Integrand size = 19, antiderivative size = 367 \[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\frac {\cot ^{-1}(d+e x) \log \left (\frac {2 e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (i-d)+\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (1-i (d+e x))}\right )}{\sqrt {b^2-4 a c}}-\frac {\cot ^{-1}(d+e x) \log \left (\frac {2 e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (i-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1-i (d+e x))}\right )}{\sqrt {b^2-4 a c}}+\frac {i \operatorname {PolyLog}\left (2,1+\frac {2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 i c-2 c d+b e-\sqrt {b^2-4 a c} e\right ) (1-i (d+e x))}\right )}{2 \sqrt {b^2-4 a c}}-\frac {i \operatorname {PolyLog}\left (2,1+\frac {2 \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c (i-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1-i (d+e x))}\right )}{2 \sqrt {b^2-4 a c}} \]
arccot(e*x+d)*ln(2*e*(b+2*c*x-(-4*a*c+b^2)^(1/2))/(1-I*(e*x+d))/(2*c*(I-d) +e*(b-(-4*a*c+b^2)^(1/2))))/(-4*a*c+b^2)^(1/2)-arccot(e*x+d)*ln(2*e*(b+2*c *x+(-4*a*c+b^2)^(1/2))/(1-I*(e*x+d))/(2*c*(I-d)+e*(b+(-4*a*c+b^2)^(1/2)))) /(-4*a*c+b^2)^(1/2)+1/2*I*polylog(2,1+2*(2*c*d-2*c*(e*x+d)-e*(b-(-4*a*c+b^ 2)^(1/2)))/(1-I*(e*x+d))/(2*I*c-2*c*d+b*e-e*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b ^2)^(1/2)-1/2*I*polylog(2,1+2*(2*c*d-2*c*(e*x+d)-e*(b+(-4*a*c+b^2)^(1/2))) /(1-I*(e*x+d))/(2*c*(I-d)+e*(b+(-4*a*c+b^2)^(1/2))))/(-4*a*c+b^2)^(1/2)
\[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx \]
Time = 1.13 (sec) , antiderivative size = 388, 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.
| \(\displaystyle \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2 c \cot ^{-1}(d+e x)}{\sqrt {b^2-4 a c} \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}-\frac {2 c \cot ^{-1}(d+e x)}{\sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i \operatorname {PolyLog}\left (2,\frac {2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (-2 d c+2 i c+b e-\sqrt {b^2-4 a c} e\right ) (1-i (d+e x))}+1\right )}{2 \sqrt {b^2-4 a c}}-\frac {i \operatorname {PolyLog}\left (2,\frac {2 \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c (i-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1-i (d+e x))}+1\right )}{2 \sqrt {b^2-4 a c}}+\frac {\cot ^{-1}(d+e x) \log \left (-\frac {2 \left (-e \left (b-\sqrt {b^2-4 a c}\right )-2 c (d+e x)+2 c d\right )}{(1-i (d+e x)) \left (-e \sqrt {b^2-4 a c}+b e-2 c d+2 i c\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {\cot ^{-1}(d+e x) \log \left (-\frac {2 \left (-e \left (\sqrt {b^2-4 a c}+b\right )-2 c (d+e x)+2 c d\right )}{(1-i (d+e x)) \left (e \left (\sqrt {b^2-4 a c}+b\right )+2 c (-d+i)\right )}\right )}{\sqrt {b^2-4 a c}}\) |
(ArcCot[d + e*x]*Log[(-2*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x )))/(((2*I)*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)*(1 - I*(d + e*x)))])/Sq rt[b^2 - 4*a*c] - (ArcCot[d + e*x]*Log[(-2*(2*c*d - (b + Sqrt[b^2 - 4*a*c] )*e - 2*c*(d + e*x)))/((2*c*(I - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/Sqrt[b^2 - 4*a*c] + ((I/2)*PolyLog[2, 1 + (2*(2*c*d - (b - Sqr t[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/(((2*I)*c - 2*c*d + b*e - Sqrt[b^2 - 4 *a*c]*e)*(1 - I*(d + e*x)))])/Sqrt[b^2 - 4*a*c] - ((I/2)*PolyLog[2, 1 + (2 *(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c*(I - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/Sqrt[b^2 - 4*a*c]
3.2.13.3.1 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 958 vs. \(2 (329 ) = 658\).
Time = 2.49 (sec) , antiderivative size = 959, normalized size of antiderivative = 2.61
| method | result | size |
| risch | \(\frac {i e \pi \arctan \left (\frac {i b e -2 i c d -2 \left (-i e x -i d +1\right ) c +2 c}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \ln \left (-i e x -i d +1\right ) \ln \left (\frac {i b e -2 i c d -2 \left (-i e x -i d +1\right ) c +\sqrt {4 a c \,e^{2}-b^{2} e^{2}}+2 c}{i b e -2 i 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 (-i e x -i d +1\right ) \ln \left (\frac {i b e -2 i c d -2 \left (-i e x -i d +1\right ) c -\sqrt {4 a c \,e^{2}-b^{2} e^{2}}+2 c}{i b e -2 i 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 {i b e -2 i c d -2 \left (-i e x -i d +1\right ) c +\sqrt {4 a c \,e^{2}-b^{2} e^{2}}+2 c}{i b e -2 i 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 {i b e -2 i c d -2 \left (-i e x -i d +1\right ) c -\sqrt {4 a c \,e^{2}-b^{2} e^{2}}+2 c}{i b e -2 i 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 (i e x +i d +1\right ) \ln \left (\frac {i b e -2 i c d +2 \left (i e x +i d +1\right ) c -\sqrt {4 a c \,e^{2}-b^{2} e^{2}}-2 c}{i b e -2 i 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 (i e x +i d +1\right ) \ln \left (\frac {i b e -2 i c d +2 \left (i e x +i d +1\right ) c +\sqrt {4 a c \,e^{2}-b^{2} e^{2}}-2 c}{i b e -2 i 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 {i b e -2 i c d +2 \left (i e x +i d +1\right ) c -\sqrt {4 a c \,e^{2}-b^{2} e^{2}}-2 c}{i b e -2 i 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 {i b e -2 i c d +2 \left (i e x +i d +1\right ) c +\sqrt {4 a c \,e^{2}-b^{2} e^{2}}-2 c}{i b e -2 i 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}}}\) | \(959\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(4611\) |
| default | \(\text {Expression too large to display}\) | \(4611\) |
I*e*Pi/(-4*a*c*e^2+b^2*e^2)^(1/2)*arctan((I*b*e-2*I*c*d-2*(1-I*d-I*e*x)*c+ 2*c)/(-4*a*c*e^2+b^2*e^2)^(1/2))-1/2*e*ln(1-I*d-I*e*x)/(4*a*c*e^2-b^2*e^2) ^(1/2)*ln((I*b*e-2*I*c*d-2*(1-I*d-I*e*x)*c+(4*a*c*e^2-b^2*e^2)^(1/2)+2*c)/ (I*b*e-2*I*c*d+2*c+(4*a*c*e^2-b^2*e^2)^(1/2)))+1/2*e*ln(1-I*d-I*e*x)/(4*a* c*e^2-b^2*e^2)^(1/2)*ln((I*b*e-2*I*c*d-2*(1-I*d-I*e*x)*c-(4*a*c*e^2-b^2*e^ 2)^(1/2)+2*c)/(I*b*e-2*I*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((I*b*e-2*I*c*d-2*(1-I*d-I*e*x)*c+(4*a*c*e^2-b^2*e ^2)^(1/2)+2*c)/(I*b*e-2*I*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((I*b*e-2*I*c*d-2*(1-I*d-I*e*x)*c-(4*a*c*e^2-b^2* e^2)^(1/2)+2*c)/(I*b*e-2*I*c*d+2*c-(4*a*c*e^2-b^2*e^2)^(1/2)))-1/2*e*ln(1+ I*d+I*e*x)/(4*a*c*e^2-b^2*e^2)^(1/2)*ln((I*b*e-2*I*c*d+2*(1+I*d+I*e*x)*c-( 4*a*c*e^2-b^2*e^2)^(1/2)-2*c)/(I*b*e-2*I*c*d-(4*a*c*e^2-b^2*e^2)^(1/2)-2*c ))+1/2*e*ln(1+I*d+I*e*x)/(4*a*c*e^2-b^2*e^2)^(1/2)*ln((I*b*e-2*I*c*d+2*(1+ I*d+I*e*x)*c+(4*a*c*e^2-b^2*e^2)^(1/2)-2*c)/(I*b*e-2*I*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((I*b*e-2*I*c*d+2*(1 +I*d+I*e*x)*c-(4*a*c*e^2-b^2*e^2)^(1/2)-2*c)/(I*b*e-2*I*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((I*b*e-2*I*c*d+2*( 1+I*d+I*e*x)*c+(4*a*c*e^2-b^2*e^2)^(1/2)-2*c)/(I*b*e-2*I*c*d+(4*a*c*e^2-b^ 2*e^2)^(1/2)-2*c))
\[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\int { \frac {\operatorname {arccot}\left (e x + d\right )}{c x^{2} + b x + a} \,d x } \]
Timed out. \[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
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
Timed out. \[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\int \frac {\mathrm {acot}\left (d+e\,x\right )}{c\,x^2+b\,x+a} \,d x \]