Integrand size = 20, antiderivative size = 752 \[ \int \frac {1}{x^2 \left (a-b \left (c+d x^2\right )^4\right )} \, dx=-\frac {1}{2 \sqrt {a} \left (\sqrt {a}-\sqrt {b} c^2\right ) x}-\frac {1}{2 \sqrt {a} \left (\sqrt {a}+\sqrt {b} c^2\right ) x}-\frac {\sqrt [8]{b} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}+\sqrt [4]{b} c}}\right )}{4 a^{3/4} \left (\sqrt [4]{a}+\sqrt [4]{b} c\right )^{3/2}}+\frac {\sqrt [8]{b} \left (2 \sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}\right ) \sqrt {d} \arctan \left (\frac {\sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}}\right )}{4 \sqrt {2} \sqrt {a} \left (\sqrt {a}+\sqrt {b} c^2\right )^{3/2} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}}-\frac {\sqrt [8]{b} \left (2 \sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}\right ) \sqrt {d} \arctan \left (\frac {\sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}+\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}}\right )}{4 \sqrt {2} \sqrt {a} \left (\sqrt {a}+\sqrt {b} c^2\right )^{3/2} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}}+\frac {\sqrt [8]{b} \sqrt {d} \text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{4 a^{3/4} \left (\sqrt [4]{a}-\sqrt [4]{b} c\right )^{3/2}}-\frac {\sqrt [8]{b} \left (2 \sqrt [4]{b} c-\sqrt {\sqrt {a}+\sqrt {b} c^2}\right ) \sqrt {d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} \sqrt {d} x}{\sqrt {\sqrt {a}+\sqrt {b} c^2}+\sqrt [4]{b} d x^2}\right )}{4 \sqrt {2} \sqrt {a} \left (\sqrt {a}+\sqrt {b} c^2\right )^{3/2} \sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}} \] Output:
-1/2/a^(1/2)/(a^(1/2)-b^(1/2)*c^2)/x-1/2/a^(1/2)/(a^(1/2)+b^(1/2)*c^2)/x-1 /4*b^(1/8)*d^(1/2)*arctan(b^(1/8)*d^(1/2)*x/(a^(1/4)+b^(1/4)*c)^(1/2))/a^( 3/4)/(a^(1/4)+b^(1/4)*c)^(3/2)+1/8*b^(1/8)*(2*b^(1/4)*c+(a^(1/2)+b^(1/2)*c ^2)^(1/2))*d^(1/2)*arctan(((-b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2)- 2^(1/2)*b^(1/8)*d^(1/2)*x)/(b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2))* 2^(1/2)/a^(1/2)/(a^(1/2)+b^(1/2)*c^2)^(3/2)/(b^(1/4)*c+(a^(1/2)+b^(1/2)*c^ 2)^(1/2))^(1/2)-1/8*b^(1/8)*(2*b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))*d^(1 /2)*arctan(((-b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2)+2^(1/2)*b^(1/8) *d^(1/2)*x)/(b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2))*2^(1/2)/a^(1/2) /(a^(1/2)+b^(1/2)*c^2)^(3/2)/(b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2) +1/4*b^(1/8)*d^(1/2)*arctanh(b^(1/8)*d^(1/2)*x/(a^(1/4)-b^(1/4)*c)^(1/2))/ a^(3/4)/(a^(1/4)-b^(1/4)*c)^(3/2)-1/8*b^(1/8)*(2*b^(1/4)*c-(a^(1/2)+b^(1/2 )*c^2)^(1/2))*d^(1/2)*arctanh(2^(1/2)*b^(1/8)*(-b^(1/4)*c+(a^(1/2)+b^(1/2) *c^2)^(1/2))^(1/2)*d^(1/2)*x/((a^(1/2)+b^(1/2)*c^2)^(1/2)+b^(1/4)*d*x^2))* 2^(1/2)/a^(1/2)/(a^(1/2)+b^(1/2)*c^2)^(3/2)/(-b^(1/4)*c+(a^(1/2)+b^(1/2)*c ^2)^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^2 \left (a-b \left (c+d x^2\right )^4\right )} \, dx=-\frac {8+x \text {RootSum}\left [a-b c^4-4 b c^3 d \text {$\#$1}^2-6 b c^2 d^2 \text {$\#$1}^4-4 b c d^3 \text {$\#$1}^6-b d^4 \text {$\#$1}^8\&,\frac {4 c^3 \log (x-\text {$\#$1})+6 c^2 d \log (x-\text {$\#$1}) \text {$\#$1}^2+4 c d^2 \log (x-\text {$\#$1}) \text {$\#$1}^4+d^3 \log (x-\text {$\#$1}) \text {$\#$1}^6}{c^3 \text {$\#$1}+3 c^2 d \text {$\#$1}^3+3 c d^2 \text {$\#$1}^5+d^3 \text {$\#$1}^7}\&\right ]}{8 a x-8 b c^4 x} \] Input:
Integrate[1/(x^2*(a - b*(c + d*x^2)^4)),x]
Output:
-((8 + x*RootSum[a - b*c^4 - 4*b*c^3*d*#1^2 - 6*b*c^2*d^2*#1^4 - 4*b*c*d^3 *#1^6 - b*d^4*#1^8 & , (4*c^3*Log[x - #1] + 6*c^2*d*Log[x - #1]*#1^2 + 4*c *d^2*Log[x - #1]*#1^4 + d^3*Log[x - #1]*#1^6)/(c^3*#1 + 3*c^2*d*#1^3 + 3*c *d^2*#1^5 + d^3*#1^7) & ])/(8*a*x - 8*b*c^4*x))
Leaf count is larger than twice the leaf count of optimal. \(3288\) vs. \(2(752)=1504\).
Time = 10.62 (sec) , antiderivative size = 3288, normalized size of antiderivative = 4.37, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {7293, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a-b \left (c+d x^2\right )^4\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {1}{x^2 \left (a-b c^4\right )}+\frac {b d \left (4 c^3+6 c^2 d x^2+4 c d^2 x^4+d^3 x^6\right )}{\left (a-b c^4\right ) \left (a \left (1-\frac {b c^4}{a}\right )-4 b c^3 d x^2-6 b c^2 d^2 x^4-4 b c d^3 x^6-b d^4 x^8\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^{7/8} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right ) c^3}{4 a^{3/4} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} \left (a-b c^4\right )}-\frac {b^{7/8} \sqrt {d} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right ) c^3}{4 \sqrt {2} \sqrt {a} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right ) \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}+\frac {b^{7/8} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right ) c^3}{4 \sqrt {2} \sqrt {a} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right ) \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}+\frac {b^{7/8} \sqrt {d} \text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right ) c^3}{4 a^{3/4} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} \left (a-b c^4\right )}-\frac {b^{7/8} \sqrt {d} \log \left (\sqrt [4]{b} d x^2-\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right ) c^3}{8 \sqrt {2} \sqrt {a} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right ) \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}+\frac {b^{7/8} \sqrt {d} \log \left (\sqrt [4]{b} d x^2+\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right ) c^3}{8 \sqrt {2} \sqrt {a} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right ) \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}-\frac {b^{5/8} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right ) c^2}{4 \sqrt {a} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} \left (a-b c^4\right )}-\frac {b^{5/8} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}} \sqrt {d} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right ) c^2}{4 \sqrt {2} \sqrt {a} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right )}+\frac {b^{5/8} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right ) c^2}{4 \sqrt {2} \sqrt {a} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right )}+\frac {b^{5/8} \sqrt {d} \text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right ) c^2}{4 \sqrt {a} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} \left (a-b c^4\right )}+\frac {b^{5/8} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} \log \left (\sqrt [4]{b} d x^2-\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right ) c^2}{8 \sqrt {2} \sqrt {a} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right )}-\frac {b^{5/8} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} \log \left (\sqrt [4]{b} d x^2+\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right ) c^2}{8 \sqrt {2} \sqrt {a} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right )}+\frac {b^{3/8} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right ) c}{4 \sqrt [4]{a} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} \left (a-b c^4\right )}+\frac {b^{3/8} \sqrt {d} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right ) c}{4 \sqrt {2} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right ) \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}-\frac {b^{3/8} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right ) c}{4 \sqrt {2} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right ) \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}+\frac {b^{3/8} \sqrt {d} \text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right ) c}{4 \sqrt [4]{a} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} \left (a-b c^4\right )}+\frac {b^{3/8} \sqrt {d} \log \left (\sqrt [4]{b} d x^2-\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right ) c}{8 \sqrt {2} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right ) \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}-\frac {b^{3/8} \sqrt {d} \log \left (\sqrt [4]{b} d x^2+\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right ) c}{8 \sqrt {2} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right ) \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}-\frac {\sqrt [8]{b} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right )}{4 \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} \left (a-b c^4\right )}+\frac {\sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}} \sqrt {d} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right )}{4 \sqrt {2} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right )}-\frac {\sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right )}{4 \sqrt {2} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right )}+\frac {\sqrt [8]{b} \sqrt {d} \text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{4 \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} \left (a-b c^4\right )}-\frac {\sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} \log \left (\sqrt [4]{b} d x^2-\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right )}{8 \sqrt {2} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right )}+\frac {\sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} \log \left (\sqrt [4]{b} d x^2+\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right )}{8 \sqrt {2} \sqrt {\sqrt {b} c^2+\sqrt {a}} \left (a-b c^4\right )}-\frac {1}{\left (a-b c^4\right ) x}\) |
Input:
Int[1/(x^2*(a - b*(c + d*x^2)^4)),x]
Output:
-(1/((a - b*c^4)*x)) - (b^(1/8)*Sqrt[d]*ArcTan[(b^(1/8)*Sqrt[d]*x)/Sqrt[a^ (1/4) + b^(1/4)*c]])/(4*Sqrt[a^(1/4) + b^(1/4)*c]*(a - b*c^4)) + (b^(3/8)* c*Sqrt[d]*ArcTan[(b^(1/8)*Sqrt[d]*x)/Sqrt[a^(1/4) + b^(1/4)*c]])/(4*a^(1/4 )*Sqrt[a^(1/4) + b^(1/4)*c]*(a - b*c^4)) - (b^(5/8)*c^2*Sqrt[d]*ArcTan[(b^ (1/8)*Sqrt[d]*x)/Sqrt[a^(1/4) + b^(1/4)*c]])/(4*Sqrt[a]*Sqrt[a^(1/4) + b^( 1/4)*c]*(a - b*c^4)) + (b^(7/8)*c^3*Sqrt[d]*ArcTan[(b^(1/8)*Sqrt[d]*x)/Sqr t[a^(1/4) + b^(1/4)*c]])/(4*a^(3/4)*Sqrt[a^(1/4) + b^(1/4)*c]*(a - b*c^4)) + (b^(3/8)*c*Sqrt[d]*ArcTan[(Sqrt[-(b^(1/4)*c) + Sqrt[Sqrt[a] + Sqrt[b]*c ^2]] - Sqrt[2]*b^(1/8)*Sqrt[d]*x)/Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]* c^2]]])/(4*Sqrt[2]*Sqrt[Sqrt[a] + Sqrt[b]*c^2]*(a - b*c^4)*Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]) - (b^(7/8)*c^3*Sqrt[d]*ArcTan[(Sqrt[-(b^(1 /4)*c) + Sqrt[Sqrt[a] + Sqrt[b]*c^2]] - Sqrt[2]*b^(1/8)*Sqrt[d]*x)/Sqrt[b^ (1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]])/(4*Sqrt[2]*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]*c^2]*(a - b*c^4)*Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]) + (b^(1/8)*Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]*Sqrt[d]*ArcTan[(Sq rt[-(b^(1/4)*c) + Sqrt[Sqrt[a] + Sqrt[b]*c^2]] - Sqrt[2]*b^(1/8)*Sqrt[d]*x )/Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]])/(4*Sqrt[2]*Sqrt[Sqrt[a] + Sqrt[b]*c^2]*(a - b*c^4)) - (b^(5/8)*c^2*Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]*Sqrt[d]*ArcTan[(Sqrt[-(b^(1/4)*c) + Sqrt[Sqrt[a] + Sqrt[b]* c^2]] - Sqrt[2]*b^(1/8)*Sqrt[d]*x)/Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt...
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.21
| method | result | size |
| default | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{8}+4 b c \,d^{3} \textit {\_Z}^{6}+6 b \,c^{2} d^{2} \textit {\_Z}^{4}+4 b \,c^{3} d \,\textit {\_Z}^{2}+b \,c^{4}-a \right )}{\sum }\frac {\left (-\textit {\_R}^{6} d^{3}-4 \textit {\_R}^{4} c \,d^{2}-6 \textit {\_R}^{2} c^{2} d -4 c^{3}\right ) \ln \left (x -\textit {\_R} \right )}{-d^{3} \textit {\_R}^{7}-3 c \,d^{2} \textit {\_R}^{5}-3 c^{2} d \,\textit {\_R}^{3}-c^{3} \textit {\_R}}}{8 \left (-b \,c^{4}+a \right )}-\frac {1}{\left (-b \,c^{4}+a \right ) x}\) | \(156\) |
| risch | \(-\frac {1}{\left (-b \,c^{4}+a \right ) x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (-a^{6} b^{3} c^{12}+3 a^{7} b^{2} c^{8}-3 a^{8} b \,c^{4}+a^{9}\right ) \textit {\_Z}^{8}+\left (-24 a^{5} b^{2} c^{7} d -40 a^{6} b \,c^{3} d \right ) \textit {\_Z}^{6}+\left (2 a^{3} b^{2} c^{6} d^{2}-42 a^{4} b \,c^{2} d^{2}\right ) \textit {\_Z}^{4}-12 a^{2} b c \,d^{3} \textit {\_Z}^{2}-b \,d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-3 a^{5} b^{4} c^{16}+18 a^{7} b^{2} c^{8}-24 a^{8} b \,c^{4}+9 a^{9}\right ) \textit {\_R}^{8}+\left (a^{3} b^{4} c^{15} d -7 a^{4} b^{3} c^{11} d -181 a^{5} b^{2} c^{7} d -325 a^{6} b \,c^{3} d \right ) \textit {\_R}^{6}+\left (-6 a^{2} b^{3} c^{10} d^{2}+28 a^{3} b^{2} c^{6} d^{2}-342 a^{4} b \,c^{2} d^{2}\right ) \textit {\_R}^{4}+\left (-b^{3} c^{9} d^{3}+2 a \,b^{2} c^{5} d^{3}-97 a^{2} b c \,d^{3}\right ) \textit {\_R}^{2}-8 b \,d^{4}\right ) x +\left (3 a^{4} b^{4} c^{16}+6 a^{5} b^{3} c^{12}-20 a^{6} b^{2} c^{8}+10 a^{7} b \,c^{4}+a^{8}\right ) \textit {\_R}^{7}+\left (16 a^{3} b^{3} c^{11} d -32 a^{4} b^{2} c^{7} d +16 a^{5} b \,c^{3} d \right ) \textit {\_R}^{5}+\left (3 a \,b^{3} c^{10} d^{2}-6 a^{2} b^{2} c^{6} d^{2}+3 a^{3} b \,c^{2} d^{2}\right ) \textit {\_R}^{3}\right )\right )}{8}\) | \(459\) |
Input:
int(1/x^2/(a-b*(d*x^2+c)^4),x,method=_RETURNVERBOSE)
Output:
-1/8/(-b*c^4+a)*sum((-_R^6*d^3-4*_R^4*c*d^2-6*_R^2*c^2*d-4*c^3)/(-_R^7*d^3 -3*_R^5*c*d^2-3*_R^3*c^2*d-_R*c^3)*ln(x-_R),_R=RootOf(_Z^8*b*d^4+4*_Z^6*b* c*d^3+6*_Z^4*b*c^2*d^2+4*_Z^2*b*c^3*d+b*c^4-a))-1/(-b*c^4+a)/x
Timed out. \[ \int \frac {1}{x^2 \left (a-b \left (c+d x^2\right )^4\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(a-b*(d*x^2+c)^4),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^2 \left (a-b \left (c+d x^2\right )^4\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x**2/(a-b*(d*x**2+c)**4),x)
Output:
Timed out
\[ \int \frac {1}{x^2 \left (a-b \left (c+d x^2\right )^4\right )} \, dx=\int { -\frac {1}{{\left ({\left (d x^{2} + c\right )}^{4} b - a\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a-b*(d*x^2+c)^4),x, algorithm="maxima")
Output:
b*d*integrate((d^3*x^6 + 4*c*d^2*x^4 + 6*c^2*d*x^2 + 4*c^3)/(b*d^4*x^8 + 4 *b*c*d^3*x^6 + 6*b*c^2*d^2*x^4 + 4*b*c^3*d*x^2 + b*c^4 - a), x)/(b*c^4 - a ) + 1/((b*c^4 - a)*x)
\[ \int \frac {1}{x^2 \left (a-b \left (c+d x^2\right )^4\right )} \, dx=\int { -\frac {1}{{\left ({\left (d x^{2} + c\right )}^{4} b - a\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a-b*(d*x^2+c)^4),x, algorithm="giac")
Output:
sage0*x
Time = 10.47 (sec) , antiderivative size = 2353, normalized size of antiderivative = 3.13 \[ \int \frac {1}{x^2 \left (a-b \left (c+d x^2\right )^4\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x^2*(a - b*(c + d*x^2)^4)),x)
Output:
symsum(log(-root(50331648*a^7*b^2*c^8*z^8 - 16777216*a^6*b^3*c^12*z^8 - 50 331648*a^8*b*c^4*z^8 + 16777216*a^9*z^8 - 6291456*a^5*b^2*c^7*d*z^6 - 1048 5760*a^6*b*c^3*d*z^6 - 172032*a^4*b*c^2*d^2*z^4 + 8192*a^3*b^2*c^6*d^2*z^4 - 768*a^2*b*c*d^3*z^2 - b*d^4, z, k)*(root(50331648*a^7*b^2*c^8*z^8 - 167 77216*a^6*b^3*c^12*z^8 - 50331648*a^8*b*c^4*z^8 + 16777216*a^9*z^8 - 62914 56*a^5*b^2*c^7*d*z^6 - 10485760*a^6*b*c^3*d*z^6 - 172032*a^4*b*c^2*d^2*z^4 + 8192*a^3*b^2*c^6*d^2*z^4 - 768*a^2*b*c*d^3*z^2 - b*d^4, z, k)*(root(503 31648*a^7*b^2*c^8*z^8 - 16777216*a^6*b^3*c^12*z^8 - 50331648*a^8*b*c^4*z^8 + 16777216*a^9*z^8 - 6291456*a^5*b^2*c^7*d*z^6 - 10485760*a^6*b*c^3*d*z^6 - 172032*a^4*b*c^2*d^2*z^4 + 8192*a^3*b^2*c^6*d^2*z^4 - 768*a^2*b*c*d^3*z ^2 - b*d^4, z, k)*(x*(1024*a^13*b^9*c^3*d^33 - 11264*a^12*b^10*c^7*d^33 + 56320*a^11*b^11*c^11*d^33 - 168960*a^10*b^12*c^15*d^33 + 337920*a^9*b^13*c ^19*d^33 - 473088*a^8*b^14*c^23*d^33 + 473088*a^7*b^15*c^27*d^33 - 337920* a^6*b^16*c^31*d^33 + 168960*a^5*b^17*c^35*d^33 - 56320*a^4*b^18*c^39*d^33 + 11264*a^3*b^19*c^43*d^33 - 1024*a^2*b^20*c^47*d^33) + root(50331648*a^7* b^2*c^8*z^8 - 16777216*a^6*b^3*c^12*z^8 - 50331648*a^8*b*c^4*z^8 + 1677721 6*a^9*z^8 - 6291456*a^5*b^2*c^7*d*z^6 - 10485760*a^6*b*c^3*d*z^6 - 172032* a^4*b*c^2*d^2*z^4 + 8192*a^3*b^2*c^6*d^2*z^4 - 768*a^2*b*c*d^3*z^2 - b*d^4 , z, k)*(root(50331648*a^7*b^2*c^8*z^8 - 16777216*a^6*b^3*c^12*z^8 - 50331 648*a^8*b*c^4*z^8 + 16777216*a^9*z^8 - 6291456*a^5*b^2*c^7*d*z^6 - 1048...
\[ \int \frac {1}{x^2 \left (a-b \left (c+d x^2\right )^4\right )} \, dx=\int \frac {1}{x^{2} \left (a -b \left (d \,x^{2}+c \right )^{4}\right )}d x \] Input:
int(1/x^2/(a-b*(d*x^2+c)^4),x)
Output:
int(1/x^2/(a-b*(d*x^2+c)^4),x)