f}-b \,d^{3} \left (\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (c^{3} f -d^{3} e -3 c^{2} f +3 f c -f \right ) \textit {\_Z}^{6}+\left (3 c^{3} f -3 d^{3} e -3 c^{2} f -3 f c +3 f \right ) \textit {\_Z}^{4}+\left (3 c^{3} f -3 d^{3} e +3 c^{2} f -3 f c -3 f \right ) \textit {\_Z}^{2}+c^{3} f -d^{3} e +3 c^{2} f +3 f c +f \right )}{\sum }\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (\frac {\textit {\_R1} -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1}^{4} c^{3} f -\textit {\_R1}^{4} d^{3} e -3 \textit {\_R1}^{4} c^{2} f +3 \textit {\_R1}^{4} c f +2 \textit {\_R1}^{2} c^{3} f -2 \textit {\_R1}^{2} d^{3} e -\textit {\_R1}^{4} f -2 \textit {\_R1}^{2} c^{2} f -2 \textit {\_R1}^{2} c f +c^{3} f -d^{3} e +2 \textit {\_R1}^{2} f +c^{2} f -f c -f}\right )}{3}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (c^{3} f -d^{3} e -3 c^{2} f +3 f c -f \right ) \textit {\_Z}^{6}+\left (3 c^{3} f -3 d^{3} e -3 c^{2} f -3 f c +3 f \right ) \textit {\_Z}^{4}+\left (3 c^{3} f -3 d^{3} e +3 c^{2} f -3 f c -3 f \right ) \textit {\_Z}^{2}+c^{3} f -d^{3} e +3 c^{2} f +3 f c +f \right )}{\sum }\frac {\textit {\_R1}^{2} \left (\operatorname {arctanh}\left (d x +c \right ) \ln \left (\frac {\textit {\_R1} -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{4} c^{3} f -\textit {\_R1}^{4} d^{3} e -3 \textit {\_R1}^{4} c^{2} f +3 \textit {\_R1}^{4} c f +2 \textit {\_R1}^{2} c^{3} f -2 \textit {\_R1}^{2} d^{3} e -\textit {\_R1}^{4} f -2 \textit {\_R1}^{2} c^{2} f -2 \textit {\_R1}^{2} c f +c^{3} f -d^{3} e +2 \textit {\_R1}^{2} f +c^{2} f -f c -f}\right )}{3}\right )}{d}\) | \(663\) |
| default | \(\frac {-\frac {a \,d^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}-3 f c \,\textit {\_Z}^{2}+3 c^{2} f \textit {\_Z} -c^{3} f +d^{3} e \right )}{\sum }\frac {\ln \left (d x -\textit {\_R} +c \right )}{-\textit {\_R}^{2}+2 \textit {\_R} c -c^{2}}\right )}{3 f}-b \,d^{3} \left (\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (c^{3} f -d^{3} e -3 c^{2} f +3 f c -f \right ) \textit {\_Z}^{6}+\left (3 c^{3} f -3 d^{3} e -3 c^{2} f -3 f c +3 f \right ) \textit {\_Z}^{4}+\left (3 c^{3} f -3 d^{3} e +3 c^{2} f -3 f c -3 f \right ) \textit {\_Z}^{2}+c^{3} f -d^{3} e +3 c^{2} f +3 f c +f \right )}{\sum }\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (\frac {\textit {\_R1} -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1}^{4} c^{3} f -\textit {\_R1}^{4} d^{3} e -3 \textit {\_R1}^{4} c^{2} f +3 \textit {\_R1}^{4} c f +2 \textit {\_R1}^{2} c^{3} f -2 \textit {\_R1}^{2} d^{3} e -\textit {\_R1}^{4} f -2 \textit {\_R1}^{2} c^{2} f -2 \textit {\_R1}^{2} c f +c^{3} f -d^{3} e +2 \textit {\_R1}^{2} f +c^{2} f -f c -f}\right )}{3}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (c^{3} f -d^{3} e -3 c^{2} f +3 f c -f \right ) \textit {\_Z}^{6}+\left (3 c^{3} f -3 d^{3} e -3 c^{2} f -3 f c +3 f \right ) \textit {\_Z}^{4}+\left (3 c^{3} f -3 d^{3} e +3 c^{2} f -3 f c -3 f \right ) \textit {\_Z}^{2}+c^{3} f -d^{3} e +3 c^{2} f +3 f c +f \right )}{\sum }\frac {\textit {\_R1}^{2} \left (\operatorname {arctanh}\left (d x +c \right ) \ln \left (\frac {\textit {\_R1} -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{4} c^{3} f -\textit {\_R1}^{4} d^{3} e -3 \textit {\_R1}^{4} c^{2} f +3 \textit {\_R1}^{4} c f +2 \textit {\_R1}^{2} c^{3} f -2 \textit {\_R1}^{2} d^{3} e -\textit {\_R1}^{4} f -2 \textit {\_R1}^{2} c^{2} f -2 \textit {\_R1}^{2} c f +c^{3} f -d^{3} e +2 \textit {\_R1}^{2} f +c^{2} f -f c -f}\right )}{3}\right )}{d}\) | \(663\) |
| parts | \(a \left (\frac {\ln \left (x +\left (\frac {e}{f}\right )^{\frac {1}{3}}\right )}{3 f \left (\frac {e}{f}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {e}{f}\right )^{\frac {1}{3}} x +\left (\frac {e}{f}\right )^{\frac {2}{3}}\right )}{6 f \left (\frac {e}{f}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {e}{f}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 f \left (\frac {e}{f}\right )^{\frac {2}{3}}}\right )-\frac {2 b \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (c^{3} f -d^{3} e -3 c^{2} f +3 f c -f \right ) \textit {\_Z}^{6}+\left (3 c^{3} f -3 d^{3} e -3 c^{2} f -3 f c +3 f \right ) \textit {\_Z}^{4}+\left (3 c^{3} f -3 d^{3} e +3 c^{2} f -3 f c -3 f \right ) \textit {\_Z}^{2}+c^{3} f -d^{3} e +3 c^{2} f +3 f c +f \right )}{\sum }\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (\frac {\textit {\_R1} -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1}^{4} c^{3} f -\textit {\_R1}^{4} d^{3} e -3 \textit {\_R1}^{4} c^{2} f +3 \textit {\_R1}^{4} c f +2 \textit {\_R1}^{2} c^{3} f -2 \textit {\_R1}^{2} d^{3} e -\textit {\_R1}^{4} f -2 \textit {\_R1}^{2} c^{2} f -2 \textit {\_R1}^{2} c f +c^{3} f -d^{3} e +2 \textit {\_R1}^{2} f +c^{2} f -f c -f}\right )}{3}-\frac {2 b \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (c^{3} f -d^{3} e -3 c^{2} f +3 f c -f \right ) \textit {\_Z}^{6}+\left (3 c^{3} f -3 d^{3} e -3 c^{2} f -3 f c +3 f \right ) \textit {\_Z}^{4}+\left (3 c^{3} f -3 d^{3} e +3 c^{2} f -3 f c -3 f \right ) \textit {\_Z}^{2}+c^{3} f -d^{3} e +3 c^{2} f +3 f c +f \right )}{\sum }\frac {\textit {\_R1}^{2} \left (\operatorname {arctanh}\left (d x +c \right ) \ln \left (\frac {\textit {\_R1} -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{4} c^{3} f -\textit {\_R1}^{4} d^{3} e -3 \textit {\_R1}^{4} c^{2} f +3 \textit {\_R1}^{4} c f +2 \textit {\_R1}^{2} c^{3} f -2 \textit {\_R1}^{2} d^{3} e -\textit {\_R1}^{4} f -2 \textit {\_R1}^{2} c^{2} f -2 \textit {\_R1}^{2} c f +c^{3} f -d^{3} e +2 \textit {\_R1}^{2} f +c^{2} f -f c -f}\right )}{3}\) | \(681\) |
Input:
int((a+b*arctanh(d*x+c))/(f*x^3+e),x,method=_RETURNVERBOSE)
Output:
-1/6*d^2*b/f*sum(1/(_R1^2+2*_R1*c+c^2-2*_R1-2*c+1)*(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))+1/3*d^2*a/f*sum(1/(_R ^2+2*_R*c+c^2-2*_R-2*c+1)*ln(-d*x-_R-c+1),_R=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))+1/6*b*d^2/f*sum(1/( _R1^2-2*_R1*c+c^2-2*_R1+2*c+1)*(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))
\[ \int \frac {a+b \text {arctanh}(c+d x)}{e+f x^3} \, dx=\int { \frac {b \operatorname {artanh}\left (d x + c\right ) + a}{f x^{3} + e} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))/(f*x^3+e),x, algorithm="fricas")
Output:
integral((b*arctanh(d*x + c) + a)/(f*x^3 + e), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c+d x)}{e+f x^3} \, dx=\text {Timed out} \] Input:
integrate((a+b*atanh(d*x+c))/(f*x**3+e),x)
Output:
Timed out
Exception generated. \[ \int \frac {a+b \text {arctanh}(c+d x)}{e+f x^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*arctanh(d*x+c))/(f*x^3+e),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \text {arctanh}(c+d x)}{e+f x^3} \, dx=\int { \frac {b \operatorname {artanh}\left (d x + c\right ) + a}{f x^{3} + e} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))/(f*x^3+e),x, algorithm="giac")
Output:
integrate((b*arctanh(d*x + c) + a)/(f*x^3 + e), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c+d x)}{e+f x^3} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c+d\,x\right )}{f\,x^3+e} \,d x \] Input:
int((a + b*atanh(c + d*x))/(e + f*x^3),x)
Output:
int((a + b*atanh(c + d*x))/(e + f*x^3), x)
\[ \int \frac {a+b \text {arctanh}(c+d x)}{e+f x^3} \, dx=\frac {-2 e^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {e^{\frac {1}{3}}-2 f^{\frac {1}{3}} x}{e^{\frac {1}{3}} \sqrt {3}}\right ) a -e^{\frac {1}{3}} \mathrm {log}\left (e^{\frac {2}{3}}-f^{\frac {1}{3}} e^{\frac {1}{3}} x +f^{\frac {2}{3}} x^{2}\right ) a +2 e^{\frac {1}{3}} \mathrm {log}\left (e^{\frac {1}{3}}+f^{\frac {1}{3}} x \right ) a +6 f^{\frac {1}{3}} \left (\int \frac {\mathit {atanh} \left (d x +c \right )}{f \,x^{3}+e}d x \right ) b e}{6 f^{\frac {1}{3}} e} \] Input:
int((a+b*atanh(d*x+c))/(f*x^3+e),x)
Output:
( - 2*e**(1/3)*sqrt(3)*atan((e**(1/3) - 2*f**(1/3)*x)/(e**(1/3)*sqrt(3)))* a - e**(1/3)*log(e**(2/3) - f**(1/3)*e**(1/3)*x + f**(2/3)*x**2)*a + 2*e** (1/3)*log(e**(1/3) + f**(1/3)*x)*a + 6*f**(1/3)*int(atanh(c + d*x)/(e + f* x**3),x)*b*e)/(6*f**(1/3)*e)