Integrand size = 24, antiderivative size = 355 \[ \int \frac {a+b \text {arctanh}(c+d x)}{x^2 \left (e-f x^2\right )} \, dx=-\frac {a+b \text {arctanh}(c+d x)}{e x}+\frac {b d \log (x)}{\left (1-c^2\right ) e}-\frac {b d \log (1-c-d x)}{2 (1-c) e}-\frac {b d \log (1+c+d x)}{2 (1+c) e}-\frac {\sqrt {f} (a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left (\sqrt {e}-\sqrt {f} x\right )}{\left (d \sqrt {e}-(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{2 e^{3/2}}+\frac {\sqrt {f} (a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left (\sqrt {e}+\sqrt {f} x\right )}{\left (d \sqrt {e}+(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{2 e^{3/2}}+\frac {b \sqrt {f} \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt {e}-\sqrt {f} x\right )}{\left (d \sqrt {e}-(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{4 e^{3/2}}-\frac {b \sqrt {f} \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt {e}+\sqrt {f} x\right )}{\left (d \sqrt {e}+(1-c) \sqrt {f}\right ) (1+c+d x)}\right )}{4 e^{3/2}} \] Output:
-(a+b*arctanh(d*x+c))/e/x+b*d*ln(x)/(-c^2+1)/e-1/2*b*d*ln(-d*x-c+1)/(1-c)/ e-1/2*b*d*ln(d*x+c+1)/(1+c)/e-1/2*f^(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^(3/2)+1/2*f^(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^(3/2)+1/4*b*f^(1/2)*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^(3/2)-1/4*b*f^(1/2)*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^(3/2)
Result contains complex when optimal does not.
Time = 7.80 (sec) , antiderivative size = 1221, normalized size of antiderivative = 3.44 \[ \int \frac {a+b \text {arctanh}(c+d x)}{x^2 \left (e-f x^2\right )} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcTanh[c + d*x])/(x^2*(e - f*x^2)),x]
Output:
-(a/(e*x)) + (a*Sqrt[f]*ArcTanh[(Sqrt[f]*x)/Sqrt[e]])/e^(3/2) - (b*((-1 + c^2 + c*d*x)*ArcTanh[c + d*x] + d*x*Log[-((d*x)/Sqrt[1 - (c + d*x)^2])]))/ ((-1 + c)*(1 + c)*e*x) - (b*Sqrt[f]*(2*d^2*e*ArcTanh[c - (d*Sqrt[e])/Sqrt[ f]]*ArcTanh[c + d*x] - 2*c^2*f*ArcTanh[c - (d*Sqrt[e])/Sqrt[f]]*ArcTanh[c + d*x] - 2*d^2*e*ArcTanh[c + (d*Sqrt[e])/Sqrt[f]]*ArcTanh[c + d*x] + 2*c^2 *f*ArcTanh[c + (d*Sqrt[e])/Sqrt[f]]*ArcTanh[c + d*x] - 2*d*Sqrt[e]*Sqrt[f] *ArcTanh[c + d*x]^2 + d*Sqrt[e]*E^ArcTanh[c + (d*Sqrt[e])/Sqrt[f]]*Sqrt[1 - c^2 - (d^2*e)/f - (2*c*d*Sqrt[e])/Sqrt[f]]*Sqrt[f]*ArcTanh[c + d*x]^2 + d*Sqrt[e]*E^ArcTanh[c - (d*Sqrt[e])/Sqrt[f]]*Sqrt[1 - c^2 - (d^2*e)/f + (2 *c*d*Sqrt[e])/Sqrt[f]]*Sqrt[f]*ArcTanh[c + d*x]^2 - c*E^ArcTanh[c + (d*Sqr t[e])/Sqrt[f]]*Sqrt[1 - c^2 - (d^2*e)/f - (2*c*d*Sqrt[e])/Sqrt[f]]*f*ArcTa nh[c + d*x]^2 + c*E^ArcTanh[c - (d*Sqrt[e])/Sqrt[f]]*Sqrt[1 - c^2 - (d^2*e )/f + (2*c*d*Sqrt[e])/Sqrt[f]]*f*ArcTanh[c + d*x]^2 + 2*d^2*e*ArcTanh[c - (d*Sqrt[e])/Sqrt[f]]*Log[1 - E^(2*ArcTanh[c - (d*Sqrt[e])/Sqrt[f]] - 2*Arc Tanh[c + d*x])] - 2*c^2*f*ArcTanh[c - (d*Sqrt[e])/Sqrt[f]]*Log[1 - E^(2*Ar cTanh[c - (d*Sqrt[e])/Sqrt[f]] - 2*ArcTanh[c + d*x])] - 2*d^2*e*ArcTanh[c + d*x]*Log[1 - E^(2*ArcTanh[c - (d*Sqrt[e])/Sqrt[f]] - 2*ArcTanh[c + d*x]) ] + 2*c^2*f*ArcTanh[c + d*x]*Log[1 - E^(2*ArcTanh[c - (d*Sqrt[e])/Sqrt[f]] - 2*ArcTanh[c + d*x])] - 2*d^2*e*ArcTanh[c + (d*Sqrt[e])/Sqrt[f]]*Log[1 - E^(2*ArcTanh[c + (d*Sqrt[e])/Sqrt[f]] - 2*ArcTanh[c + d*x])] + 2*c^2*f...
Time = 1.21 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.58, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c+d x)}{x^2 \left (e-f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {a}{x^2 \left (e-f x^2\right )}+\frac {b \text {arctanh}(c+d x)}{x^2 \left (e-f x^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{3/2}}-\frac {a}{e x}-\frac {b \text {arctanh}(c+d x)}{e x}+\frac {b d \log (x)}{\left (1-c^2\right ) e}+\frac {b \sqrt {f} \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} (-c-d x+1)}{d \sqrt {e}-(1-c) \sqrt {f}}\right )}{4 e^{3/2}}-\frac {b \sqrt {f} \operatorname {PolyLog}\left (2,\frac {\sqrt {f} (-c-d x+1)}{\sqrt {f} (1-c)+d \sqrt {e}}\right )}{4 e^{3/2}}+\frac {b \sqrt {f} \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} (c+d x+1)}{d \sqrt {e}-(c+1) \sqrt {f}}\right )}{4 e^{3/2}}-\frac {b \sqrt {f} \operatorname {PolyLog}\left (2,\frac {\sqrt {f} (c+d x+1)}{\sqrt {f} (c+1)+d \sqrt {e}}\right )}{4 e^{3/2}}+\frac {b \sqrt {f} \log (-c-d x+1) \log \left (\frac {d \left (\sqrt {e}-\sqrt {f} x\right )}{d \sqrt {e}-(1-c) \sqrt {f}}\right )}{4 e^{3/2}}-\frac {b \sqrt {f} \log (c+d x+1) \log \left (\frac {d \left (\sqrt {e}-\sqrt {f} x\right )}{(c+1) \sqrt {f}+d \sqrt {e}}\right )}{4 e^{3/2}}-\frac {b \sqrt {f} \log (-c-d x+1) \log \left (\frac {d \left (\sqrt {e}+\sqrt {f} x\right )}{(1-c) \sqrt {f}+d \sqrt {e}}\right )}{4 e^{3/2}}+\frac {b \sqrt {f} \log (c+d x+1) \log \left (\frac {d \left (\sqrt {e}+\sqrt {f} x\right )}{d \sqrt {e}-(c+1) \sqrt {f}}\right )}{4 e^{3/2}}-\frac {b d \log (-c-d x+1)}{2 (1-c) e}-\frac {b d \log (c+d x+1)}{2 (c+1) e}\) |
Input:
Int[(a + b*ArcTanh[c + d*x])/(x^2*(e - f*x^2)),x]
Output:
-(a/(e*x)) + (a*Sqrt[f]*ArcTanh[(Sqrt[f]*x)/Sqrt[e]])/e^(3/2) - (b*ArcTanh [c + d*x])/(e*x) + (b*d*Log[x])/((1 - c^2)*e) - (b*d*Log[1 - c - d*x])/(2* (1 - c)*e) - (b*d*Log[1 + c + d*x])/(2*(1 + c)*e) + (b*Sqrt[f]*Log[1 - c - d*x]*Log[(d*(Sqrt[e] - Sqrt[f]*x))/(d*Sqrt[e] - (1 - c)*Sqrt[f])])/(4*e^( 3/2)) - (b*Sqrt[f]*Log[1 + c + d*x]*Log[(d*(Sqrt[e] - Sqrt[f]*x))/(d*Sqrt[ e] + (1 + c)*Sqrt[f])])/(4*e^(3/2)) - (b*Sqrt[f]*Log[1 - c - d*x]*Log[(d*( Sqrt[e] + Sqrt[f]*x))/(d*Sqrt[e] + (1 - c)*Sqrt[f])])/(4*e^(3/2)) + (b*Sqr t[f]*Log[1 + c + d*x]*Log[(d*(Sqrt[e] + Sqrt[f]*x))/(d*Sqrt[e] - (1 + c)*S qrt[f])])/(4*e^(3/2)) + (b*Sqrt[f]*PolyLog[2, -((Sqrt[f]*(1 - c - d*x))/(d *Sqrt[e] - (1 - c)*Sqrt[f]))])/(4*e^(3/2)) - (b*Sqrt[f]*PolyLog[2, (Sqrt[f ]*(1 - c - d*x))/(d*Sqrt[e] + (1 - c)*Sqrt[f])])/(4*e^(3/2)) + (b*Sqrt[f]* PolyLog[2, -((Sqrt[f]*(1 + c + d*x))/(d*Sqrt[e] - (1 + c)*Sqrt[f]))])/(4*e ^(3/2)) - (b*Sqrt[f]*PolyLog[2, (Sqrt[f]*(1 + c + d*x))/(d*Sqrt[e] + (1 + c)*Sqrt[f])])/(4*e^(3/2))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(677\) vs. \(2(295)=590\).
Time = 2.18 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.91
| method | result | size |
| risch | \(-\frac {d b \ln \left (-d x \right )}{2 e \left (-1+c \right )}+\frac {d b \ln \left (-d x -c +1\right )}{2 e \left (-1+c \right )}+\frac {b \ln \left (-d x -c +1\right ) c}{2 e \left (-1+c \right ) x}-\frac {b \ln \left (-d x -c +1\right )}{2 e \left (-1+c \right ) x}-\frac {b f \ln \left (-d x -c +1\right ) \ln \left (\frac {d \sqrt {e f}-\left (-d x -c +1\right ) f -f c +f}{d \sqrt {e f}-f c +f}\right )}{4 e \sqrt {e f}}+\frac {b f \ln \left (-d x -c +1\right ) \ln \left (\frac {d \sqrt {e f}+\left (-d x -c +1\right ) f +f c -f}{d \sqrt {e f}+f c -f}\right )}{4 e \sqrt {e f}}-\frac {b f \operatorname {dilog}\left (\frac {d \sqrt {e f}-\left (-d x -c +1\right ) f -f c +f}{d \sqrt {e f}-f c +f}\right )}{4 e \sqrt {e f}}+\frac {b f \operatorname {dilog}\left (\frac {d \sqrt {e f}+\left (-d x -c +1\right ) f +f c -f}{d \sqrt {e f}+f c -f}\right )}{4 e \sqrt {e f}}-\frac {a}{e x}-\frac {a f \,\operatorname {arctanh}\left (\frac {2 \left (-d x -c +1\right ) f +2 f c -2 f}{2 d \sqrt {e f}}\right )}{e \sqrt {e f}}+\frac {b d \ln \left (d x \right )}{2 e \left (1+c \right )}-\frac {b d \ln \left (d x +c +1\right )}{2 \left (1+c \right ) e}-\frac {b \ln \left (d x +c +1\right ) c}{2 e \left (1+c \right ) x}-\frac {b \ln \left (d x +c +1\right )}{2 e \left (1+c \right ) x}-\frac {b f \ln \left (d x +c +1\right ) \ln \left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )}{4 e \sqrt {e f}}+\frac {b f \ln \left (d x +c +1\right ) \ln \left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{4 e \sqrt {e f}}-\frac {b f \operatorname {dilog}\left (\frac {d \sqrt {e f}+f c -f \left (d x +c +1\right )+f}{d \sqrt {e f}+f c +f}\right )}{4 e \sqrt {e f}}+\frac {b f \operatorname {dilog}\left (\frac {d \sqrt {e f}-f c +f \left (d x +c +1\right )-f}{d \sqrt {e f}-f c -f}\right )}{4 e \sqrt {e f}}\) | \(678\) |
| parts | \(\text {Expression too large to display}\) | \(19082\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(19160\) |
| default | \(\text {Expression too large to display}\) | \(19160\) |
Input:
int((a+b*arctanh(d*x+c))/x^2/(-f*x^2+e),x,method=_RETURNVERBOSE)
Output:
-1/2*d*b/e/(-1+c)*ln(-d*x)+1/2*d*b/e*ln(-d*x-c+1)/(-1+c)+1/2*b/e*ln(-d*x-c +1)/(-1+c)/x*c-1/2*b/e*ln(-d*x-c+1)/(-1+c)/x-1/4*b*f/e*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*f /e*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*f/e/(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*f/e/(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/x-a*f/e/(e*f)^(1/2)*arctanh(1/2 *(2*(-d*x-c+1)*f+2*f*c-2*f)/d/(e*f)^(1/2))+1/2*b*d/e/(1+c)*ln(d*x)-1/2*b*d *ln(d*x+c+1)/(1+c)/e-1/2*b/e*ln(d*x+c+1)/(1+c)/x*c-1/2*b/e*ln(d*x+c+1)/(1+ c)/x-1/4*b*f/e*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*f/e*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*f/e/(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*f/e/(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))
\[ \int \frac {a+b \text {arctanh}(c+d x)}{x^2 \left (e-f x^2\right )} \, dx=\int { -\frac {b \operatorname {artanh}\left (d x + c\right ) + a}{{\left (f x^{2} - e\right )} x^{2}} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))/x^2/(-f*x^2+e),x, algorithm="fricas")
Output:
integral(-(b*arctanh(d*x + c) + a)/(f*x^4 - e*x^2), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c+d x)}{x^2 \left (e-f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((a+b*atanh(d*x+c))/x**2/(-f*x**2+e),x)
Output:
Timed out
Exception generated. \[ \int \frac {a+b \text {arctanh}(c+d x)}{x^2 \left (e-f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*arctanh(d*x+c))/x^2/(-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
\[ \int \frac {a+b \text {arctanh}(c+d x)}{x^2 \left (e-f x^2\right )} \, dx=\int { -\frac {b \operatorname {artanh}\left (d x + c\right ) + a}{{\left (f x^{2} - e\right )} x^{2}} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))/x^2/(-f*x^2+e),x, algorithm="giac")
Output:
integrate(-(b*arctanh(d*x + c) + a)/((f*x^2 - e)*x^2), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c+d x)}{x^2 \left (e-f x^2\right )} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c+d\,x\right )}{x^2\,\left (e-f\,x^2\right )} \,d x \] Input:
int((a + b*atanh(c + d*x))/(x^2*(e - f*x^2)),x)
Output:
int((a + b*atanh(c + d*x))/(x^2*(e - f*x^2)), x)
\[ \int \frac {a+b \text {arctanh}(c+d x)}{x^2 \left (e-f x^2\right )} \, dx=\frac {\sqrt {f}\, \sqrt {e}\, \mathrm {log}\left (-\sqrt {f}\, \sqrt {e}-f x \right ) a x -\sqrt {f}\, \sqrt {e}\, \mathrm {log}\left (\sqrt {f}\, \sqrt {e}-f x \right ) a x +2 \left (\int \frac {\mathit {atanh} \left (d x +c \right )}{-f \,x^{4}+e \,x^{2}}d x \right ) b \,e^{2} x -2 a e}{2 e^{2} x} \] Input:
int((a+b*atanh(d*x+c))/x^2/(-f*x^2+e),x)
Output:
(sqrt(f)*sqrt(e)*log( - sqrt(f)*sqrt(e) - f*x)*a*x - sqrt(f)*sqrt(e)*log(s qrt(f)*sqrt(e) - f*x)*a*x + 2*int(atanh(c + d*x)/(e*x**2 - f*x**4),x)*b*e* *2*x - 2*a*e)/(2*e**2*x)