Integrand size = 25, antiderivative size = 1199 \[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx =\text {Too large to display} \] Output:
-1/2*g^(1/2)*(a+b*arccot(d*x+c))*ln(-2*d*((-f-(-4*e*g+f^2)^(1/2))^(1/2)-g^ (1/2)*x*2^(1/2))/(2^(1/2)*(I-c)*g^(1/2)-d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/( 1-I*(d*x+c)))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f-(-4*e*g+f^2)^(1/2))^(1/2)+1/2 *g^(1/2)*(a+b*arccot(d*x+c))*ln(-2*d*((-f+(-4*e*g+f^2)^(1/2))^(1/2)-g^(1/2 )*x*2^(1/2))/(2^(1/2)*(I-c)*g^(1/2)-d*(-f+(-4*e*g+f^2)^(1/2))^(1/2))/(1-I* (d*x+c)))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f+(-4*e*g+f^2)^(1/2))^(1/2)+1/2*g^( 1/2)*(a+b*arccot(d*x+c))*ln(2*d*((-f-(-4*e*g+f^2)^(1/2))^(1/2)+g^(1/2)*x*2 ^(1/2))/(2^(1/2)*(I-c)*g^(1/2)+d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/(1-I*(d*x+ c)))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f-(-4*e*g+f^2)^(1/2))^(1/2)-1/2*g^(1/2)* (a+b*arccot(d*x+c))*ln(2*d*((-f+(-4*e*g+f^2)^(1/2))^(1/2)+g^(1/2)*x*2^(1/2 ))/(2^(1/2)*(I-c)*g^(1/2)+d*(-f+(-4*e*g+f^2)^(1/2))^(1/2))/(1-I*(d*x+c)))* 2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f+(-4*e*g+f^2)^(1/2))^(1/2)-1/4*I*b*g^(1/2)*p olylog(2,1+2*d*((-f-(-4*e*g+f^2)^(1/2))^(1/2)-g^(1/2)*x*2^(1/2))/(2^(1/2)* (I-c)*g^(1/2)-d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/(1-I*(d*x+c)))*2^(1/2)/(-4* e*g+f^2)^(1/2)/(-f-(-4*e*g+f^2)^(1/2))^(1/2)+1/4*I*b*g^(1/2)*polylog(2,1+2 *d*((-f+(-4*e*g+f^2)^(1/2))^(1/2)-g^(1/2)*x*2^(1/2))/(2^(1/2)*(I-c)*g^(1/2 )-d*(-f+(-4*e*g+f^2)^(1/2))^(1/2))/(1-I*(d*x+c)))*2^(1/2)/(-4*e*g+f^2)^(1/ 2)/(-f+(-4*e*g+f^2)^(1/2))^(1/2)+1/4*I*b*g^(1/2)*polylog(2,1-2*d*((-f-(-4* e*g+f^2)^(1/2))^(1/2)+g^(1/2)*x*2^(1/2))/(2^(1/2)*(I-c)*g^(1/2)+d*(-f-(-4* e*g+f^2)^(1/2))^(1/2))/(1-I*(d*x+c)))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f-(-...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3211\) vs. \(2(1199)=2398\).
Time = 4.60 (sec) , antiderivative size = 3211, normalized size of antiderivative = 2.68 \[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*ArcCot[c + d*x])/(e + f*x^2 + g*x^4),x]
Output:
(Sqrt[g]*(4*a*Sqrt[-f + Sqrt[f^2 - 4*e*g]]*Sqrt[-(f + Sqrt[f^2 - 4*e*g])^2 ]*ArcTan[(Sqrt[2]*Sqrt[g]*x)/Sqrt[f - Sqrt[f^2 - 4*e*g]]] - 4*a*Sqrt[-f - Sqrt[f^2 - 4*e*g]]*Sqrt[-(f - Sqrt[f^2 - 4*e*g])^2]*ArcTan[(Sqrt[2]*Sqrt[g ]*x)/Sqrt[f + Sqrt[f^2 - 4*e*g]]] + I*b*Sqrt[-(f - Sqrt[f^2 - 4*e*g])^2]*S qrt[f + Sqrt[f^2 - 4*e*g]]*Log[(Sqrt[2]*Sqrt[g]*(-I + c + d*x))/(Sqrt[2]*( -I + c)*Sqrt[g] + d*Sqrt[-f - Sqrt[f^2 - 4*e*g]])]*Log[Sqrt[-f - Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x] - I*b*Sqrt[-(f - Sqrt[f^2 - 4*e*g])^2]*Sqrt [f + Sqrt[f^2 - 4*e*g]]*Log[(-I + c + d*x)/(c + d*x)]*Log[Sqrt[-f - Sqrt[f ^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x] - I*b*Sqrt[-(f - Sqrt[f^2 - 4*e*g])^2]*S qrt[f + Sqrt[f^2 - 4*e*g]]*Log[(Sqrt[2]*Sqrt[g]*(I + c + d*x))/(Sqrt[2]*(I + c)*Sqrt[g] + d*Sqrt[-f - Sqrt[f^2 - 4*e*g]])]*Log[Sqrt[-f - Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x] + I*b*Sqrt[-(f - Sqrt[f^2 - 4*e*g])^2]*Sqrt[f + Sqrt[f^2 - 4*e*g]]*Log[(I + c + d*x)/(c + d*x)]*Log[Sqrt[-f - Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x] - I*b*Sqrt[f - Sqrt[f^2 - 4*e*g]]*Sqrt[-(f + Sqrt[f^2 - 4*e*g])^2]*Log[(Sqrt[2]*Sqrt[g]*(-I + c + d*x))/(Sqrt[2]*(-I + c)*Sqrt[g] + d*Sqrt[-f + Sqrt[f^2 - 4*e*g]])]*Log[Sqrt[-f + Sqrt[f^2 - 4 *e*g]] - Sqrt[2]*Sqrt[g]*x] + I*b*Sqrt[f - Sqrt[f^2 - 4*e*g]]*Sqrt[-(f + S qrt[f^2 - 4*e*g])^2]*Log[(-I + c + d*x)/(c + d*x)]*Log[Sqrt[-f + Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x] + I*b*Sqrt[f - Sqrt[f^2 - 4*e*g]]*Sqrt[-(f + Sqrt[f^2 - 4*e*g])^2]*Log[(Sqrt[2]*Sqrt[g]*(I + c + d*x))/(Sqrt[2]*(I...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2808\) vs. \(2(1199)=2398\).
Time = 7.95 (sec) , antiderivative size = 2808, normalized size of antiderivative = 2.34, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
\(\displaystyle \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {a}{e+f x^2+g x^4}+\frac {b \cot ^{-1}(c+d x)}{e+f x^2+g x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {2} a \sqrt {g} \arctan \left (\frac {\sqrt {2} \sqrt {g} x}{\sqrt {f-\sqrt {f^2-4 e g}}}\right )}{\sqrt {f^2-4 e g} \sqrt {f-\sqrt {f^2-4 e g}}}-\frac {\sqrt {2} a \sqrt {g} \arctan \left (\frac {\sqrt {2} \sqrt {g} x}{\sqrt {f+\sqrt {f^2-4 e g}}}\right )}{\sqrt {f^2-4 e g} \sqrt {f+\sqrt {f^2-4 e g}}}-\frac {b \sqrt {g} \log \left (-\frac {-c-d x+i}{c+d x}\right ) \log \left (1-\frac {\left (2 g c^2+d^2 \left (f-\sqrt {f^2-4 e g}\right )\right ) (-c-d x+i)}{\left (-\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2\right )-\sqrt {2} \sqrt {g} \sqrt {f-\sqrt {f^2-4 e g}} d+2 (i-c) c g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f-\sqrt {f^2-4 e g}}}+\frac {b \sqrt {g} \log \left (-\frac {-c-d x+i}{c+d x}\right ) \log \left (1-\frac {\left (2 g c^2+d^2 \left (f-\sqrt {f^2-4 e g}\right )\right ) (-c-d x+i)}{\left (-\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2\right )+\sqrt {2} \sqrt {g} \sqrt {f-\sqrt {f^2-4 e g}} d+2 (i-c) c g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f-\sqrt {f^2-4 e g}}}+\frac {b \sqrt {g} \log \left (-\frac {-c-d x+i}{c+d x}\right ) \log \left (1-\frac {\left (2 g c^2+d^2 \left (f+\sqrt {f^2-4 e g}\right )\right ) (-c-d x+i)}{\left (-\left (\left (f+\sqrt {f^2-4 e g}\right ) d^2\right )-\sqrt {2} \sqrt {g} \sqrt {f+\sqrt {f^2-4 e g}} d+2 (i-c) c g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f+\sqrt {f^2-4 e g}}}-\frac {b \sqrt {g} \log \left (-\frac {-c-d x+i}{c+d x}\right ) \log \left (1-\frac {\left (2 g c^2+d^2 \left (f+\sqrt {f^2-4 e g}\right )\right ) (-c-d x+i)}{\left (-\left (\left (f+\sqrt {f^2-4 e g}\right ) d^2\right )+\sqrt {2} \sqrt {g} \sqrt {f+\sqrt {f^2-4 e g}} d+2 (i-c) c g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f+\sqrt {f^2-4 e g}}}+\frac {b \sqrt {g} \log \left (\frac {c+d x+i}{c+d x}\right ) \log \left (1-\frac {\left (2 g c^2+d^2 \left (f-\sqrt {f^2-4 e g}\right )\right ) (c+d x+i)}{\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2-\sqrt {2} \sqrt {g} \sqrt {f-\sqrt {f^2-4 e g}} d+2 c (c+i) g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f-\sqrt {f^2-4 e g}}}-\frac {b \sqrt {g} \log \left (\frac {c+d x+i}{c+d x}\right ) \log \left (1-\frac {\left (2 g c^2+d^2 \left (f-\sqrt {f^2-4 e g}\right )\right ) (c+d x+i)}{\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2+\sqrt {2} \sqrt {g} \sqrt {f-\sqrt {f^2-4 e g}} d+2 c (c+i) g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f-\sqrt {f^2-4 e g}}}-\frac {b \sqrt {g} \log \left (\frac {c+d x+i}{c+d x}\right ) \log \left (1-\frac {\left (2 g c^2+d^2 \left (f+\sqrt {f^2-4 e g}\right )\right ) (c+d x+i)}{\left (\left (f+\sqrt {f^2-4 e g}\right ) d^2-\sqrt {2} \sqrt {g} \sqrt {f+\sqrt {f^2-4 e g}} d+2 c (c+i) g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f+\sqrt {f^2-4 e g}}}+\frac {b \sqrt {g} \log \left (\frac {c+d x+i}{c+d x}\right ) \log \left (1-\frac {\left (2 g c^2+d^2 \left (f+\sqrt {f^2-4 e g}\right )\right ) (c+d x+i)}{\left (\left (f+\sqrt {f^2-4 e g}\right ) d^2+\sqrt {2} \sqrt {g} \sqrt {f+\sqrt {f^2-4 e g}} d+2 c (c+i) g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f+\sqrt {f^2-4 e g}}}-\frac {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\left (2 g c^2+d^2 \left (f-\sqrt {f^2-4 e g}\right )\right ) (-c-d x+i)}{\left (-\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2\right )-\sqrt {2} \sqrt {g} \sqrt {f-\sqrt {f^2-4 e g}} d+2 (i-c) c g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f-\sqrt {f^2-4 e g}}}+\frac {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\left (2 g c^2+d^2 \left (f-\sqrt {f^2-4 e g}\right )\right ) (-c-d x+i)}{\left (-\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2\right )+\sqrt {2} \sqrt {g} \sqrt {f-\sqrt {f^2-4 e g}} d+2 (i-c) c g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f-\sqrt {f^2-4 e g}}}+\frac {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\left (2 g c^2+d^2 \left (f+\sqrt {f^2-4 e g}\right )\right ) (-c-d x+i)}{\left (-\left (\left (f+\sqrt {f^2-4 e g}\right ) d^2\right )-\sqrt {2} \sqrt {g} \sqrt {f+\sqrt {f^2-4 e g}} d+2 (i-c) c g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f+\sqrt {f^2-4 e g}}}-\frac {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\left (2 g c^2+d^2 \left (f+\sqrt {f^2-4 e g}\right )\right ) (-c-d x+i)}{\left (-\left (\left (f+\sqrt {f^2-4 e g}\right ) d^2\right )+\sqrt {2} \sqrt {g} \sqrt {f+\sqrt {f^2-4 e g}} d+2 (i-c) c g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f+\sqrt {f^2-4 e g}}}+\frac {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\left (2 g c^2+d^2 \left (f-\sqrt {f^2-4 e g}\right )\right ) (c+d x+i)}{\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2-\sqrt {2} \sqrt {g} \sqrt {f-\sqrt {f^2-4 e g}} d+2 c (c+i) g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f-\sqrt {f^2-4 e g}}}-\frac {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\left (2 g c^2+d^2 \left (f-\sqrt {f^2-4 e g}\right )\right ) (c+d x+i)}{\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2+\sqrt {2} \sqrt {g} \sqrt {f-\sqrt {f^2-4 e g}} d+2 c (c+i) g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f-\sqrt {f^2-4 e g}}}-\frac {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\left (2 g c^2+d^2 \left (f+\sqrt {f^2-4 e g}\right )\right ) (c+d x+i)}{\left (\left (f+\sqrt {f^2-4 e g}\right ) d^2-\sqrt {2} \sqrt {g} \sqrt {f+\sqrt {f^2-4 e g}} d+2 c (c+i) g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f+\sqrt {f^2-4 e g}}}+\frac {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\left (2 g c^2+d^2 \left (f+\sqrt {f^2-4 e g}\right )\right ) (c+d x+i)}{\left (\left (f+\sqrt {f^2-4 e g}\right ) d^2+\sqrt {2} \sqrt {g} \sqrt {f+\sqrt {f^2-4 e g}} d+2 c (c+i) g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {f+\sqrt {f^2-4 e g}}}\) |
Input:
Int[(a + b*ArcCot[c + d*x])/(e + f*x^2 + g*x^4),x]
Output:
(Sqrt[2]*a*Sqrt[g]*ArcTan[(Sqrt[2]*Sqrt[g]*x)/Sqrt[f - Sqrt[f^2 - 4*e*g]]] )/(Sqrt[f^2 - 4*e*g]*Sqrt[f - Sqrt[f^2 - 4*e*g]]) - (Sqrt[2]*a*Sqrt[g]*Arc Tan[(Sqrt[2]*Sqrt[g]*x)/Sqrt[f + Sqrt[f^2 - 4*e*g]]])/(Sqrt[f^2 - 4*e*g]*S qrt[f + Sqrt[f^2 - 4*e*g]]) - (b*Sqrt[g]*Log[-((I - c - d*x)/(c + d*x))]*L og[1 - ((2*c^2*g + d^2*(f - Sqrt[f^2 - 4*e*g]))*(I - c - d*x))/((2*(I - c) *c*g - Sqrt[2]*d*Sqrt[g]*Sqrt[f - Sqrt[f^2 - 4*e*g]] - d^2*(f - Sqrt[f^2 - 4*e*g]))*(c + d*x))])/(2*Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt[f - Sqrt[f^2 - 4* e*g]]) + (b*Sqrt[g]*Log[-((I - c - d*x)/(c + d*x))]*Log[1 - ((2*c^2*g + d^ 2*(f - Sqrt[f^2 - 4*e*g]))*(I - c - d*x))/((2*(I - c)*c*g + Sqrt[2]*d*Sqrt [g]*Sqrt[f - Sqrt[f^2 - 4*e*g]] - d^2*(f - Sqrt[f^2 - 4*e*g]))*(c + d*x))] )/(2*Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt[f - Sqrt[f^2 - 4*e*g]]) + (b*Sqrt[g]*L og[-((I - c - d*x)/(c + d*x))]*Log[1 - ((2*c^2*g + d^2*(f + Sqrt[f^2 - 4*e *g]))*(I - c - d*x))/((2*(I - c)*c*g - Sqrt[2]*d*Sqrt[g]*Sqrt[f + Sqrt[f^2 - 4*e*g]] - d^2*(f + Sqrt[f^2 - 4*e*g]))*(c + d*x))])/(2*Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt[f + Sqrt[f^2 - 4*e*g]]) - (b*Sqrt[g]*Log[-((I - c - d*x)/(c + d*x))]*Log[1 - ((2*c^2*g + d^2*(f + Sqrt[f^2 - 4*e*g]))*(I - c - d*x))/ ((2*(I - c)*c*g + Sqrt[2]*d*Sqrt[g]*Sqrt[f + Sqrt[f^2 - 4*e*g]] - d^2*(f + Sqrt[f^2 - 4*e*g]))*(c + d*x))])/(2*Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt[f + Sq rt[f^2 - 4*e*g]]) + (b*Sqrt[g]*Log[(I + c + d*x)/(c + d*x)]*Log[1 - ((2*c^ 2*g + d^2*(f - Sqrt[f^2 - 4*e*g]))*(I + c + d*x))/((2*c*(I + c)*g - Sqr...
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.42 (sec) , antiderivative size = 1287, normalized size of antiderivative = 1.07
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1287\) |
parts | \(\text {Expression too large to display}\) | \(2083\) |
derivativedivides | \(\text {Expression too large to display}\) | \(2090\) |
default | \(\text {Expression too large to display}\) | \(2090\) |
Input:
int((a+b*arccot(d*x+c))/(g*x^4+f*x^2+e),x,method=_RETURNVERBOSE)
Output:
1/4*I*d^3*Pi*sum(1/(6*I*_R^2*c*g-2*I*c^3*g-I*c*d^2*f-12*I*_R*c*g+2*_R^3*g- 6*_R*c^2*g-_R*d^2*f+6*I*c*g-6*g*_R^2+6*c^2*g+d^2*f+6*g*_R-2*g)*ln(-I*d*x-I *c+1-_R),_R=RootOf(g*_Z^4+(4*RootOf(_Z^2+1,index=1)*c*g-4*g)*_Z^3+(-12*Roo tOf(_Z^2+1,index=1)*c*g-6*c^2*g-d^2*f+6*g)*_Z^2+(-4*RootOf(_Z^2+1,index=1) *c^3*g-2*RootOf(_Z^2+1,index=1)*c*d^2*f+12*RootOf(_Z^2+1,index=1)*c*g+12*c ^2*g+2*d^2*f-4*g)*_Z+4*RootOf(_Z^2+1,index=1)*c^3*g+2*RootOf(_Z^2+1,index= 1)*c*d^2*f+c^4*g+c^2*d^2*f+e*d^4-4*RootOf(_Z^2+1,index=1)*c*g-6*c^2*g-d^2* f+g))*b+1/2*I*d^3*a*sum(1/(6*I*_R^2*c*g-2*I*c^3*g-I*c*d^2*f-12*I*_R*c*g+2* _R^3*g-6*_R*c^2*g-_R*d^2*f+6*I*c*g-6*g*_R^2+6*c^2*g+d^2*f+6*g*_R-2*g)*ln(- I*d*x-I*c+1-_R),_R=RootOf(g*_Z^4+(4*RootOf(_Z^2+1,index=1)*c*g-4*g)*_Z^3+( -12*RootOf(_Z^2+1,index=1)*c*g-6*c^2*g-d^2*f+6*g)*_Z^2+(-4*RootOf(_Z^2+1,i ndex=1)*c^3*g-2*RootOf(_Z^2+1,index=1)*c*d^2*f+12*RootOf(_Z^2+1,index=1)*c *g+12*c^2*g+2*d^2*f-4*g)*_Z+4*RootOf(_Z^2+1,index=1)*c^3*g+2*RootOf(_Z^2+1 ,index=1)*c*d^2*f+c^4*g+c^2*d^2*f+e*d^4-4*RootOf(_Z^2+1,index=1)*c*g-6*c^2 *g-d^2*f+g))+1/4*d^3*b*sum(1/(6*I*_R1^2*c*g-2*I*c^3*g-I*c*d^2*f-12*I*_R1*c *g+2*_R1^3*g-6*_R1*c^2*g-_R1*d^2*f+6*I*c*g-6*_R1^2*g+6*c^2*g+d^2*f+6*_R1*g -2*g)*(ln(1-I*c-I*d*x)*ln((_R1+I*d*x+I*c-1)/_R1)+dilog((_R1+I*d*x+I*c-1)/_ R1)),_R1=RootOf(g*_Z^4+(4*RootOf(_Z^2+1,index=1)*c*g-4*g)*_Z^3+(-12*RootOf (_Z^2+1,index=1)*c*g-6*c^2*g-d^2*f+6*g)*_Z^2+(-4*RootOf(_Z^2+1,index=1)*c^ 3*g-2*RootOf(_Z^2+1,index=1)*c*d^2*f+12*RootOf(_Z^2+1,index=1)*c*g+12*c...
\[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\int { \frac {b \operatorname {arccot}\left (d x + c\right ) + a}{g x^{4} + f x^{2} + e} \,d x } \] Input:
integrate((a+b*arccot(d*x+c))/(g*x^4+f*x^2+e),x, algorithm="fricas")
Output:
integral((b*arccot(d*x + c) + a)/(g*x^4 + f*x^2 + e), x)
Timed out. \[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\text {Timed out} \] Input:
integrate((a+b*acot(d*x+c))/(g*x**4+f*x**2+e),x)
Output:
Timed out
\[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\int { \frac {b \operatorname {arccot}\left (d x + c\right ) + a}{g x^{4} + f x^{2} + e} \,d x } \] Input:
integrate((a+b*arccot(d*x+c))/(g*x^4+f*x^2+e),x, algorithm="maxima")
Output:
integrate((b*arccot(d*x + c) + a)/(g*x^4 + f*x^2 + e), x)
Timed out. \[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\text {Timed out} \] Input:
integrate((a+b*arccot(d*x+c))/(g*x^4+f*x^2+e),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\int \frac {a+b\,\mathrm {acot}\left (c+d\,x\right )}{g\,x^4+f\,x^2+e} \,d x \] Input:
int((a + b*acot(c + d*x))/(e + f*x^2 + g*x^4),x)
Output:
int((a + b*acot(c + d*x))/(e + f*x^2 + g*x^4), x)
\[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\frac {2 \sqrt {e}\, \sqrt {2 \sqrt {g}\, \sqrt {e}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {g}\, \sqrt {e}-f}-2 \sqrt {g}\, x}{\sqrt {2 \sqrt {g}\, \sqrt {e}+f}}\right ) a f -4 \sqrt {g}\, \sqrt {2 \sqrt {g}\, \sqrt {e}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {g}\, \sqrt {e}-f}-2 \sqrt {g}\, x}{\sqrt {2 \sqrt {g}\, \sqrt {e}+f}}\right ) a e -2 \sqrt {e}\, \sqrt {2 \sqrt {g}\, \sqrt {e}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {g}\, \sqrt {e}-f}+2 \sqrt {g}\, x}{\sqrt {2 \sqrt {g}\, \sqrt {e}+f}}\right ) a f +4 \sqrt {g}\, \sqrt {2 \sqrt {g}\, \sqrt {e}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {g}\, \sqrt {e}-f}+2 \sqrt {g}\, x}{\sqrt {2 \sqrt {g}\, \sqrt {e}+f}}\right ) a e -\sqrt {e}\, \sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, \mathrm {log}\left (-\sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, x +\sqrt {e}+\sqrt {g}\, x^{2}\right ) a f +\sqrt {e}\, \sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, \mathrm {log}\left (\sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, x +\sqrt {e}+\sqrt {g}\, x^{2}\right ) a f -2 \sqrt {g}\, \sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, \mathrm {log}\left (-\sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, x +\sqrt {e}+\sqrt {g}\, x^{2}\right ) a e +2 \sqrt {g}\, \sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, \mathrm {log}\left (\sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, x +\sqrt {e}+\sqrt {g}\, x^{2}\right ) a e +16 \left (\int \frac {\mathit {acot} \left (d x +c \right )}{g \,x^{4}+f \,x^{2}+e}d x \right ) b \,e^{2} g -4 \left (\int \frac {\mathit {acot} \left (d x +c \right )}{g \,x^{4}+f \,x^{2}+e}d x \right ) b e \,f^{2}}{4 e \left (4 e g -f^{2}\right )} \] Input:
int((a+b*acot(d*x+c))/(g*x^4+f*x^2+e),x)
Output:
(2*sqrt(e)*sqrt(2*sqrt(g)*sqrt(e) + f)*atan((sqrt(2*sqrt(g)*sqrt(e) - f) - 2*sqrt(g)*x)/sqrt(2*sqrt(g)*sqrt(e) + f))*a*f - 4*sqrt(g)*sqrt(2*sqrt(g)* sqrt(e) + f)*atan((sqrt(2*sqrt(g)*sqrt(e) - f) - 2*sqrt(g)*x)/sqrt(2*sqrt( g)*sqrt(e) + f))*a*e - 2*sqrt(e)*sqrt(2*sqrt(g)*sqrt(e) + f)*atan((sqrt(2* sqrt(g)*sqrt(e) - f) + 2*sqrt(g)*x)/sqrt(2*sqrt(g)*sqrt(e) + f))*a*f + 4*s qrt(g)*sqrt(2*sqrt(g)*sqrt(e) + f)*atan((sqrt(2*sqrt(g)*sqrt(e) - f) + 2*s qrt(g)*x)/sqrt(2*sqrt(g)*sqrt(e) + f))*a*e - sqrt(e)*sqrt(2*sqrt(g)*sqrt(e ) - f)*log( - sqrt(2*sqrt(g)*sqrt(e) - f)*x + sqrt(e) + sqrt(g)*x**2)*a*f + sqrt(e)*sqrt(2*sqrt(g)*sqrt(e) - f)*log(sqrt(2*sqrt(g)*sqrt(e) - f)*x + sqrt(e) + sqrt(g)*x**2)*a*f - 2*sqrt(g)*sqrt(2*sqrt(g)*sqrt(e) - f)*log( - sqrt(2*sqrt(g)*sqrt(e) - f)*x + sqrt(e) + sqrt(g)*x**2)*a*e + 2*sqrt(g)*s qrt(2*sqrt(g)*sqrt(e) - f)*log(sqrt(2*sqrt(g)*sqrt(e) - f)*x + sqrt(e) + s qrt(g)*x**2)*a*e + 16*int(acot(c + d*x)/(e + f*x**2 + g*x**4),x)*b*e**2*g - 4*int(acot(c + d*x)/(e + f*x**2 + g*x**4),x)*b*e*f**2)/(4*e*(4*e*g - f** 2))