Optimal. Leaf size=72 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \sqrt {e}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2} \sqrt {e}}+\frac {x}{4 d^2 \left (d+e x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1150, 414, 522, 208, 205} \[ \frac {x}{4 d^2 \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \sqrt {e}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2} \sqrt {e}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 208
Rule 414
Rule 522
Rule 1150
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx &=\int \frac {1}{\left (d-e x^2\right ) \left (d+e x^2\right )^2} \, dx\\ &=\frac {x}{4 d^2 \left (d+e x^2\right )}-\frac {\int \frac {-3 d e+e^2 x^2}{\left (d-e x^2\right ) \left (d+e x^2\right )} \, dx}{4 d^2 e}\\ &=\frac {x}{4 d^2 \left (d+e x^2\right )}+\frac {\int \frac {1}{d-e x^2} \, dx}{4 d^2}+\frac {\int \frac {1}{d+e x^2} \, dx}{2 d^2}\\ &=\frac {x}{4 d^2 \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \sqrt {e}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2} \sqrt {e}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 65, normalized size = 0.90 \[ \frac {\frac {\sqrt {d} x}{d+e x^2}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}}{4 d^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.69, size = 189, normalized size = 2.62 \[ \left [\frac {2 \, d e x + 4 \, {\left (e x^{2} + d\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + {\left (e x^{2} + d\right )} \sqrt {d e} \log \left (\frac {e x^{2} + 2 \, \sqrt {d e} x + d}{e x^{2} - d}\right )}{8 \, {\left (d^{3} e^{2} x^{2} + d^{4} e\right )}}, \frac {d e x - {\left (e x^{2} + d\right )} \sqrt {-d e} \arctan \left (\frac {\sqrt {-d e} x}{d}\right ) - {\left (e x^{2} + d\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right )}{4 \, {\left (d^{3} e^{2} x^{2} + d^{4} e\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 55, normalized size = 0.76 \[ \frac {x}{4 \left (e \,x^{2}+d \right ) d^{2}}+\frac {\arctanh \left (\frac {e x}{\sqrt {d e}}\right )}{4 \sqrt {d e}\, d^{2}}+\frac {\arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}\, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.44, size = 71, normalized size = 0.99 \[ \frac {x}{4 \, {\left (d^{2} e x^{2} + d^{3}\right )}} + \frac {\arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d^{2}} - \frac {\log \left (\frac {e x - \sqrt {d e}}{e x + \sqrt {d e}}\right )}{8 \, \sqrt {d e} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.16, size = 74, normalized size = 1.03 \[ \frac {x}{4\,d^2\,\left (e\,x^2+d\right )}+\frac {\mathrm {atanh}\left (\frac {x\,\sqrt {d^5\,e}}{d^3}\right )\,\sqrt {d^5\,e}}{4\,d^5\,e}-\frac {\mathrm {atanh}\left (\frac {x\,\sqrt {-d^5\,e}}{d^3}\right )\,\sqrt {-d^5\,e}}{2\,d^5\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.45, size = 226, normalized size = 3.14 \[ \frac {x}{4 d^{3} + 4 d^{2} e x^{2}} - \frac {\sqrt {\frac {1}{d^{5} e}} \log {\left (- \frac {d^{8} e \left (\frac {1}{d^{5} e}\right )^{\frac {3}{2}}}{10} - \frac {9 d^{3} \sqrt {\frac {1}{d^{5} e}}}{10} + x \right )}}{8} + \frac {\sqrt {\frac {1}{d^{5} e}} \log {\left (\frac {d^{8} e \left (\frac {1}{d^{5} e}\right )^{\frac {3}{2}}}{10} + \frac {9 d^{3} \sqrt {\frac {1}{d^{5} e}}}{10} + x \right )}}{8} - \frac {\sqrt {- \frac {1}{d^{5} e}} \log {\left (- \frac {4 d^{8} e \left (- \frac {1}{d^{5} e}\right )^{\frac {3}{2}}}{5} - \frac {9 d^{3} \sqrt {- \frac {1}{d^{5} e}}}{5} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{d^{5} e}} \log {\left (\frac {4 d^{8} e \left (- \frac {1}{d^{5} e}\right )^{\frac {3}{2}}}{5} + \frac {9 d^{3} \sqrt {- \frac {1}{d^{5} e}}}{5} + x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________