Integrand size = 24, antiderivative size = 381 \[ \int \frac {x^3 (a+b \text {arctanh}(c+d x))}{e-f x^2} \, dx=-\frac {b x}{2 d f}-\frac {x^2 (a+b \text {arctanh}(c+d x))}{2 f}-\frac {b (1-c)^2 \log (1-c-d x)}{4 d^2 f}+\frac {e (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1+c+d x}\right )}{f^2}+\frac {b (1+c)^2 \log (1+c+d x)}{4 d^2 f}-\frac {e (a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left (\sqrt {e}-\sqrt {f} x\right )}{\left (d \sqrt {e}-(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{2 f^2}-\frac {e (a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left (\sqrt {e}+\sqrt {f} x\right )}{\left (d \sqrt {e}+(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{2 f^2}-\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f^2}+\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt {e}-\sqrt {f} x\right )}{\left (d \sqrt {e}-(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{4 f^2}+\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt {e}+\sqrt {f} x\right )}{\left (d \sqrt {e}+(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{4 f^2} \] Output:
-1/2*b*x/d/f-1/2*x^2*(a+b*arctanh(d*x+c))/f-1/4*b*(1-c)^2*ln(-d*x-c+1)/d^2 /f+e*(a+b*arctanh(d*x+c))*ln(2/(d*x+c+1))/f^2+1/4*b*(1+c)^2*ln(d*x+c+1)/d^ 2/f-1/2*e*(a+b*arctanh(d*x+c))*ln(2*d*(e^(1/2)-f^(1/2)*x)/(d*e^(1/2)-(1-c) *f^(1/2))/(d*x+c+1))/f^2-1/2*e*(a+b*arctanh(d*x+c))*ln(2*d*(e^(1/2)+f^(1/2 )*x)/(d*e^(1/2)+(1-c)*f^(1/2))/(d*x+c+1))/f^2-1/2*b*e*polylog(2,1-2/(d*x+c +1))/f^2+1/4*b*e*polylog(2,1-2*d*(e^(1/2)-f^(1/2)*x)/(d*e^(1/2)-(1-c)*f^(1 /2))/(d*x+c+1))/f^2+1/4*b*e*polylog(2,1-2*d*(e^(1/2)+f^(1/2)*x)/(d*e^(1/2) +(1-c)*f^(1/2))/(d*x+c+1))/f^2
Time = 25.80 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.82 \[ \int \frac {x^3 (a+b \text {arctanh}(c+d x))}{e-f x^2} \, dx=-\frac {2 a d^2 \left (f x^2+e \log \left (e-f x^2\right )\right )+b \left (2 d f x+2 d^2 f x^2 \text {arctanh}(c+d x)+f \log (1-c-d x)-2 c f \log (1-c-d x)+c^2 f \log (1-c-d x)-d^2 e \log \left (-\frac {\sqrt {e}}{\sqrt {f}}+x\right ) \log (1-c-d x)-d^2 e \log \left (\frac {\sqrt {e}}{\sqrt {f}}+x\right ) \log (1-c-d x)+d^2 e \log \left (\frac {\sqrt {e}}{\sqrt {f}}+x\right ) \log \left (-\frac {\sqrt {f} (-1+c+d x)}{d \sqrt {e}-(-1+c) \sqrt {f}}\right )+d^2 e \log \left (-\frac {\sqrt {e}}{\sqrt {f}}+x\right ) \log \left (\frac {\sqrt {f} (-1+c+d x)}{d \sqrt {e}+(-1+c) \sqrt {f}}\right )-f \log (1+c+d x)-2 c f \log (1+c+d x)-c^2 f \log (1+c+d x)+d^2 e \log \left (-\frac {\sqrt {e}}{\sqrt {f}}+x\right ) \log (1+c+d x)+d^2 e \log \left (\frac {\sqrt {e}}{\sqrt {f}}+x\right ) \log (1+c+d x)-d^2 e \log \left (\frac {\sqrt {e}}{\sqrt {f}}+x\right ) \log \left (-\frac {\sqrt {f} (1+c+d x)}{d \sqrt {e}-(1+c) \sqrt {f}}\right )-d^2 e \log \left (-\frac {\sqrt {e}}{\sqrt {f}}+x\right ) \log \left (\frac {\sqrt {f} (1+c+d x)}{d \sqrt {e}+(1+c) \sqrt {f}}\right )+2 d^2 e \text {arctanh}(c+d x) \log \left (e-f x^2\right )+d^2 e \log (1-c-d x) \log \left (e-f x^2\right )-d^2 e \log (1+c+d x) \log \left (e-f x^2\right )+d^2 e \operatorname {PolyLog}\left (2,\frac {d \left (\frac {\sqrt {e}}{\sqrt {f}}+x\right )}{1-c+\frac {d \sqrt {e}}{\sqrt {f}}}\right )+d^2 e \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {e}-\sqrt {f} x\right )}{d \sqrt {e}+(-1+c) \sqrt {f}}\right )-d^2 e \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {e}-\sqrt {f} x\right )}{d \sqrt {e}+(1+c) \sqrt {f}}\right )-d^2 e \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {e}+\sqrt {f} x\right )}{d \sqrt {e}-(1+c) \sqrt {f}}\right )\right )}{4 d^2 f^2} \] Input:
Integrate[(x^3*(a + b*ArcTanh[c + d*x]))/(e - f*x^2),x]
Output:
-1/4*(2*a*d^2*(f*x^2 + e*Log[e - f*x^2]) + b*(2*d*f*x + 2*d^2*f*x^2*ArcTan h[c + d*x] + f*Log[1 - c - d*x] - 2*c*f*Log[1 - c - d*x] + c^2*f*Log[1 - c - d*x] - d^2*e*Log[-(Sqrt[e]/Sqrt[f]) + x]*Log[1 - c - d*x] - d^2*e*Log[S qrt[e]/Sqrt[f] + x]*Log[1 - c - d*x] + d^2*e*Log[Sqrt[e]/Sqrt[f] + x]*Log[ -((Sqrt[f]*(-1 + c + d*x))/(d*Sqrt[e] - (-1 + c)*Sqrt[f]))] + d^2*e*Log[-( Sqrt[e]/Sqrt[f]) + x]*Log[(Sqrt[f]*(-1 + c + d*x))/(d*Sqrt[e] + (-1 + c)*S qrt[f])] - f*Log[1 + c + d*x] - 2*c*f*Log[1 + c + d*x] - c^2*f*Log[1 + c + d*x] + d^2*e*Log[-(Sqrt[e]/Sqrt[f]) + x]*Log[1 + c + d*x] + d^2*e*Log[Sqr t[e]/Sqrt[f] + x]*Log[1 + c + d*x] - d^2*e*Log[Sqrt[e]/Sqrt[f] + x]*Log[-( (Sqrt[f]*(1 + c + d*x))/(d*Sqrt[e] - (1 + c)*Sqrt[f]))] - d^2*e*Log[-(Sqrt [e]/Sqrt[f]) + x]*Log[(Sqrt[f]*(1 + c + d*x))/(d*Sqrt[e] + (1 + c)*Sqrt[f] )] + 2*d^2*e*ArcTanh[c + d*x]*Log[e - f*x^2] + d^2*e*Log[1 - c - d*x]*Log[ e - f*x^2] - d^2*e*Log[1 + c + d*x]*Log[e - f*x^2] + d^2*e*PolyLog[2, (d*( Sqrt[e]/Sqrt[f] + x))/(1 - c + (d*Sqrt[e])/Sqrt[f])] + d^2*e*PolyLog[2, (d *(Sqrt[e] - Sqrt[f]*x))/(d*Sqrt[e] + (-1 + c)*Sqrt[f])] - d^2*e*PolyLog[2, (d*(Sqrt[e] - Sqrt[f]*x))/(d*Sqrt[e] + (1 + c)*Sqrt[f])] - d^2*e*PolyLog[ 2, (d*(Sqrt[e] + Sqrt[f]*x))/(d*Sqrt[e] - (1 + c)*Sqrt[f])]))/(d^2*f^2)
Time = 1.26 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
| \(\displaystyle \int \frac {x^3 (a+b \text {arctanh}(c+d x))}{e-f x^2} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {a x^3}{f x^2-e}-\frac {b x^3 \text {arctanh}(c+d x)}{f x^2-e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a e \log \left (e-f x^2\right )}{2 f^2}-\frac {a x^2}{2 f}+\frac {b e \text {arctanh}(c+d x) \log \left (\frac {2}{c+d x+1}\right )}{f^2}-\frac {b e \text {arctanh}(c+d x) \log \left (\frac {2 d \left (\sqrt {e}-\sqrt {f} x\right )}{(c+d x+1) \left (d \sqrt {e}-(1-c) \sqrt {f}\right )}\right )}{2 f^2}-\frac {b e \text {arctanh}(c+d x) \log \left (\frac {2 d \left (\sqrt {e}+\sqrt {f} x\right )}{(c+d x+1) \left ((1-c) \sqrt {f}+d \sqrt {e}\right )}\right )}{2 f^2}-\frac {b x^2 \text {arctanh}(c+d x)}{2 f}-\frac {b (1-c)^2 \log (-c-d x+1)}{4 d^2 f}+\frac {b (c+1)^2 \log (c+d x+1)}{4 d^2 f}-\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right )}{2 f^2}+\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt {e}-\sqrt {f} x\right )}{\left (d \sqrt {e}-(1-c) \sqrt {f}\right ) (c+d x+1)}\right )}{4 f^2}+\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt {f} x+\sqrt {e}\right )}{\left (\sqrt {f} (1-c)+d \sqrt {e}\right ) (c+d x+1)}\right )}{4 f^2}-\frac {b x}{2 d f}\) |
Input:
Int[(x^3*(a + b*ArcTanh[c + d*x]))/(e - f*x^2),x]
Output:
-1/2*(b*x)/(d*f) - (a*x^2)/(2*f) - (b*x^2*ArcTanh[c + d*x])/(2*f) - (b*(1 - c)^2*Log[1 - c - d*x])/(4*d^2*f) + (b*e*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/f^2 + (b*(1 + c)^2*Log[1 + c + d*x])/(4*d^2*f) - (b*e*ArcTanh[c + d *x]*Log[(2*d*(Sqrt[e] - Sqrt[f]*x))/((d*Sqrt[e] - (1 - c)*Sqrt[f])*(1 + c + d*x))])/(2*f^2) - (b*e*ArcTanh[c + d*x]*Log[(2*d*(Sqrt[e] + Sqrt[f]*x))/ ((d*Sqrt[e] + (1 - c)*Sqrt[f])*(1 + c + d*x))])/(2*f^2) - (a*e*Log[e - f*x ^2])/(2*f^2) - (b*e*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f^2) + (b*e*PolyLo g[2, 1 - (2*d*(Sqrt[e] - Sqrt[f]*x))/((d*Sqrt[e] - (1 - c)*Sqrt[f])*(1 + c + d*x))])/(4*f^2) + (b*e*PolyLog[2, 1 - (2*d*(Sqrt[e] + Sqrt[f]*x))/((d*S qrt[e] + (1 - c)*Sqrt[f])*(1 + c + d*x))])/(4*f^2)
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]
Time = 0.44 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.61
| method | result | size |
| parts | \(-\frac {a \,x^{2}}{2 f}-\frac {a e \ln \left (f \,x^{2}-e \right )}{2 f^{2}}+\frac {b \left (-\frac {d^{2} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{2}}{2 f}+\frac {d^{2} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right ) c}{f}-\frac {d^{4} \operatorname {arctanh}\left (d x +c \right ) e \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2 f^{2}}+\frac {d^{2} \left (-\frac {d x +c +\frac {\left (-2 c +1\right ) \ln \left (d x +c -1\right )}{2}-\frac {\left (2 c +1\right ) \ln \left (d x +c +1\right )}{2}}{f}-\frac {e \,d^{2} \left (-\frac {\ln \left (d x +c +1\right ) \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2}+f \left (\frac {\ln \left (d x +c +1\right ) \left (\ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )+\ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )\right )}{2 f}+\frac {\operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )+\operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{2 f}\right )+\frac {\ln \left (d x +c -1\right ) \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2}-f \left (\frac {\ln \left (d x +c -1\right ) \left (\ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c -1\right )-f}{d \sqrt {e f}+f c -f}\right )+\ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c -1\right )+f}{d \sqrt {e f}-f c +f}\right )\right )}{2 f}+\frac {\operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c -1\right )-f}{d \sqrt {e f}+f c -f}\right )+\operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c -1\right )+f}{d \sqrt {e f}-f c +f}\right )}{2 f}\right )\right )}{f^{2}}\right )}{2}\right )}{d^{4}}\) | \(615\) |
| derivativedivides | \(\frac {a \,d^{2} \left (\frac {c \left (d x +c \right )-\frac {\left (d x +c \right )^{2}}{2}}{f}-\frac {e \,d^{2} \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2 f^{2}}\right )+b \,d^{2} \left (\frac {\operatorname {arctanh}\left (d x +c \right ) c \left (d x +c \right )}{f}-\frac {\operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{2}}{2 f}-\frac {\operatorname {arctanh}\left (d x +c \right ) e \,d^{2} \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2 f^{2}}-\frac {e \,d^{2} \left (\frac {\ln \left (d x +c -1\right ) \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2}+f \left (-\frac {\ln \left (d x +c -1\right ) \left (\ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c -1\right )-f}{d \sqrt {e f}+f c -f}\right )+\ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c -1\right )+f}{d \sqrt {e f}-f c +f}\right )\right )}{2 f}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c -1\right )-f}{d \sqrt {e f}+f c -f}\right )+\operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c -1\right )+f}{d \sqrt {e f}-f c +f}\right )}{2 f}\right )-\frac {\ln \left (d x +c +1\right ) \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2}-f \left (-\frac {\ln \left (d x +c +1\right ) \left (\ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )+\ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )\right )}{2 f}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )+\operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{2 f}\right )\right )}{2 f^{2}}+\frac {-d x -c +\frac {\left (2 c -1\right ) \ln \left (d x +c -1\right )}{2}-\frac {\left (-2 c -1\right ) \ln \left (d x +c +1\right )}{2}}{2 f}\right )}{d^{4}}\) | \(651\) |
| default | \(\frac {a \,d^{2} \left (\frac {c \left (d x +c \right )-\frac {\left (d x +c \right )^{2}}{2}}{f}-\frac {e \,d^{2} \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2 f^{2}}\right )+b \,d^{2} \left (\frac {\operatorname {arctanh}\left (d x +c \right ) c \left (d x +c \right )}{f}-\frac {\operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{2}}{2 f}-\frac {\operatorname {arctanh}\left (d x +c \right ) e \,d^{2} \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2 f^{2}}-\frac {e \,d^{2} \left (\frac {\ln \left (d x +c -1\right ) \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2}+f \left (-\frac {\ln \left (d x +c -1\right ) \left (\ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c -1\right )-f}{d \sqrt {e f}+f c -f}\right )+\ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c -1\right )+f}{d \sqrt {e f}-f c +f}\right )\right )}{2 f}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c -1\right )-f}{d \sqrt {e f}+f c -f}\right )+\operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c -1\right )+f}{d \sqrt {e f}-f c +f}\right )}{2 f}\right )-\frac {\ln \left (d x +c +1\right ) \ln \left (c^{2} f -2 c f \left (d x +c \right )-e \,d^{2}+f \left (d x +c \right )^{2}\right )}{2}-f \left (-\frac {\ln \left (d x +c +1\right ) \left (\ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )+\ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )\right )}{2 f}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )+\operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{2 f}\right )\right )}{2 f^{2}}+\frac {-d x -c +\frac {\left (2 c -1\right ) \ln \left (d x +c -1\right )}{2}-\frac {\left (-2 c -1\right ) \ln \left (d x +c +1\right )}{2}}{2 f}\right )}{d^{4}}\) | \(651\) |
| risch | \(-\frac {a \,x^{2}}{2 f}+\frac {b \ln \left (-d x -c +1\right ) x^{2}}{4 f}-\frac {b \ln \left (-d x -c +1\right )}{4 d^{2} f}+\frac {b e \operatorname {dilog}\left (\frac {d \sqrt {e f}-\left (-d x -c +1\right ) f -f c +f}{d \sqrt {e f}-f c +f}\right )}{4 f^{2}}+\frac {b e \operatorname {dilog}\left (\frac {d \sqrt {e f}+\left (-d x -c +1\right ) f +f c -f}{d \sqrt {e f}+f c -f}\right )}{4 f^{2}}-\frac {a e \ln \left (f \left (-d x -c +1\right )^{2}+2 \left (-d x -c +1\right ) c f +c^{2} f -e \,d^{2}-2 \left (-d x -c +1\right ) f -2 f c +f \right )}{2 f^{2}}-\frac {b \ln \left (d x +c +1\right ) x^{2}}{4 f}+\frac {b \ln \left (d x +c +1\right )}{4 d^{2} f}-\frac {b e \operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )}{4 f^{2}}-\frac {b e \operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{4 f^{2}}-\frac {b \ln \left (-d x -c +1\right ) c^{2}}{4 d^{2} f}+\frac {b \ln \left (-d x -c +1\right ) c}{2 d^{2} f}+\frac {b e \ln \left (-d x -c +1\right ) \ln \left (\frac {d \sqrt {e f}-\left (-d x -c +1\right ) f -f c +f}{d \sqrt {e f}-f c +f}\right )}{4 f^{2}}+\frac {b e \ln \left (-d x -c +1\right ) \ln \left (\frac {d \sqrt {e f}+\left (-d x -c +1\right ) f +f c -f}{d \sqrt {e f}+f c -f}\right )}{4 f^{2}}+\frac {b \ln \left (d x +c +1\right ) c^{2}}{4 d^{2} f}+\frac {b \ln \left (d x +c +1\right ) c}{2 d^{2} f}-\frac {b e \ln \left (d x +c +1\right ) \ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )}{4 f^{2}}-\frac {b e \ln \left (d x +c +1\right ) \ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{4 f^{2}}-\frac {b x}{2 d f}-\frac {3 b c}{2 d^{2} f}+\frac {a \,c^{2}}{2 d^{2} f}-\frac {a c}{d^{2} f}+\frac {a}{2 d^{2} f}\) | \(691\) |
Input:
int(x^3*(a+b*arctanh(d*x+c))/(-f*x^2+e),x,method=_RETURNVERBOSE)
Output:
-1/2*a/f*x^2-1/2*a*e/f^2*ln(f*x^2-e)+b/d^4*(-1/2*d^2*arctanh(d*x+c)/f*(d*x +c)^2+d^2*arctanh(d*x+c)/f*(d*x+c)*c-1/2*d^4*arctanh(d*x+c)*e/f^2*ln(c^2*f -2*c*f*(d*x+c)-e*d^2+f*(d*x+c)^2)+1/2*d^2*(-1/f*(d*x+c+1/2*(-2*c+1)*ln(d*x +c-1)-1/2*(2*c+1)*ln(d*x+c+1))-e*d^2/f^2*(-1/2*ln(d*x+c+1)*ln(c^2*f-2*c*f* (d*x+c)-e*d^2+f*(d*x+c)^2)+f*(1/2*ln(d*x+c+1)*(ln((d*(e*f)^(1/2)+f*c-f*(d* x+c+1)+f)/(d*(e*f)^(1/2)+f*c+f))+ln((d*(e*f)^(1/2)-f*c+f*(d*x+c+1)-f)/(d*( e*f)^(1/2)-f*c-f)))/f+1/2*(dilog((d*(e*f)^(1/2)+f*c-f*(d*x+c+1)+f)/(d*(e*f )^(1/2)+f*c+f))+dilog((d*(e*f)^(1/2)-f*c+f*(d*x+c+1)-f)/(d*(e*f)^(1/2)-f*c -f)))/f)+1/2*ln(d*x+c-1)*ln(c^2*f-2*c*f*(d*x+c)-e*d^2+f*(d*x+c)^2)-f*(1/2* ln(d*x+c-1)*(ln((d*(e*f)^(1/2)+f*c-f*(d*x+c-1)-f)/(d*(e*f)^(1/2)+f*c-f))+l n((d*(e*f)^(1/2)-f*c+f*(d*x+c-1)+f)/(d*(e*f)^(1/2)-f*c+f)))/f+1/2*(dilog(( d*(e*f)^(1/2)+f*c-f*(d*x+c-1)-f)/(d*(e*f)^(1/2)+f*c-f))+dilog((d*(e*f)^(1/ 2)-f*c+f*(d*x+c-1)+f)/(d*(e*f)^(1/2)-f*c+f)))/f))))
\[ \int \frac {x^3 (a+b \text {arctanh}(c+d x))}{e-f x^2} \, dx=\int { -\frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )} x^{3}}{f x^{2} - e} \,d x } \] Input:
integrate(x^3*(a+b*arctanh(d*x+c))/(-f*x^2+e),x, algorithm="fricas")
Output:
integral(-(b*x^3*arctanh(d*x + c) + a*x^3)/(f*x^2 - e), x)
Timed out. \[ \int \frac {x^3 (a+b \text {arctanh}(c+d x))}{e-f x^2} \, dx=\text {Timed out} \] Input:
integrate(x**3*(a+b*atanh(d*x+c))/(-f*x**2+e),x)
Output:
Timed out
\[ \int \frac {x^3 (a+b \text {arctanh}(c+d x))}{e-f x^2} \, dx=\int { -\frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )} x^{3}}{f x^{2} - e} \,d x } \] Input:
integrate(x^3*(a+b*arctanh(d*x+c))/(-f*x^2+e),x, algorithm="maxima")
Output:
-1/2*a*(x^2/f + e*log(f*x^2 - e)/f^2) - 1/2*b*integrate(x^3*(log(d*x + c + 1) - log(-d*x - c + 1))/(f*x^2 - e), x)
\[ \int \frac {x^3 (a+b \text {arctanh}(c+d x))}{e-f x^2} \, dx=\int { -\frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )} x^{3}}{f x^{2} - e} \,d x } \] Input:
integrate(x^3*(a+b*arctanh(d*x+c))/(-f*x^2+e),x, algorithm="giac")
Output:
integrate(-(b*arctanh(d*x + c) + a)*x^3/(f*x^2 - e), x)
Timed out. \[ \int \frac {x^3 (a+b \text {arctanh}(c+d x))}{e-f x^2} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}{e-f\,x^2} \,d x \] Input:
int((x^3*(a + b*atanh(c + d*x)))/(e - f*x^2),x)
Output:
int((x^3*(a + b*atanh(c + d*x)))/(e - f*x^2), x)
\[ \int \frac {x^3 (a+b \text {arctanh}(c+d x))}{e-f x^2} \, dx =\text {Too large to display} \] Input:
int(x^3*(a+b*atanh(d*x+c))/(-f*x^2+e),x)
Output:
(3*atanh(c + d*x)**2*b*c**2*d**2*e*f + atanh(c + d*x)**2*b*d**4*e**2 + ata nh(c + d*x)**2*b*d**2*e*f + 2*atanh(c + d*x)*b*c**3*f**2 + 4*atanh(c + d*x )*b*c**2*f**2 - 2*atanh(c + d*x)*b*c*d**2*e*f - 2*atanh(c + d*x)*b*c*d**2* f**2*x**2 + 2*atanh(c + d*x)*b*c*f**2 - 2*atanh(c + d*x)*b*d**3*e*f*x - 2* atanh(c + d*x)*b*d**2*e*f + 8*int(atanh(c + d*x)/(c**2*e - c**2*f*x**2 + 2 *c*d*e*x - 2*c*d*f*x**3 + d**2*e*x**2 - d**2*f*x**4 - e + f*x**2),x)*b*c** 2*d**3*e**2*f + 2*int(atanh(c + d*x)/(c**2*e - c**2*f*x**2 + 2*c*d*e*x - 2 *c*d*f*x**3 + d**2*e*x**2 - d**2*f*x**4 - e + f*x**2),x)*b*d**5*e**3 - 2*i nt((atanh(c + d*x)*x**4)/(c**2*e - c**2*f*x**2 + 2*c*d*e*x - 2*c*d*f*x**3 + d**2*e*x**2 - d**2*f*x**4 - e + f*x**2),x)*b*d**5*e*f**2 + 4*int((atanh( c + d*x)*x)/(c**2*e - c**2*f*x**2 + 2*c*d*e*x - 2*c*d*f*x**3 + d**2*e*x**2 - d**2*f*x**4 - e + f*x**2),x)*b*c**3*d**2*e*f**2 + 4*int((atanh(c + d*x) *x)/(c**2*e - c**2*f*x**2 + 2*c*d*e*x - 2*c*d*f*x**3 + d**2*e*x**2 - d**2* f*x**4 - e + f*x**2),x)*b*c*d**4*e**2*f - 4*int((atanh(c + d*x)*x)/(c**2*e - c**2*f*x**2 + 2*c*d*e*x - 2*c*d*f*x**3 + d**2*e*x**2 - d**2*f*x**4 - e + f*x**2),x)*b*c*d**2*e*f**2 - 2*log( - sqrt(f)*sqrt(e) - f*x)*a*c*d**2*e* f - 2*log(sqrt(f)*sqrt(e) - f*x)*a*c*d**2*e*f + 4*log(c + d*x - 1)*b*c**2* f**2 - 2*log(c + d*x - 1)*b*d**2*e*f - 2*a*c*d**2*f**2*x**2 - 2*b*c*d*f**2 *x)/(4*c*d**2*f**3)