Integrand size = 23, antiderivative size = 791 \[ \int \frac {a+b \text {arctanh}(c+d x)}{x^3 \left (e+f x^3\right )} \, dx=-\frac {b d}{2 \left (1-c^2\right ) e x}-\frac {a+b \text {arctanh}(c+d x)}{2 e x^2}+\frac {b c d^2 \log (x)}{\left (1-c^2\right )^2 e}-\frac {b d^2 \log (1-c-d x)}{4 (1-c)^2 e}+\frac {f^{2/3} (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1+c+d x}\right )}{3 e^{5/3}}-\frac {\sqrt [3]{-1} f^{2/3} (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1+c+d x}\right )}{3 e^{5/3}}+\frac {(-1)^{2/3} f^{2/3} (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1+c+d x}\right )}{3 e^{5/3}}+\frac {b d^2 \log (1+c+d x)}{4 (1+c)^2 e}-\frac {f^{2/3} (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^{5/3}}-\frac {(-1)^{2/3} f^{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^{5/3}}+\frac {\sqrt [3]{-1} f^{2/3} (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^{5/3}}-\frac {b f^{2/3} \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{6 e^{5/3}}+\frac {\sqrt [3]{-1} b f^{2/3} \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{6 e^{5/3}}-\frac {(-1)^{2/3} b f^{2/3} \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{6 e^{5/3}}+\frac {b f^{2/3} \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^{5/3}}+\frac {(-1)^{2/3} b f^{2/3} \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^{5/3}}-\frac {\sqrt [3]{-1} b f^{2/3} \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^{5/3}} \] Output:
-1/2*b*d/(-c^2+1)/e/x-1/2*(a+b*arctanh(d*x+c))/e/x^2+b*c*d^2*ln(x)/(-c^2+1 )^2/e-1/4*b*d^2*ln(-d*x-c+1)/(1-c)^2/e+1/3*f^(2/3)*(a+b*arctanh(d*x+c))*ln (2/(d*x+c+1))/e^(5/3)-1/3*(-1)^(1/3)*f^(2/3)*(a+b*arctanh(d*x+c))*ln(2/(d* x+c+1))/e^(5/3)+1/3*(-1)^(2/3)*f^(2/3)*(a+b*arctanh(d*x+c))*ln(2/(d*x+c+1) )/e^(5/3)+1/4*b*d^2*ln(d*x+c+1)/(1+c)^2/e-1/3*f^(2/3)*(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^(5/3)-1 /3*(-1)^(2/3)*f^(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^(5/3)+1/3*(-1)^(1 /3)*f^(2/3)*(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^(5/3)-1/6*b*f^(2/3)*polylog (2,1-2/(d*x+c+1))/e^(5/3)+1/6*(-1)^(1/3)*b*f^(2/3)*polylog(2,1-2/(d*x+c+1) )/e^(5/3)-1/6*(-1)^(2/3)*b*f^(2/3)*polylog(2,1-2/(d*x+c+1))/e^(5/3)+1/6*b* f^(2/3)*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^(5/3)+1/6*(-1)^(2/3)*b*f^(2/3)*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^(5/3)-1/6*( -1)^(1/3)*b*f^(2/3)*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^(5/3)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 7.37 (sec) , antiderivative size = 1564, normalized size of antiderivative = 1.98 \[ \int \frac {a+b \text {arctanh}(c+d x)}{x^3 \left (e+f x^3\right )} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcTanh[c + d*x])/(x^3*(e + f*x^3)),x]
Output:
((-3*a*e^(2/3))/x^2 + 2*Sqrt[3]*a*f^(2/3)*ArcTan[(1 - (2*f^(1/3)*x)/e^(1/3 ))/Sqrt[3]] - 2*a*f^(2/3)*Log[e^(1/3) + f^(1/3)*x] + a*f^(2/3)*Log[e^(2/3) - e^(1/3)*f^(1/3)*x + f^(2/3)*x^2] + (3*b*e^(2/3)*(c*(-1 - c^4 + d^2*x^2 + c^2*(2 + d^2*x^2))*ArcTanh[c + d*x] + d*x*((-1 + c^2)*(c + d*x) + 2*c^2* d*x*Log[-((d*x)/Sqrt[1 - (c + d*x)^2])])))/((-1 + c)^2*c*(1 + c)^2*x^2) - b*d^2*e^(2/3)*f*(4*ArcTanh[c + d*x]*RootSum[d^3*e - f - 3*c*f - 3*c^2*f - c^3*f + 3*d^3*e*#1 + 3*f*#1 + 3*c*f*#1 - 3*c^2*f*#1 - 3*c^3*f*#1 + 3*d^3*e *#1^2 - 3*f*#1^2 + 3*c*f*#1^2 + 3*c^2*f*#1^2 - 3*c^3*f*#1^2 + d^3*e*#1^3 + f*#1^3 - 3*c*f*#1^3 + 3*c^2*f*#1^3 - c^3*f*#1^3 & , (ArcTanh[c + d*x] + L og[(-1 - c - d*x + #1 - c*#1 - d*x*#1)/Sqrt[1 - (c + d*x)^2]] + ArcTanh[c + d*x]*#1 + Log[(-1 - c - d*x + #1 - c*#1 - d*x*#1)/Sqrt[1 - (c + d*x)^2]] *#1)/(d^3*e + f + c*f - c^2*f - c^3*f + 2*d^3*e*#1 - 2*f*#1 + 2*c*f*#1 + 2 *c^2*f*#1 - 2*c^3*f*#1 + d^3*e*#1^2 + f*#1^2 - 3*c*f*#1^2 + 3*c^2*f*#1^2 - c^3*f*#1^2) & ] + RootSum[d^3*e - f - 3*c*f - 3*c^2*f - c^3*f + 3*d^3*e*# 1 + 3*f*#1 + 3*c*f*#1 - 3*c^2*f*#1 - 3*c^3*f*#1 + 3*d^3*e*#1^2 - 3*f*#1^2 + 3*c*f*#1^2 + 3*c^2*f*#1^2 - 3*c^3*f*#1^2 + d^3*e*#1^3 + f*#1^3 - 3*c*f*# 1^3 + 3*c^2*f*#1^3 - c^3*f*#1^3 & , (I*Pi*ArcTanh[c + d*x] + 2*ArcTanh[c + d*x]*ArcTanh[(1 - #1)/(1 + #1)] - I*Pi*Log[1 + E^(2*ArcTanh[c + d*x])] + 2*ArcTanh[c + d*x]*Log[1 - E^(-2*(ArcTanh[c + d*x] + ArcTanh[(1 - #1)/(1 + #1)]))] + 2*ArcTanh[(1 - #1)/(1 + #1)]*Log[1 - E^(-2*(ArcTanh[c + d*x]...
Time = 2.17 (sec) , antiderivative size = 1028, normalized size of antiderivative = 1.30, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c+d x)}{x^3 \left (e+f x^3\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {a}{x^3 \left (e+f x^3\right )}+\frac {b \text {arctanh}(c+d x)}{x^3 \left (e+f x^3\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b c \log (x) d^2}{\left (1-c^2\right )^2 e}-\frac {b \log (-c-d x+1) d^2}{4 (1-c)^2 e}+\frac {b \log (c+d x+1) d^2}{4 (c+1)^2 e}-\frac {b d}{2 \left (1-c^2\right ) e x}+\frac {a f^{2/3} \arctan \left (\frac {\sqrt [3]{e}-2 \sqrt [3]{f} x}{\sqrt {3} \sqrt [3]{e}}\right )}{\sqrt {3} e^{5/3}}-\frac {b \text {arctanh}(c+d x)}{2 e x^2}-\frac {a f^{2/3} \log \left (\sqrt [3]{f} x+\sqrt [3]{e}\right )}{3 e^{5/3}}+\frac {b f^{2/3} \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^{5/3}}-\frac {b f^{2/3} \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^{5/3}}+\frac {(-1)^{2/3} b f^{2/3} \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^{5/3}}-\frac {(-1)^{2/3} b f^{2/3} \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^{5/3}}-\frac {\sqrt [3]{-1} b f^{2/3} \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^{5/3}}+\frac {\sqrt [3]{-1} b f^{2/3} \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^{5/3}}+\frac {a f^{2/3} \log \left (f^{2/3} x^2-\sqrt [3]{e} \sqrt [3]{f} x+e^{2/3}\right )}{6 e^{5/3}}+\frac {b f^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{f} (-c-d x+1)}{\sqrt [3]{f} (1-c)+d \sqrt [3]{e}}\right )}{6 e^{5/3}}+\frac {(-1)^{2/3} b f^{2/3} \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^{5/3}}-\frac {\sqrt [3]{-1} b f^{2/3} \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^{5/3}}-\frac {b f^{2/3} \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{f} (c+d x+1)}{d \sqrt [3]{e}-(c+1) \sqrt [3]{f}}\right )}{6 e^{5/3}}-\frac {(-1)^{2/3} b f^{2/3} \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^{5/3}}+\frac {\sqrt [3]{-1} b f^{2/3} \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{f} (c+d x+1)}{d \sqrt [3]{e}-(-1)^{2/3} (c+1) \sqrt [3]{f}}\right )}{6 e^{5/3}}-\frac {a}{2 e x^2}\) |
Input:
Int[(a + b*ArcTanh[c + d*x])/(x^3*(e + f*x^3)),x]
Output:
-1/2*a/(e*x^2) - (b*d)/(2*(1 - c^2)*e*x) + (a*f^(2/3)*ArcTan[(e^(1/3) - 2* f^(1/3)*x)/(Sqrt[3]*e^(1/3))])/(Sqrt[3]*e^(5/3)) - (b*ArcTanh[c + d*x])/(2 *e*x^2) + (b*c*d^2*Log[x])/((1 - c^2)^2*e) - (b*d^2*Log[1 - c - d*x])/(4*( 1 - c)^2*e) + (b*d^2*Log[1 + c + d*x])/(4*(1 + c)^2*e) - (a*f^(2/3)*Log[e^ (1/3) + f^(1/3)*x])/(3*e^(5/3)) + (b*f^(2/3)*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^(5/3)) - (b*f^(2/3) *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^(5/3)) + ((-1)^(2/3)*b*f^(2/3)*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^ (5/3)) - ((-1)^(2/3)*b*f^(2/3)*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^(5/3)) - (( -1)^(1/3)*b*f^(2/3)*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^(5/3)) + ((-1)^(1/3)*b *f^(2/3)*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^(5/3)) + (a*f^(2/3)*Log[e^(2/3) - e^(1/3)*f^(1/3)*x + f^(2/3)*x^2])/(6*e^(5/3)) + (b*f^(2/3)*PolyLog[2, (f^ (1/3)*(1 - c - d*x))/(d*e^(1/3) + (1 - c)*f^(1/3))])/(6*e^(5/3)) + ((-1)^( 2/3)*b*f^(2/3)*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^(5/3)) - ((-1)^(1/3)*b*f^(2/3)*PolyL og[2, ((-1)^(2/3)*f^(1/3)*(1 - c - d*x))/(d*e^(1/3) + (-1)^(2/3)*(1 - c...
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 = 0.53 (sec) , antiderivative size = 662, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(\frac {d^{2} b \ln \left (-d x \right )}{4 e \left (-1+c \right )^{2}}-\frac {d b}{4 e \left (-1+c \right )^{2} x}+\frac {d b c}{4 e \left (-1+c \right )^{2} x}-\frac {d^{2} b \ln \left (-d x -c +1\right )}{4 e \left (-1+c \right )^{2}}+\frac {b \ln \left (-d x -c +1\right ) c^{2}}{4 e \,x^{2} \left (-1+c \right )^{2}}-\frac {b \ln \left (-d x -c +1\right ) c}{2 e \,x^{2} \left (-1+c \right )^{2}}+\frac {b \ln \left (-d x -c +1\right )}{4 e \,x^{2} \left (-1+c \right )^{2}}+\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 e}-\frac {a}{2 e \,x^{2}}-\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 e}-\frac {b \,d^{2} \ln \left (d x \right )}{4 e \left (1+c \right )^{2}}-\frac {b d}{4 e \left (1+c \right )^{2} x}-\frac {b d c}{4 e \left (1+c \right )^{2} x}+\frac {b \,d^{2} \ln \left (d x +c +1\right )}{4 \left (1+c \right )^{2} e}-\frac {b \ln \left (d x +c +1\right ) c^{2}}{4 e \,x^{2} \left (1+c \right )^{2}}-\frac {b \ln \left (d x +c +1\right ) c}{2 e \,x^{2} \left (1+c \right )^{2}}-\frac {b \ln \left (d x +c +1\right )}{4 e \,x^{2} \left (1+c \right )^{2}}-\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 e}\) | \(662\) |
| derivativedivides | \(d^{2} \left (\frac {a \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 e}-\frac {a}{2 e \,d^{2} x^{2}}-\frac {b \,\operatorname {arctanh}\left (d x +c \right )}{2 e \,d^{2} x^{2}}-\frac {b \ln \left (d x +c -1\right )}{4 e \left (-1+c \right )^{2}}+\frac {b \ln \left (d x +c +1\right )}{4 e \left (1+c \right )^{2}}+\frac {b}{2 e \left (-1+c \right ) \left (1+c \right ) d x}+\frac {b c \ln \left (-d x \right )}{e \left (-1+c \right )^{2} \left (1+c \right )^{2}}+\frac {2 b f \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 e}+\frac {2 b f \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 e}\right )\) | \(772\) |
| default | \(d^{2} \left (\frac {a \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 e}-\frac {a}{2 e \,d^{2} x^{2}}-\frac {b \,\operatorname {arctanh}\left (d x +c \right )}{2 e \,d^{2} x^{2}}-\frac {b \ln \left (d x +c -1\right )}{4 e \left (-1+c \right )^{2}}+\frac {b \ln \left (d x +c +1\right )}{4 e \left (1+c \right )^{2}}+\frac {b}{2 e \left (-1+c \right ) \left (1+c \right ) d x}+\frac {b c \ln \left (-d x \right )}{e \left (-1+c \right )^{2} \left (1+c \right )^{2}}+\frac {2 b f \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 e}+\frac {2 b f \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 e}\right )\) | \(772\) |
| parts | \(-\frac {a \ln \left (x +\left (\frac {e}{f}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {e}{f}\right )^{\frac {2}{3}}}+\frac {a \ln \left (x^{2}-\left (\frac {e}{f}\right )^{\frac {1}{3}} x +\left (\frac {e}{f}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {e}{f}\right )^{\frac {2}{3}}}-\frac {a \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {e}{f}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {e}{f}\right )^{\frac {2}{3}}}-\frac {a}{2 e \,x^{2}}-\frac {b \,\operatorname {arctanh}\left (d x +c \right )}{2 e \,x^{2}}+\frac {b \,d^{2} \ln \left (d x +c +1\right )}{4 \left (1+c \right )^{2} e}+\frac {b d}{2 e \left (-1+c \right ) \left (1+c \right ) x}+\frac {b \,d^{2} c \ln \left (d x \right )}{e \left (-1+c \right )^{2} \left (1+c \right )^{2}}-\frac {b \,d^{2} \ln \left (d x +c -1\right )}{4 e \left (-1+c \right )^{2}}+\frac {2 b \,d^{2} f \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 e}+\frac {2 b \,d^{2} f \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 e}\) | \(798\) |
Input:
int((a+b*arctanh(d*x+c))/x^3/(f*x^3+e),x,method=_RETURNVERBOSE)
Output:
1/4*d^2*b/e/(-1+c)^2*ln(-d*x)-1/4*d*b/e/(-1+c)^2/x+1/4*d*b/e/(-1+c)^2/x*c- 1/4*d^2*b/e*ln(-d*x-c+1)/(-1+c)^2+1/4*b/e*ln(-d*x-c+1)/x^2/(-1+c)^2*c^2-1/ 2*b/e*ln(-d*x-c+1)/x^2/(-1+c)^2*c+1/4*b/e*ln(-d*x-c+1)/x^2/(-1+c)^2+1/6*d^ 2*b/e*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))-1/2*a/e/x^2-1/3*d^2*a*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))/e-1/4*b*d^2/e/(1 +c)^2*ln(d*x)-1/4*b*d/e/(1+c)^2/x-1/4*b*d/e/(1+c)^2/x*c+1/4*b*d^2*ln(d*x+c +1)/(1+c)^2/e-1/4*b/e*ln(d*x+c+1)/x^2/(1+c)^2*c^2-1/2*b/e*ln(d*x+c+1)/x^2/ (1+c)^2*c-1/4*b/e*ln(d*x+c+1)/x^2/(1+c)^2-1/6*b*d^2/e*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))
\[ \int \frac {a+b \text {arctanh}(c+d x)}{x^3 \left (e+f x^3\right )} \, dx=\int { \frac {b \operatorname {artanh}\left (d x + c\right ) + a}{{\left (f x^{3} + e\right )} x^{3}} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))/x^3/(f*x^3+e),x, algorithm="fricas")
Output:
integral((b*arctanh(d*x + c) + a)/(f*x^6 + e*x^3), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c+d x)}{x^3 \left (e+f x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((a+b*atanh(d*x+c))/x**3/(f*x**3+e),x)
Output:
Timed out
Exception generated. \[ \int \frac {a+b \text {arctanh}(c+d x)}{x^3 \left (e+f x^3\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*arctanh(d*x+c))/x^3/(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
\[ \int \frac {a+b \text {arctanh}(c+d x)}{x^3 \left (e+f x^3\right )} \, dx=\int { \frac {b \operatorname {artanh}\left (d x + c\right ) + a}{{\left (f x^{3} + e\right )} x^{3}} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))/x^3/(f*x^3+e),x, algorithm="giac")
Output:
integrate((b*arctanh(d*x + c) + a)/((f*x^3 + e)*x^3), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c+d x)}{x^3 \left (e+f x^3\right )} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c+d\,x\right )}{x^3\,\left (f\,x^3+e\right )} \,d x \] Input:
int((a + b*atanh(c + d*x))/(x^3*(e + f*x^3)),x)
Output:
int((a + b*atanh(c + d*x))/(x^3*(e + f*x^3)), x)
\[ \int \frac {a+b \text {arctanh}(c+d x)}{x^3 \left (e+f x^3\right )} \, dx=\text {too large to display} \] Input:
int((a+b*atanh(d*x+c))/x^3/(f*x^3+e),x)
Output:
(6*e**(1/3)*sqrt(3)*atan((e**(1/3) - 2*f**(1/3)*x)/(e**(1/3)*sqrt(3)))*a*c **4*f*x**2 - 4*e**(1/3)*sqrt(3)*atan((e**(1/3) - 2*f**(1/3)*x)/(e**(1/3)*s qrt(3)))*a*c**2*f*x**2 - 2*e**(1/3)*sqrt(3)*atan((e**(1/3) - 2*f**(1/3)*x) /(e**(1/3)*sqrt(3)))*a*f*x**2 + 6*f**(1/3)*atanh(c + d*x)**2*b*c**3*d**2*e *x**2 - 6*f**(1/3)*atanh(c + d*x)**2*b*c*d**2*e*x**2 + 3*f**(1/3)*atanh(c + d*x)*b*c**4*e - 12*f**(1/3)*atanh(c + d*x)*b*c**3*d*e*x - 15*f**(1/3)*at anh(c + d*x)*b*c**2*d**2*e*x**2 - 6*f**(1/3)*atanh(c + d*x)*b*c**2*e + 18* f**(1/3)*atanh(c + d*x)*b*c*d**2*e*x**2 + 12*f**(1/3)*atanh(c + d*x)*b*c*d *e*x - 3*f**(1/3)*atanh(c + d*x)*b*d**2*e*x**2 + 3*f**(1/3)*atanh(c + d*x) *b*e + 3*e**(1/3)*log(e**(2/3) - f**(1/3)*e**(1/3)*x + f**(2/3)*x**2)*a*c* *4*f*x**2 - 2*e**(1/3)*log(e**(2/3) - f**(1/3)*e**(1/3)*x + f**(2/3)*x**2) *a*c**2*f*x**2 - e**(1/3)*log(e**(2/3) - f**(1/3)*e**(1/3)*x + f**(2/3)*x* *2)*a*f*x**2 - 6*e**(1/3)*log(e**(1/3) + f**(1/3)*x)*a*c**4*f*x**2 + 4*e** (1/3)*log(e**(1/3) + f**(1/3)*x)*a*c**2*f*x**2 + 2*e**(1/3)*log(e**(1/3) + f**(1/3)*x)*a*f*x**2 + 72*f**(1/3)*int(atanh(c + d*x)/(3*c**4*e*x**3 + 3* c**4*f*x**6 + 6*c**3*d*e*x**4 + 6*c**3*d*f*x**7 + 3*c**2*d**2*e*x**5 + 3*c **2*d**2*f*x**8 - 2*c**2*e*x**3 - 2*c**2*f*x**6 + 2*c*d*e*x**4 + 2*c*d*f*x **7 + d**2*e*x**5 + d**2*f*x**8 - e*x**3 - f*x**6),x)*b*c**8*e**2*x**2 - 1 20*f**(1/3)*int(atanh(c + d*x)/(3*c**4*e*x**3 + 3*c**4*f*x**6 + 6*c**3*d*e *x**4 + 6*c**3*d*f*x**7 + 3*c**2*d**2*e*x**5 + 3*c**2*d**2*f*x**8 - 2*c...