Integrand size = 19, antiderivative size = 335 \[ \int \frac {\text {arctanh}(d+e x)}{a+b x+c x^2} \, dx=\frac {\text {arctanh}(d+e x) \log \left (\frac {2 e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (1-d)+\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\text {arctanh}(d+e x) \log \left (\frac {2 e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\operatorname {PolyLog}\left (2,1+\frac {2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c-2 c d+b e-\sqrt {b^2-4 a c} e\right ) (1+d+e x)}\right )}{2 \sqrt {b^2-4 a c}}+\frac {\operatorname {PolyLog}\left (2,1+\frac {2 \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{2 \sqrt {b^2-4 a c}} \]
arctanh(e*x+d)*ln(2*e*(b+2*c*x-(-4*a*c+b^2)^(1/2))/(e*x+d+1)/(2*c*(1-d)+e* (b-(-4*a*c+b^2)^(1/2))))/(-4*a*c+b^2)^(1/2)-arctanh(e*x+d)*ln(2*e*(b+2*c*x +(-4*a*c+b^2)^(1/2))/(e*x+d+1)/(2*c*(1-d)+e*(b+(-4*a*c+b^2)^(1/2))))/(-4*a *c+b^2)^(1/2)-1/2*polylog(2,1+2*(2*c*d-2*c*(e*x+d)-e*(b-(-4*a*c+b^2)^(1/2) ))/(e*x+d+1)/(2*c-2*c*d+b*e-e*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(1/2)+1/2* polylog(2,1+2*(2*c*d-2*c*(e*x+d)-e*(b+(-4*a*c+b^2)^(1/2)))/(e*x+d+1)/(2*c* (1-d)+e*(b+(-4*a*c+b^2)^(1/2))))/(-4*a*c+b^2)^(1/2)
Time = 0.36 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.20 \[ \int \frac {\text {arctanh}(d+e x)}{a+b x+c x^2} \, dx=\frac {-\log \left (\frac {e \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{2 c (-1+d)+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right ) \log (1-d-e x)+\log \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c (-1+d)+\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) \log (1-d-e x)+\log \left (\frac {e \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{2 c (1+d)+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right ) \log (1+d+e x)-\log \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c (1+d)+\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) \log (1+d+e x)-\operatorname {PolyLog}\left (2,\frac {2 c (-1+d+e x)}{2 c (-1+d)+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )+\operatorname {PolyLog}\left (2,\frac {2 c (-1+d+e x)}{2 c (-1+d)-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )+\operatorname {PolyLog}\left (2,\frac {2 c (1+d+e x)}{2 c (1+d)+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )-\operatorname {PolyLog}\left (2,\frac {2 c (1+d+e x)}{2 c (1+d)-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{2 \sqrt {b^2-4 a c}} \]
(-(Log[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*(-1 + d) + (-b + Sqrt[b^2 - 4*a*c])*e)]*Log[1 - d - e*x]) + Log[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)) /(-2*c*(-1 + d) + (b + Sqrt[b^2 - 4*a*c])*e)]*Log[1 - d - e*x] + Log[(e*(- b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*(1 + d) + (-b + Sqrt[b^2 - 4*a*c])*e) ]*Log[1 + d + e*x] - Log[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(1 + d) + (b + Sqrt[b^2 - 4*a*c])*e)]*Log[1 + d + e*x] - PolyLog[2, (2*c*(-1 + d + e*x))/(2*c*(-1 + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] + PolyLog[2, (2*c*(-1 + d + e*x))/(2*c*(-1 + d) - (b + Sqrt[b^2 - 4*a*c])*e)] + PolyLog[2, (2*c *(1 + d + e*x))/(2*c*(1 + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] - PolyLog[2, ( 2*c*(1 + d + e*x))/(2*c*(1 + d) - (b + Sqrt[b^2 - 4*a*c])*e)])/(2*Sqrt[b^2 - 4*a*c])
Time = 1.05 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {7279, 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.
| \(\displaystyle \int \frac {\text {arctanh}(d+e x)}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2 c \text {arctanh}(d+e x)}{\sqrt {b^2-4 a c} \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}-\frac {2 c \text {arctanh}(d+e x)}{\sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\text {arctanh}(d+e x) \log \left (-\frac {2 \left (-e \left (b-\sqrt {b^2-4 a c}\right )-2 c (d+e x)+2 c d\right )}{(d+e x+1) \left (-e \sqrt {b^2-4 a c}+b e-2 c d+2 c\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {\text {arctanh}(d+e x) \log \left (-\frac {2 \left (-e \left (\sqrt {b^2-4 a c}+b\right )-2 c (d+e x)+2 c d\right )}{(d+e x+1) \left (e \left (\sqrt {b^2-4 a c}+b\right )+2 c (1-d)\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {\operatorname {PolyLog}\left (2,\frac {2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (-2 d c+2 c+b e-\sqrt {b^2-4 a c} e\right ) (d+e x+1)}+1\right )}{2 \sqrt {b^2-4 a c}}+\frac {\operatorname {PolyLog}\left (2,\frac {2 \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (d+e x+1)}+1\right )}{2 \sqrt {b^2-4 a c}}\) |
(ArcTanh[d + e*x]*Log[(-2*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e* x)))/((2*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)*(1 + d + e*x))])/Sqrt[b^2 - 4*a*c] - (ArcTanh[d + e*x]*Log[(-2*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c*(1 - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 + d + e*x))] )/Sqrt[b^2 - 4*a*c] - PolyLog[2, 1 + (2*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)*(1 + d + e*x ))]/(2*Sqrt[b^2 - 4*a*c]) + PolyLog[2, 1 + (2*(2*c*d - (b + Sqrt[b^2 - 4*a *c])*e - 2*c*(d + e*x)))/((2*c*(1 - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 + d + e*x))]/(2*Sqrt[b^2 - 4*a*c])
3.1.61.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(767\) vs. \(2(307)=614\).
Time = 0.69 (sec) , antiderivative size = 768, normalized size of antiderivative = 2.29
| method | result | size |
| risch | \(\frac {e \ln \left (-e x -d +1\right ) \ln \left (\frac {-2 \left (-e x -d +1\right ) c +b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}{b e -2 c d +2 c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \ln \left (-e x -d +1\right ) \ln \left (\frac {2 \left (-e x -d +1\right ) c -b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}{-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \operatorname {dilog}\left (\frac {-2 \left (-e x -d +1\right ) c +b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}{b e -2 c d +2 c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \operatorname {dilog}\left (\frac {2 \left (-e x -d +1\right ) c -b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}{-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \ln \left (e x +d +1\right ) \ln \left (\frac {-2 \left (e x +d +1\right ) c -b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}{-b e +2 c d +2 c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \ln \left (e x +d +1\right ) \ln \left (\frac {2 \left (e x +d +1\right ) c +b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}{b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \operatorname {dilog}\left (\frac {-2 \left (e x +d +1\right ) c -b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}{-b e +2 c d +2 c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \operatorname {dilog}\left (\frac {2 \left (e x +d +1\right ) c +b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}{b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\) | \(768\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2132\) |
| default | \(\text {Expression too large to display}\) | \(2132\) |
1/2*e*ln(-e*x-d+1)/(-4*a*c*e^2+b^2*e^2)^(1/2)*ln((-2*(-e*x-d+1)*c+b*e-2*c* d+(-4*a*c*e^2+b^2*e^2)^(1/2)+2*c)/(b*e-2*c*d+2*c+(-4*a*c*e^2+b^2*e^2)^(1/2 )))-1/2*e*ln(-e*x-d+1)/(-4*a*c*e^2+b^2*e^2)^(1/2)*ln((2*(-e*x-d+1)*c-b*e+2 *c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)-2*c)/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2 )-2*c))+1/2*e/(-4*a*c*e^2+b^2*e^2)^(1/2)*dilog((-2*(-e*x-d+1)*c+b*e-2*c*d+ (-4*a*c*e^2+b^2*e^2)^(1/2)+2*c)/(b*e-2*c*d+2*c+(-4*a*c*e^2+b^2*e^2)^(1/2)) )-1/2*e/(-4*a*c*e^2+b^2*e^2)^(1/2)*dilog((2*(-e*x-d+1)*c-b*e+2*c*d+(-4*a*c *e^2+b^2*e^2)^(1/2)-2*c)/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)-2*c))+1/2* e*ln(e*x+d+1)/(-4*a*c*e^2+b^2*e^2)^(1/2)*ln((-2*(e*x+d+1)*c-b*e+2*c*d+(-4* a*c*e^2+b^2*e^2)^(1/2)+2*c)/(-b*e+2*c*d+2*c+(-4*a*c*e^2+b^2*e^2)^(1/2)))-1 /2*e*ln(e*x+d+1)/(-4*a*c*e^2+b^2*e^2)^(1/2)*ln((2*(e*x+d+1)*c+b*e-2*c*d+(- 4*a*c*e^2+b^2*e^2)^(1/2)-2*c)/(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)-2*c))+ 1/2*e/(-4*a*c*e^2+b^2*e^2)^(1/2)*dilog((-2*(e*x+d+1)*c-b*e+2*c*d+(-4*a*c*e ^2+b^2*e^2)^(1/2)+2*c)/(-b*e+2*c*d+2*c+(-4*a*c*e^2+b^2*e^2)^(1/2)))-1/2*e/ (-4*a*c*e^2+b^2*e^2)^(1/2)*dilog((2*(e*x+d+1)*c+b*e-2*c*d+(-4*a*c*e^2+b^2* e^2)^(1/2)-2*c)/(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)-2*c))
\[ \int \frac {\text {arctanh}(d+e x)}{a+b x+c x^2} \, dx=\int { \frac {\operatorname {artanh}\left (e x + d\right )}{c x^{2} + b x + a} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(d+e x)}{a+b x+c x^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\text {arctanh}(d+e x)}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
\[ \int \frac {\text {arctanh}(d+e x)}{a+b x+c x^2} \, dx=\int { \frac {\operatorname {artanh}\left (e x + d\right )}{c x^{2} + b x + a} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(d+e x)}{a+b x+c x^2} \, dx=\int \frac {\mathrm {atanh}\left (d+e\,x\right )}{c\,x^2+b\,x+a} \,d x \]