Integrand size = 27, antiderivative size = 349 \[ \int \frac {d+e x^4}{d^2-b x^4+e^2 x^8} \, dx=-\frac {\sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}-\frac {\sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}-\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}-\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}} \]
-1/2*arctan(x*2^(1/2)*e^(1/2)/((-2*d*e+b)^(1/2)-(2*d*e+b)^(1/2))^(1/2))*e^ (1/2)*2^(1/2)/(-2*d*e+b)^(1/2)/((-2*d*e+b)^(1/2)-(2*d*e+b)^(1/2))^(1/2)-1/ 2*arctanh(x*2^(1/2)*e^(1/2)/((-2*d*e+b)^(1/2)-(2*d*e+b)^(1/2))^(1/2))*e^(1 /2)*2^(1/2)/(-2*d*e+b)^(1/2)/((-2*d*e+b)^(1/2)-(2*d*e+b)^(1/2))^(1/2)-1/2* arctan(x*2^(1/2)*e^(1/2)/((-2*d*e+b)^(1/2)+(2*d*e+b)^(1/2))^(1/2))*e^(1/2) *2^(1/2)/(-2*d*e+b)^(1/2)/((-2*d*e+b)^(1/2)+(2*d*e+b)^(1/2))^(1/2)-1/2*arc tanh(x*2^(1/2)*e^(1/2)/((-2*d*e+b)^(1/2)+(2*d*e+b)^(1/2))^(1/2))*e^(1/2)*2 ^(1/2)/(-2*d*e+b)^(1/2)/((-2*d*e+b)^(1/2)+(2*d*e+b)^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.20 \[ \int \frac {d+e x^4}{d^2-b x^4+e^2 x^8} \, dx=\frac {1}{4} \text {RootSum}\left [d^2-b \text {$\#$1}^4+e^2 \text {$\#$1}^8\&,\frac {d \log (x-\text {$\#$1})+e \log (x-\text {$\#$1}) \text {$\#$1}^4}{-b \text {$\#$1}^3+2 e^2 \text {$\#$1}^7}\&\right ] \]
RootSum[d^2 - b*#1^4 + e^2*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/( -(b*#1^3) + 2*e^2*#1^7) & ]/4
Time = 0.50 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1749, 1406, 216, 220}
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.
\(\displaystyle \int \frac {d+e x^4}{-b x^4+d^2+e^2 x^8} \, dx\) |
\(\Big \downarrow \) 1749 |
\(\displaystyle \frac {\int \frac {1}{x^4-\frac {\sqrt {b+2 d e} x^2}{e}+\frac {d}{e}}dx}{2 e}+\frac {\int \frac {1}{x^4+\frac {\sqrt {b+2 d e} x^2}{e}+\frac {d}{e}}dx}{2 e}\) |
\(\Big \downarrow \) 1406 |
\(\displaystyle \frac {\frac {e \int \frac {1}{x^2-\frac {\sqrt {b-2 d e}+\sqrt {b+2 d e}}{2 e}}dx}{\sqrt {b-2 d e}}-\frac {e \int \frac {1}{x^2+\frac {\sqrt {b-2 d e}-\sqrt {b+2 d e}}{2 e}}dx}{\sqrt {b-2 d e}}}{2 e}+\frac {\frac {e \int \frac {1}{x^2-\frac {\sqrt {b-2 d e}-\sqrt {b+2 d e}}{2 e}}dx}{\sqrt {b-2 d e}}-\frac {e \int \frac {1}{x^2+\frac {\sqrt {b-2 d e}+\sqrt {b+2 d e}}{2 e}}dx}{\sqrt {b-2 d e}}}{2 e}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {e \int \frac {1}{x^2-\frac {\sqrt {b-2 d e}-\sqrt {b+2 d e}}{2 e}}dx}{\sqrt {b-2 d e}}-\frac {\sqrt {2} e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}\right )}{\sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}}{2 e}+\frac {\frac {e \int \frac {1}{x^2-\frac {\sqrt {b-2 d e}+\sqrt {b+2 d e}}{2 e}}dx}{\sqrt {b-2 d e}}-\frac {\sqrt {2} e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}\right )}{\sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}}{2 e}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {-\frac {\sqrt {2} e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}\right )}{\sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}-\frac {\sqrt {2} e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}\right )}{\sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}}{2 e}+\frac {-\frac {\sqrt {2} e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}\right )}{\sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}-\frac {\sqrt {2} e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}\right )}{\sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}}{2 e}\) |
(-((Sqrt[2]*e^(3/2)*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] + Sqrt [b + 2*d*e]]])/(Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]])) - (Sqrt[2]*e^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] - Sqrt [b + 2*d*e]]])/(Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]))/ (2*e) + (-((Sqrt[2]*e^(3/2)*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e ] - Sqrt[b + 2*d*e]]])/(Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2* d*e]])) - (Sqrt[2]*e^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e ] + Sqrt[b + 2*d*e]]])/(Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2* d*e]]))/(2*e)
3.1.7.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x _Symbol] :> With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x^(n/2) + x^n, x], x], x] + Simp[e/(2*c) Int[1/Simp[d/e - q*x^(n/2) + x^n, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d, e*Rt[a/c, 2]]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.16
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (e^{2} \textit {\_Z}^{8}-\textit {\_Z}^{4} b +d^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{4} e -d \right ) \ln \left (x -\textit {\_R} \right )}{-2 \textit {\_R}^{7} e^{2}+\textit {\_R}^{3} b}\right )}{4}\) | \(57\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (e^{2} \textit {\_Z}^{8}-\textit {\_Z}^{4} b +d^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{4} e -d \right ) \ln \left (x -\textit {\_R} \right )}{-2 \textit {\_R}^{7} e^{2}+\textit {\_R}^{3} b}\right )}{4}\) | \(57\) |
Leaf count of result is larger than twice the leaf count of optimal. 2453 vs. \(2 (261) = 522\).
Time = 0.33 (sec) , antiderivative size = 2453, normalized size of antiderivative = 7.03 \[ \int \frac {d+e x^4}{d^2-b x^4+e^2 x^8} \, dx=\text {Too large to display} \]
1/4*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4 *b*d^3*e + b^2*d^2)))*log(e*x + 1/2*(2*d*e + (4*d^4*e^2 - 4*b*d^3*e + b^2* d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))) - 1/4*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^ 3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))*log(e*x - 1/2*(2*d*e + (4 *d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^ 2 + 6*b^2*d^5*e - b^3*d^4)) - b)*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3 *e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))) + 1/4*sqrt(-sqrt(1/2)* sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12* b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2) ))*log(e*x + 1/2*(2*d*e + (4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)*sqrt(-sqrt(1/ 2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d ^2)))) - 1/4*sqrt(-sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sq...
Time = 19.11 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.39 \[ \int \frac {d+e x^4}{d^2-b x^4+e^2 x^8} \, dx=\operatorname {RootSum} {\left (t^{8} \cdot \left (65536 b^{4} d^{2} - 524288 b^{3} d^{3} e + 1572864 b^{2} d^{4} e^{2} - 2097152 b d^{5} e^{3} + 1048576 d^{6} e^{4}\right ) + t^{4} \left (- 256 b^{3} + 1024 b^{2} d e - 1024 b d^{2} e^{2}\right ) + e^{2}, \left ( t \mapsto t \log {\left (x + \frac {1024 t^{5} b^{2} d^{2} - 4096 t^{5} b d^{3} e + 4096 t^{5} d^{4} e^{2} - 4 t b + 4 t d e}{e} \right )} \right )\right )} \]
RootSum(_t**8*(65536*b**4*d**2 - 524288*b**3*d**3*e + 1572864*b**2*d**4*e* *2 - 2097152*b*d**5*e**3 + 1048576*d**6*e**4) + _t**4*(-256*b**3 + 1024*b* *2*d*e - 1024*b*d**2*e**2) + e**2, Lambda(_t, _t*log(x + (1024*_t**5*b**2* d**2 - 4096*_t**5*b*d**3*e + 4096*_t**5*d**4*e**2 - 4*_t*b + 4*_t*d*e)/e)) )
\[ \int \frac {d+e x^4}{d^2-b x^4+e^2 x^8} \, dx=\int { \frac {e x^{4} + d}{e^{2} x^{8} - b x^{4} + d^{2}} \,d x } \]
\[ \int \frac {d+e x^4}{d^2-b x^4+e^2 x^8} \, dx=\int { \frac {e x^{4} + d}{e^{2} x^{8} - b x^{4} + d^{2}} \,d x } \]
Time = 9.78 (sec) , antiderivative size = 10337, normalized size of antiderivative = 29.62 \[ \int \frac {d+e x^4}{d^2-b x^4+e^2 x^8} \, dx=\text {Too large to display} \]
2*atan(((x*(32*b*d^5*e^13 + 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4* e^12) - ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d* e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^ 2)))^(1/4)*((x*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 204 8*b^6*d^3*e^9 + 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^ 12 - 65536*b^2*d^7*e^13) - ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b *d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5* e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 + 262144*b*d^9*e^14 - 4096 *b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 - 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i)*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8 *b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i - 256*d^7*e^14 - 25 6*b*d^6*e^13 + 16*b^4*d^3*e^10 + 64*b^3*d^4*e^11)*1i)*((b^3 + ((b - 2*d*e) ^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^ 4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4) + (x*(32*b*d^5*e^ 13 + 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4*e^12) - ((b^3 + ((b - 2 *d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d ^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((x*(65536*d ^9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b ^5*d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e...