Optimal. Leaf size=75 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1162, 617, 204} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 617
Rule 1162
Rubi steps
\begin {align*} \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx &=\frac {\int \frac {1}{\frac {d}{e}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt {e}}+x^2} \, dx}{2 e}+\frac {\int \frac {1}{\frac {d}{e}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt {e}}+x^2} \, dx}{2 e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}\\ &=-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 60, normalized size = 0.80 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}+1\right )-\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.50, size = 137, normalized size = 1.83 \begin {gather*} \left [-\frac {\sqrt {2} \sqrt {-d e} \log \left (\frac {e^{2} x^{4} - 4 \, d e x^{2} - 2 \, \sqrt {2} {\left (e x^{3} - d x\right )} \sqrt {-d e} + d^{2}}{e^{2} x^{4} + d^{2}}\right )}{4 \, d e}, \frac {\sqrt {2} \sqrt {d e} \arctan \left (\frac {\sqrt {2} \sqrt {d e} x}{2 \, d}\right ) + \sqrt {2} \sqrt {d e} \arctan \left (\frac {\sqrt {2} {\left (e x^{3} + d x\right )} \sqrt {d e}}{2 \, d^{2}}\right )}{2 \, d e}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.17, size = 222, normalized size = 2.96 \begin {gather*} \frac {\sqrt {2} {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {11}{2}} + {\left (d^{2}\right )}^{\frac {3}{4}} e^{\frac {11}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} + 2 \, x\right )} e^{\frac {1}{2}}}{2 \, {\left (d^{2}\right )}^{\frac {1}{4}}}\right ) e^{\left (-6\right )}}{4 \, d^{2}} + \frac {\sqrt {2} {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {11}{2}} + {\left (d^{2}\right )}^{\frac {3}{4}} e^{\frac {11}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} - 2 \, x\right )} e^{\frac {1}{2}}}{2 \, {\left (d^{2}\right )}^{\frac {1}{4}}}\right ) e^{\left (-6\right )}}{4 \, d^{2}} + \frac {\sqrt {2} {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {11}{2}} - {\left (d^{2}\right )}^{\frac {3}{4}} e^{\frac {11}{2}}\right )} e^{\left (-6\right )} \log \left (\sqrt {2} {\left (d^{2}\right )}^{\frac {1}{4}} x e^{\left (-\frac {1}{2}\right )} + x^{2} + \sqrt {d^{2}} e^{\left (-1\right )}\right )}{8 \, d^{2}} - \frac {\sqrt {2} {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {11}{2}} - {\left (d^{2}\right )}^{\frac {3}{4}} e^{\frac {11}{2}}\right )} e^{\left (-6\right )} \log \left (-\sqrt {2} {\left (d^{2}\right )}^{\frac {1}{4}} x e^{\left (-\frac {1}{2}\right )} + x^{2} + \sqrt {d^{2}} e^{\left (-1\right )}\right )}{8 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 290, normalized size = 3.87 \begin {gather*} \frac {\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}-1\right )}{4 d}+\frac {\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}+1\right )}{4 d}+\frac {\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {d^{2}}{e^{2}}}}{x^{2}-\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {d^{2}}{e^{2}}}}\right )}{8 d}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} e}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} e}+\frac {\sqrt {2}\, \ln \left (\frac {x^{2}-\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {d^{2}}{e^{2}}}}{x^{2}+\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {d^{2}}{e^{2}}}}\right )}{8 \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.48, size = 74, normalized size = 0.99 \begin {gather*} \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, e x + \sqrt {2} \sqrt {d} \sqrt {e}\right )}}{2 \, \sqrt {d e}}\right )}{2 \, \sqrt {d e}} + \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, e x - \sqrt {2} \sqrt {d} \sqrt {e}\right )}}{2 \, \sqrt {d e}}\right )}{2 \, \sqrt {d e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.41, size = 57, normalized size = 0.76 \begin {gather*} \frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e}\,x}{2\,\sqrt {d}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,e^{3/2}\,x^3}{2\,d^{3/2}}+\frac {\sqrt {2}\,\sqrt {e}\,x}{2\,\sqrt {d}}\right )\right )}{4\,\sqrt {d}\,\sqrt {e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.22, size = 87, normalized size = 1.16 \begin {gather*} - \frac {\sqrt {2} \sqrt {- \frac {1}{d e}} \log {\left (- \sqrt {2} d x \sqrt {- \frac {1}{d e}} - \frac {d}{e} + x^{2} \right )}}{4} + \frac {\sqrt {2} \sqrt {- \frac {1}{d e}} \log {\left (\sqrt {2} d x \sqrt {- \frac {1}{d e}} - \frac {d}{e} + x^{2} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________