Integrand size = 21, antiderivative size = 273 \[ \int \frac {a+b \text {arctanh}(c+d x)}{e-f x^2} \, dx=-\frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left (\sqrt {e}-\sqrt {f} x\right )}{\left (d \sqrt {e}-(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{2 \sqrt {e} \sqrt {f}}+\frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left (\sqrt {e}+\sqrt {f} x\right )}{\left (d \sqrt {e}+(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{2 \sqrt {e} \sqrt {f}}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt {e}-\sqrt {f} x\right )}{\left (d \sqrt {e}-(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{4 \sqrt {e} \sqrt {f}}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt {e}+\sqrt {f} x\right )}{\left (d \sqrt {e}+(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{4 \sqrt {e} \sqrt {f}} \] Output:
-1/2*(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))/e^(1/2)/f^(1/2)+1/2*(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))/e^(1/2)/f^(1/2)+1/4*b*polyl og(2,1-2*d*(e^(1/2)-f^(1/2)*x)/(d*e^(1/2)-(1-c)*f^(1/2))/(d*x+c+1))/e^(1/2 )/f^(1/2)-1/4*b*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))/e^(1/2)/f^(1/2)
Time = 0.16 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.34 \[ \int \frac {a+b \text {arctanh}(c+d x)}{e-f x^2} \, dx=\frac {4 a \text {arctanh}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )+b \log (1-c-d x) \log \left (\frac {d \left (\sqrt {e}-\sqrt {f} x\right )}{d \sqrt {e}+(-1+c) \sqrt {f}}\right )-b \log (1+c+d x) \log \left (\frac {d \left (\sqrt {e}-\sqrt {f} x\right )}{d \sqrt {e}+(1+c) \sqrt {f}}\right )-b \log (1-c-d x) \log \left (\frac {d \left (\sqrt {e}+\sqrt {f} x\right )}{d \sqrt {e}-(-1+c) \sqrt {f}}\right )+b \log (1+c+d x) \log \left (\frac {d \left (\sqrt {e}+\sqrt {f} x\right )}{d \sqrt {e}-(1+c) \sqrt {f}}\right )-b \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} (-1+c+d x)}{d \sqrt {e}-(-1+c) \sqrt {f}}\right )+b \operatorname {PolyLog}\left (2,\frac {\sqrt {f} (-1+c+d x)}{d \sqrt {e}+(-1+c) \sqrt {f}}\right )+b \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} (1+c+d x)}{d \sqrt {e}-(1+c) \sqrt {f}}\right )-b \operatorname {PolyLog}\left (2,\frac {\sqrt {f} (1+c+d x)}{d \sqrt {e}+(1+c) \sqrt {f}}\right )}{4 \sqrt {e} \sqrt {f}} \] Input:
Integrate[(a + b*ArcTanh[c + d*x])/(e - f*x^2),x]
Output:
(4*a*ArcTanh[(Sqrt[f]*x)/Sqrt[e]] + b*Log[1 - c - d*x]*Log[(d*(Sqrt[e] - S qrt[f]*x))/(d*Sqrt[e] + (-1 + c)*Sqrt[f])] - b*Log[1 + c + d*x]*Log[(d*(Sq rt[e] - Sqrt[f]*x))/(d*Sqrt[e] + (1 + c)*Sqrt[f])] - b*Log[1 - c - d*x]*Lo g[(d*(Sqrt[e] + Sqrt[f]*x))/(d*Sqrt[e] - (-1 + c)*Sqrt[f])] + b*Log[1 + c + d*x]*Log[(d*(Sqrt[e] + Sqrt[f]*x))/(d*Sqrt[e] - (1 + c)*Sqrt[f])] - b*Po lyLog[2, -((Sqrt[f]*(-1 + c + d*x))/(d*Sqrt[e] - (-1 + c)*Sqrt[f]))] + b*P olyLog[2, (Sqrt[f]*(-1 + c + d*x))/(d*Sqrt[e] + (-1 + c)*Sqrt[f])] + b*Pol yLog[2, -((Sqrt[f]*(1 + c + d*x))/(d*Sqrt[e] - (1 + c)*Sqrt[f]))] - b*Poly Log[2, (Sqrt[f]*(1 + c + d*x))/(d*Sqrt[e] + (1 + c)*Sqrt[f])])/(4*Sqrt[e]* Sqrt[f])
Time = 0.83 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.74, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c+d x)}{e-f x^2} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {a}{e-f x^2}+\frac {b \text {arctanh}(c+d x)}{e-f x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \text {arctanh}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} (-c-d x+1)}{d \sqrt {e}-(1-c) \sqrt {f}}\right )}{4 \sqrt {e} \sqrt {f}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {f} (-c-d x+1)}{\sqrt {f} (1-c)+d \sqrt {e}}\right )}{4 \sqrt {e} \sqrt {f}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} (c+d x+1)}{d \sqrt {e}-(c+1) \sqrt {f}}\right )}{4 \sqrt {e} \sqrt {f}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {f} (c+d x+1)}{\sqrt {f} (c+1)+d \sqrt {e}}\right )}{4 \sqrt {e} \sqrt {f}}+\frac {b \log (-c-d x+1) \log \left (\frac {d \left (\sqrt {e}-\sqrt {f} x\right )}{d \sqrt {e}-(1-c) \sqrt {f}}\right )}{4 \sqrt {e} \sqrt {f}}-\frac {b \log (c+d x+1) \log \left (\frac {d \left (\sqrt {e}-\sqrt {f} x\right )}{(c+1) \sqrt {f}+d \sqrt {e}}\right )}{4 \sqrt {e} \sqrt {f}}-\frac {b \log (-c-d x+1) \log \left (\frac {d \left (\sqrt {e}+\sqrt {f} x\right )}{(1-c) \sqrt {f}+d \sqrt {e}}\right )}{4 \sqrt {e} \sqrt {f}}+\frac {b \log (c+d x+1) \log \left (\frac {d \left (\sqrt {e}+\sqrt {f} x\right )}{d \sqrt {e}-(c+1) \sqrt {f}}\right )}{4 \sqrt {e} \sqrt {f}}\) |
Input:
Int[(a + b*ArcTanh[c + d*x])/(e - f*x^2),x]
Output:
(a*ArcTanh[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]) + (b*Log[1 - c - d*x]*L og[(d*(Sqrt[e] - Sqrt[f]*x))/(d*Sqrt[e] - (1 - c)*Sqrt[f])])/(4*Sqrt[e]*Sq rt[f]) - (b*Log[1 + c + d*x]*Log[(d*(Sqrt[e] - Sqrt[f]*x))/(d*Sqrt[e] + (1 + c)*Sqrt[f])])/(4*Sqrt[e]*Sqrt[f]) - (b*Log[1 - c - d*x]*Log[(d*(Sqrt[e] + Sqrt[f]*x))/(d*Sqrt[e] + (1 - c)*Sqrt[f])])/(4*Sqrt[e]*Sqrt[f]) + (b*Lo g[1 + c + d*x]*Log[(d*(Sqrt[e] + Sqrt[f]*x))/(d*Sqrt[e] - (1 + c)*Sqrt[f]) ])/(4*Sqrt[e]*Sqrt[f]) + (b*PolyLog[2, -((Sqrt[f]*(1 - c - d*x))/(d*Sqrt[e ] - (1 - c)*Sqrt[f]))])/(4*Sqrt[e]*Sqrt[f]) - (b*PolyLog[2, (Sqrt[f]*(1 - c - d*x))/(d*Sqrt[e] + (1 - c)*Sqrt[f])])/(4*Sqrt[e]*Sqrt[f]) + (b*PolyLog [2, -((Sqrt[f]*(1 + c + d*x))/(d*Sqrt[e] - (1 + c)*Sqrt[f]))])/(4*Sqrt[e]* Sqrt[f]) - (b*PolyLog[2, (Sqrt[f]*(1 + c + d*x))/(d*Sqrt[e] + (1 + c)*Sqrt [f])])/(4*Sqrt[e]*Sqrt[f])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(466\) vs. \(2(217)=434\).
Time = 0.62 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.71
| method | result | size |
| risch | \(-\frac {b \ln \left (-d x -c +1\right ) \ln \left (\frac {d \sqrt {e f}-\left (-d x -c +1\right ) f -f c +f}{d \sqrt {e f}-f c +f}\right )}{4 \sqrt {e f}}+\frac {b \ln \left (-d x -c +1\right ) \ln \left (\frac {d \sqrt {e f}+\left (-d x -c +1\right ) f +f c -f}{d \sqrt {e f}+f c -f}\right )}{4 \sqrt {e f}}-\frac {b \operatorname {dilog}\left (\frac {d \sqrt {e f}-\left (-d x -c +1\right ) f -f c +f}{d \sqrt {e f}-f c +f}\right )}{4 \sqrt {e f}}+\frac {b \operatorname {dilog}\left (\frac {d \sqrt {e f}+\left (-d x -c +1\right ) f +f c -f}{d \sqrt {e f}+f c -f}\right )}{4 \sqrt {e f}}-\frac {a \,\operatorname {arctanh}\left (\frac {2 \left (-d x -c +1\right ) f +2 f c -2 f}{2 d \sqrt {e f}}\right )}{\sqrt {e f}}-\frac {b \ln \left (d x +c +1\right ) \ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )}{4 \sqrt {e f}}+\frac {b \ln \left (d x +c +1\right ) \ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{4 \sqrt {e f}}-\frac {b \operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )}{4 \sqrt {e f}}+\frac {b \operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{4 \sqrt {e f}}\) | \(467\) |
| parts | \(\text {Expression too large to display}\) | \(1285\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1315\) |
| default | \(\text {Expression too large to display}\) | \(1315\) |
Input:
int((a+b*arctanh(d*x+c))/(-f*x^2+e),x,method=_RETURNVERBOSE)
Output:
-1/4*b*ln(-d*x-c+1)/(e*f)^(1/2)*ln((d*(e*f)^(1/2)-(-d*x-c+1)*f-f*c+f)/(d*( e*f)^(1/2)-f*c+f))+1/4*b*ln(-d*x-c+1)/(e*f)^(1/2)*ln((d*(e*f)^(1/2)+(-d*x- c+1)*f+f*c-f)/(d*(e*f)^(1/2)+f*c-f))-1/4*b/(e*f)^(1/2)*dilog((d*(e*f)^(1/2 )-(-d*x-c+1)*f-f*c+f)/(d*(e*f)^(1/2)-f*c+f))+1/4*b/(e*f)^(1/2)*dilog((d*(e *f)^(1/2)+(-d*x-c+1)*f+f*c-f)/(d*(e*f)^(1/2)+f*c-f))-a/(e*f)^(1/2)*arctanh (1/2*(2*(-d*x-c+1)*f+2*f*c-2*f)/d/(e*f)^(1/2))-1/4*b*ln(d*x+c+1)/(e*f)^(1/ 2)*ln((d*(e*f)^(1/2)+f*c-f*(d*x+c+1)+f)/(d*(e*f)^(1/2)+f*c+f))+1/4*b*ln(d* x+c+1)/(e*f)^(1/2)*ln((d*(e*f)^(1/2)-f*c+f*(d*x+c+1)-f)/(d*(e*f)^(1/2)-f*c -f))-1/4*b/(e*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))+1/4*b/(e*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))
\[ \int \frac {a+b \text {arctanh}(c+d x)}{e-f x^2} \, dx=\int { -\frac {b \operatorname {artanh}\left (d x + c\right ) + a}{f x^{2} - e} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))/(-f*x^2+e),x, algorithm="fricas")
Output:
integral(-(b*arctanh(d*x + c) + a)/(f*x^2 - e), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c+d x)}{e-f x^2} \, dx=\text {Timed out} \] Input:
integrate((a+b*atanh(d*x+c))/(-f*x**2+e),x)
Output:
Timed out
Exception generated. \[ \int \frac {a+b \text {arctanh}(c+d x)}{e-f x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*arctanh(d*x+c))/(-f*x^2+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
\[ \int \frac {a+b \text {arctanh}(c+d x)}{e-f x^2} \, dx=\int { -\frac {b \operatorname {artanh}\left (d x + c\right ) + a}{f x^{2} - e} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))/(-f*x^2+e),x, algorithm="giac")
Output:
integrate(-(b*arctanh(d*x + c) + a)/(f*x^2 - e), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c+d x)}{e-f x^2} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c+d\,x\right )}{e-f\,x^2} \,d x \] Input:
int((a + b*atanh(c + d*x))/(e - f*x^2),x)
Output:
int((a + b*atanh(c + d*x))/(e - f*x^2), x)
\[ \int \frac {a+b \text {arctanh}(c+d x)}{e-f x^2} \, dx=\frac {\sqrt {f}\, \sqrt {e}\, \mathrm {log}\left (-\sqrt {f}\, \sqrt {e}-f x \right ) a -\sqrt {f}\, \sqrt {e}\, \mathrm {log}\left (\sqrt {f}\, \sqrt {e}-f x \right ) a +2 \left (\int \frac {\mathit {atanh} \left (d x +c \right )}{-f \,x^{2}+e}d x \right ) b e f}{2 e f} \] Input:
int((a+b*atanh(d*x+c))/(-f*x^2+e),x)
Output:
(sqrt(f)*sqrt(e)*log( - sqrt(f)*sqrt(e) - f*x)*a - sqrt(f)*sqrt(e)*log(sqr t(f)*sqrt(e) - f*x)*a + 2*int(atanh(c + d*x)/(e - f*x**2),x)*b*e*f)/(2*e*f )