Integrand size = 25, antiderivative size = 1135 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx =\text {Too large to display} \] Output:
-1/2*g^(1/2)*(a+b*arccoth(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)*(1-c)*g^(1/2)-d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/ (d*x+c+1))*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*arccoth(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)*(1-c)*g^(1/2)-d*(-f+(-4*e*g+f^2)^(1/2))^(1/2))/(d*x+c+ 1))*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*arccoth(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)*(1-c)*g^(1/2)+d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/(d*x+c+1))*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*arcc oth(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)*(1-c)*g^(1/2)+d*(-f+(-4*e*g+f^2)^(1/2))^(1/2))/(d*x+c+1))*2^(1/2)/(-4* e*g+f^2)^(1/2)/(-f+(-4*e*g+f^2)^(1/2))^(1/2)+1/4*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)*(1-c)*g^(1/2)- d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/(d*x+c+1))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f -(-4*e*g+f^2)^(1/2))^(1/2)-1/4*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)*(1-c)*g^(1/2)-d*(-f+(-4*e*g+f^2) ^(1/2))^(1/2))/(d*x+c+1))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f+(-4*e*g+f^2)^(1/2 ))^(1/2)-1/4*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)*(1-c)*g^(1/2)+d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/(d* x+c+1))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f-(-4*e*g+f^2)^(1/2))^(1/2)+1/4*b*...
Leaf count is larger than twice the leaf count of optimal. \(3087\) vs. \(2(1135)=2270\).
Time = 4.76 (sec) , antiderivative size = 3087, normalized size of antiderivative = 2.72 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*ArcCoth[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]]] - b*Sqrt[-(f - Sqrt[f^2 - 4*e*g])^2]*Sqr t[f + Sqrt[f^2 - 4*e*g]]*Log[(Sqrt[2]*Sqrt[g]*(-1 + c + d*x))/(Sqrt[2]*(-1 + 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] + b*Sqrt[-(f - Sqrt[f^2 - 4*e*g])^2]*Sqrt[f + Sqrt[f^2 - 4*e*g]]*Log[(-1 + c + d*x)/(c + d*x)]*Log[Sqrt[-f - Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x] + b*Sqrt[-(f - Sqrt[f^2 - 4*e*g])^2]*Sqrt[f + Sqrt[f^2 - 4*e*g]]*Log[(Sqrt[2]*Sqrt[g]*(1 + c + d*x))/(Sqrt[2]*(1 + 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] - b*Sqrt[-(f - Sqrt[f^2 - 4*e*g])^2]*Sqrt[f + Sqrt[ f^2 - 4*e*g]]*Log[(1 + c + d*x)/(c + d*x)]*Log[Sqrt[-f - Sqrt[f^2 - 4*e*g] ] - Sqrt[2]*Sqrt[g]*x] + b*Sqrt[f - Sqrt[f^2 - 4*e*g]]*Sqrt[-(f + Sqrt[f^2 - 4*e*g])^2]*Log[(Sqrt[2]*Sqrt[g]*(-1 + c + d*x))/(Sqrt[2]*(-1 + c)*Sqrt[ g] + d*Sqrt[-f + Sqrt[f^2 - 4*e*g]])]*Log[Sqrt[-f + Sqrt[f^2 - 4*e*g]] - S qrt[2]*Sqrt[g]*x] - b*Sqrt[f - Sqrt[f^2 - 4*e*g]]*Sqrt[-(f + Sqrt[f^2 - 4* e*g])^2]*Log[(-1 + c + d*x)/(c + d*x)]*Log[Sqrt[-f + Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x] - b*Sqrt[f - Sqrt[f^2 - 4*e*g]]*Sqrt[-(f + Sqrt[f^2 - 4 *e*g])^2]*Log[(Sqrt[2]*Sqrt[g]*(1 + c + d*x))/(Sqrt[2]*(1 + c)*Sqrt[g] ...
Leaf count is larger than twice the leaf count of optimal. \(2792\) vs. \(2(1135)=2270\).
Time = 7.81 (sec) , antiderivative size = 2792, normalized size of antiderivative = 2.46, 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.
| \(\displaystyle \int \frac {a+b \coth ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {a}{e+f x^2+g x^4}+\frac {b \coth ^{-1}(c+d x)}{e+f x^2+g x^4}\right )dx\) |
| \(\Big \downarrow \) 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+1}{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+1)}{\left (-\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2\right )-\sqrt {2} \sqrt {g} \sqrt {\sqrt {f^2-4 e g}-f} d+2 (1-c) c g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}+\frac {b \sqrt {g} \log \left (-\frac {-c-d x+1}{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+1)}{\left (-\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2\right )+\sqrt {2} \sqrt {g} \sqrt {\sqrt {f^2-4 e g}-f} d+2 (1-c) c g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}+\frac {b \sqrt {g} \log \left (-\frac {-c-d x+1}{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+1)}{\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 (1-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+1}{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+1)}{\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 (1-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+1}{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+1)}{\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2-\sqrt {2} \sqrt {g} \sqrt {\sqrt {f^2-4 e g}-f} d+2 c (c+1) g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}-\frac {b \sqrt {g} \log \left (\frac {c+d x+1}{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+1)}{\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2+\sqrt {2} \sqrt {g} \sqrt {\sqrt {f^2-4 e g}-f} d+2 c (c+1) g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}-\frac {b \sqrt {g} \log \left (\frac {c+d x+1}{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+1)}{\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+1) 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+1}{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+1)}{\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+1) 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+1)}{\left (-\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2\right )-\sqrt {2} \sqrt {g} \sqrt {\sqrt {f^2-4 e g}-f} d+2 (1-c) c g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}+\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+1)}{\left (-\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2\right )+\sqrt {2} \sqrt {g} \sqrt {\sqrt {f^2-4 e g}-f} d+2 (1-c) c g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}+\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+1)}{\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 (1-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+1)}{\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 (1-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+1)}{\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2-\sqrt {2} \sqrt {g} \sqrt {\sqrt {f^2-4 e g}-f} d+2 c (c+1) g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}-\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+1)}{\left (\left (f-\sqrt {f^2-4 e g}\right ) d^2+\sqrt {2} \sqrt {g} \sqrt {\sqrt {f^2-4 e g}-f} d+2 c (c+1) g\right ) (c+d x)}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}-\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+1)}{\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+1) 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+1)}{\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+1) 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*ArcCoth[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[-((1 - c - d*x)/(c + d*x))]*L og[1 - ((2*c^2*g + d^2*(f - Sqrt[f^2 - 4*e*g]))*(1 - c - d*x))/((2*(1 - c) *c*g - d^2*(f - Sqrt[f^2 - 4*e*g]) - Sqrt[2]*d*Sqrt[g]*Sqrt[-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[-((1 - c - d*x)/(c + d*x))]*Log[1 - ((2*c^2*g + d^2*(f - Sqrt[f^2 - 4*e*g]))*(1 - c - d*x))/((2*(1 - c)*c*g - d^2*(f - Sqr t[f^2 - 4*e*g]) + Sqrt[2]*d*Sqrt[g]*Sqrt[-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[-((1 - c - d*x)/(c + d*x))]*Log[1 - ((2*c^2*g + d^2*(f + Sqrt[f^2 - 4*e*g]))*(1 - c - d*x))/((2*(1 - c)*c*g - Sqrt[2]*d*Sqrt[g]*Sqrt[-f - Sqr t[f^2 - 4*e*g]] - d^2*(f + Sqrt[f^2 - 4*e*g]))*(c + d*x))])/(2*Sqrt[2]*Sqr t[f^2 - 4*e*g]*Sqrt[-f - Sqrt[f^2 - 4*e*g]]) - (b*Sqrt[g]*Log[-((1 - c - d *x)/(c + d*x))]*Log[1 - ((2*c^2*g + d^2*(f + Sqrt[f^2 - 4*e*g]))*(1 - c - d*x))/((2*(1 - 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[(1 + c + d*x)/(c + d*x)]*Log[1 - ((2*c^2*g + d^2*(f - Sqrt[f^2 - 4*e*g]))*(1 + c + d*x))/((2*c*(1 + c)...
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 = 2.19 (sec) , antiderivative size = 718, normalized size of antiderivative = 0.63
| method | result | size |
| risch | \(\frac {d^{3} a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (g \,\textit {\_Z}^{4}+\left (-4 c g +4 g \right ) \textit {\_Z}^{3}+\left (6 c^{2} g +d^{2} f -12 c g +6 g \right ) \textit {\_Z}^{2}+\left (-4 c^{3} g -2 c \,d^{2} f +12 c^{2} g +2 d^{2} f -12 c g +4 g \right ) \textit {\_Z} +c^{4} g +c^{2} d^{2} f +e \,d^{4}-4 c^{3} g -2 c \,d^{2} f +6 c^{2} g +d^{2} f -4 c g +g \right )}{\sum }\frac {\ln \left (d x -\textit {\_R} +c -1\right )}{2 \textit {\_R}^{3} g -6 \textit {\_R}^{2} c g +6 \textit {\_R} \,c^{2} g +\textit {\_R} \,d^{2} f -2 c^{3} g -c \,d^{2} f +6 g \,\textit {\_R}^{2}-12 \textit {\_R} c g +6 c^{2} g +d^{2} f +6 g \textit {\_R} -6 c g +2 g}\right )}{2}-\frac {d^{3} b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (g \,\textit {\_Z}^{4}+\left (-4 c g +4 g \right ) \textit {\_Z}^{3}+\left (6 c^{2} g +d^{2} f -12 c g +6 g \right ) \textit {\_Z}^{2}+\left (-4 c^{3} g -2 c \,d^{2} f +12 c^{2} g +2 d^{2} f -12 c g +4 g \right ) \textit {\_Z} +c^{4} g +c^{2} d^{2} f +e \,d^{4}-4 c^{3} g -2 c \,d^{2} f +6 c^{2} g +d^{2} f -4 c g +g \right )}{\sum }\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )}{2 \textit {\_R1}^{3} g -6 \textit {\_R1}^{2} c g +6 \textit {\_R1} \,c^{2} g +\textit {\_R1} \,d^{2} f -2 c^{3} g -c \,d^{2} f +6 \textit {\_R1}^{2} g -12 \textit {\_R1} c g +6 c^{2} g +d^{2} f +6 \textit {\_R1} g -6 c g +2 g}\right )}{4}+\frac {b \,d^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (g \,\textit {\_Z}^{4}+\left (-4 c g -4 g \right ) \textit {\_Z}^{3}+\left (6 c^{2} g +d^{2} f +12 c g +6 g \right ) \textit {\_Z}^{2}+\left (-4 c^{3} g -2 c \,d^{2} f -12 c^{2} g -2 d^{2} f -12 c g -4 g \right ) \textit {\_Z} +c^{4} g +c^{2} d^{2} f +e \,d^{4}+4 c^{3} g +2 c \,d^{2} f +6 c^{2} g +d^{2} f +4 c g +g \right )}{\sum }\frac {\ln \left (d x +c +1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )}{2 \textit {\_R1}^{3} g -6 \textit {\_R1}^{2} c g +6 \textit {\_R1} \,c^{2} g +\textit {\_R1} \,d^{2} f -2 c^{3} g -c \,d^{2} f -6 \textit {\_R1}^{2} g +12 \textit {\_R1} c g -6 c^{2} g -d^{2} f +6 \textit {\_R1} g -6 c g -2 g}\right )}{4}\) | \(718\) |
| parts | \(\text {Expression too large to display}\) | \(1994\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2000\) |
| default | \(\text {Expression too large to display}\) | \(2000\) |
Input:
int((a+b*arccoth(d*x+c))/(g*x^4+f*x^2+e),x,method=_RETURNVERBOSE)
Output:
1/2*d^3*a*sum(1/(2*_R^3*g-6*_R^2*c*g+6*_R*c^2*g+_R*d^2*f-2*c^3*g-c*d^2*f+6 *_R^2*g-12*_R*c*g+6*c^2*g+d^2*f+6*_R*g-6*c*g+2*g)*ln(d*x-_R+c-1),_R=RootOf (g*_Z^4+(-4*c*g+4*g)*_Z^3+(6*c^2*g+d^2*f-12*c*g+6*g)*_Z^2+(-4*c^3*g-2*c*d^ 2*f+12*c^2*g+2*d^2*f-12*c*g+4*g)*_Z+c^4*g+c^2*d^2*f+e*d^4-4*c^3*g-2*c*d^2* f+6*c^2*g+d^2*f-4*c*g+g))-1/4*d^3*b*sum(1/(2*_R1^3*g-6*_R1^2*c*g+6*_R1*c^2 *g+_R1*d^2*f-2*c^3*g-c*d^2*f+6*_R1^2*g-12*_R1*c*g+6*c^2*g+d^2*f+6*_R1*g-6* c*g+2*g)*(ln(d*x+c-1)*ln((-d*x+_R1-c+1)/_R1)+dilog((-d*x+_R1-c+1)/_R1)),_R 1=RootOf(g*_Z^4+(-4*c*g+4*g)*_Z^3+(6*c^2*g+d^2*f-12*c*g+6*g)*_Z^2+(-4*c^3* g-2*c*d^2*f+12*c^2*g+2*d^2*f-12*c*g+4*g)*_Z+c^4*g+c^2*d^2*f+e*d^4-4*c^3*g- 2*c*d^2*f+6*c^2*g+d^2*f-4*c*g+g))+1/4*b*d^3*sum(1/(2*_R1^3*g-6*_R1^2*c*g+6 *_R1*c^2*g+_R1*d^2*f-2*c^3*g-c*d^2*f-6*_R1^2*g+12*_R1*c*g-6*c^2*g-d^2*f+6* _R1*g-6*c*g-2*g)*(ln(d*x+c+1)*ln((-d*x+_R1-c-1)/_R1)+dilog((-d*x+_R1-c-1)/ _R1)),_R1=RootOf(g*_Z^4+(-4*c*g-4*g)*_Z^3+(6*c^2*g+d^2*f+12*c*g+6*g)*_Z^2+ (-4*c^3*g-2*c*d^2*f-12*c^2*g-2*d^2*f-12*c*g-4*g)*_Z+c^4*g+c^2*d^2*f+e*d^4+ 4*c^3*g+2*c*d^2*f+6*c^2*g+d^2*f+4*c*g+g))
\[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\int { \frac {b \operatorname {arcoth}\left (d x + c\right ) + a}{g x^{4} + f x^{2} + e} \,d x } \] Input:
integrate((a+b*arccoth(d*x+c))/(g*x^4+f*x^2+e),x, algorithm="fricas")
Output:
integral((b*arccoth(d*x + c) + a)/(g*x^4 + f*x^2 + e), x)
Timed out. \[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\text {Timed out} \] Input:
integrate((a+b*acoth(d*x+c))/(g*x**4+f*x**2+e),x)
Output:
Timed out
\[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\int { \frac {b \operatorname {arcoth}\left (d x + c\right ) + a}{g x^{4} + f x^{2} + e} \,d x } \] Input:
integrate((a+b*arccoth(d*x+c))/(g*x^4+f*x^2+e),x, algorithm="maxima")
Output:
integrate((b*arccoth(d*x + c) + a)/(g*x^4 + f*x^2 + e), x)
\[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\int { \frac {b \operatorname {arcoth}\left (d x + c\right ) + a}{g x^{4} + f x^{2} + e} \,d x } \] Input:
integrate((a+b*arccoth(d*x+c))/(g*x^4+f*x^2+e),x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\int \frac {a+b\,\mathrm {acoth}\left (c+d\,x\right )}{g\,x^4+f\,x^2+e} \,d x \] Input:
int((a + b*acoth(c + d*x))/(e + f*x^2 + g*x^4),x)
Output:
int((a + b*acoth(c + d*x))/(e + f*x^2 + g*x^4), x)
\[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f x^2+g x^4} \, dx=\frac {2 \sqrt {e}\, \sqrt {2 \sqrt {g}\, \sqrt {e}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {g}\, \sqrt {e}-f}-2 \sqrt {g}\, x}{\sqrt {2 \sqrt {g}\, \sqrt {e}+f}}\right ) a f -4 \sqrt {g}\, \sqrt {2 \sqrt {g}\, \sqrt {e}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {g}\, \sqrt {e}-f}-2 \sqrt {g}\, x}{\sqrt {2 \sqrt {g}\, \sqrt {e}+f}}\right ) a e -2 \sqrt {e}\, \sqrt {2 \sqrt {g}\, \sqrt {e}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {g}\, \sqrt {e}-f}+2 \sqrt {g}\, x}{\sqrt {2 \sqrt {g}\, \sqrt {e}+f}}\right ) a f +4 \sqrt {g}\, \sqrt {2 \sqrt {g}\, \sqrt {e}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {g}\, \sqrt {e}-f}+2 \sqrt {g}\, x}{\sqrt {2 \sqrt {g}\, \sqrt {e}+f}}\right ) a e -\sqrt {e}\, \sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, \mathrm {log}\left (-\sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, x +\sqrt {e}+\sqrt {g}\, x^{2}\right ) a f +\sqrt {e}\, \sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, \mathrm {log}\left (\sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, x +\sqrt {e}+\sqrt {g}\, x^{2}\right ) a f -2 \sqrt {g}\, \sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, \mathrm {log}\left (-\sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, x +\sqrt {e}+\sqrt {g}\, x^{2}\right ) a e +2 \sqrt {g}\, \sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, \mathrm {log}\left (\sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, x +\sqrt {e}+\sqrt {g}\, x^{2}\right ) a e +16 \left (\int \frac {\mathit {acoth} \left (d x +c \right )}{g \,x^{4}+f \,x^{2}+e}d x \right ) b \,e^{2} g -4 \left (\int \frac {\mathit {acoth} \left (d x +c \right )}{g \,x^{4}+f \,x^{2}+e}d x \right ) b e \,f^{2}}{4 e \left (4 e g -f^{2}\right )} \] Input:
int((a+b*acoth(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(acoth(c + d*x)/(e + f*x**2 + g*x**4),x)*b*e**2*g - 4*int(acoth(c + d*x)/(e + f*x**2 + g*x**4),x)*b*e*f**2)/(4*e*(4*e*g - f **2))