Optimal. Leaf size=119 \[ -\frac {3 (-e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{40 d^{5/2}}-\frac {3 (-e)^{3/2} \sqrt {d+e x^2}}{40 d^2 x^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}-\frac {\sqrt {-e} \sqrt {d+e x^2}}{20 d x^4} \]
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Rubi [A] time = 0.06, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5151, 266, 51, 63, 208} \[ -\frac {3 (-e)^{3/2} \sqrt {d+e x^2}}{40 d^2 x^2}-\frac {3 (-e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{40 d^{5/2}}-\frac {\sqrt {-e} \sqrt {d+e x^2}}{20 d x^4}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 5151
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^6} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}+\frac {1}{5} \sqrt {-e} \int \frac {1}{x^5 \sqrt {d+e x^2}} \, dx\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}+\frac {1}{10} \sqrt {-e} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{20 d x^4}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}+\frac {\left (3 (-e)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {d+e x}} \, dx,x,x^2\right )}{40 d}\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{20 d x^4}-\frac {3 (-e)^{3/2} \sqrt {d+e x^2}}{40 d^2 x^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}+\frac {\left (3 (-e)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{80 d^2}\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{20 d x^4}-\frac {3 (-e)^{3/2} \sqrt {d+e x^2}}{40 d^2 x^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}-\frac {\left (3 (-e)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{40 d^2}\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{20 d x^4}-\frac {3 (-e)^{3/2} \sqrt {d+e x^2}}{40 d^2 x^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}-\frac {3 (-e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{40 d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 114, normalized size = 0.96 \[ -\frac {3 e^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {-e}}{\sqrt {e} \sqrt {d+e x^2}}\right )}{40 d^{5/2}}+\sqrt {-e} \left (\frac {3 e}{40 d^2 x^2}-\frac {1}{20 d x^4}\right ) \sqrt {d+e x^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{5 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 228, normalized size = 1.92 \[ \left [\frac {3 \, e^{2} x^{5} \sqrt {-\frac {e}{d}} \log \left (-\frac {e^{2} x^{2} + 2 \, \sqrt {e x^{2} + d} d \sqrt {-e} \sqrt {-\frac {e}{d}} + 2 \, d e}{x^{2}}\right ) - 16 \, d^{2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + 2 \, {\left (3 \, e x^{3} - 2 \, d x\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{80 \, d^{2} x^{5}}, -\frac {3 \, e^{2} x^{5} \sqrt {\frac {e}{d}} \arctan \left (\frac {\sqrt {e x^{2} + d} d \sqrt {-e} \sqrt {\frac {e}{d}}}{e^{2} x^{2} + d e}\right ) + 8 \, d^{2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (3 \, e x^{3} - 2 \, d x\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{40 \, d^{2} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 108, normalized size = 0.91 \[ -\frac {1}{40} \, {\left (\frac {3 \, \arctan \left (\frac {\sqrt {-x^{2} e^{2} - d e} e^{\left (-\frac {1}{2}\right )}}{\sqrt {d}}\right ) e^{\frac {7}{2}}}{d^{\frac {5}{2}}} + \frac {{\left (5 \, \sqrt {-x^{2} e^{2} - d e} d e^{5} + 3 \, {\left (-x^{2} e^{2} - d e\right )}^{\frac {3}{2}} e^{4}\right )} e^{\left (-4\right )}}{d^{2} x^{4}}\right )} e^{\left (-1\right )} - \frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right )}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 150, normalized size = 1.26 \[ -\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{5 x^{5}}-\frac {\sqrt {-e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{20 d^{2} x^{4}}+\frac {\sqrt {-e}\, e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{40 d^{3} x^{2}}-\frac {3 \sqrt {-e}\, e^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )}{40 d^{\frac {5}{2}}}-\frac {\sqrt {-e}\, e^{2} \sqrt {e \,x^{2}+d}}{40 d^{3}}+\frac {\sqrt {-e}\, e \sqrt {e \,x^{2}+d}}{10 d^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.73, size = 148, normalized size = 1.24 \[ - \frac {\operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{5 x^{5}} - \frac {\sqrt {- e}}{20 \sqrt {e} x^{5} \sqrt {\frac {d}{e x^{2}} + 1}} + \frac {\sqrt {e} \sqrt {- e}}{40 d x^{3} \sqrt {\frac {d}{e x^{2}} + 1}} + \frac {3 e^{\frac {3}{2}} \sqrt {- e}}{40 d^{2} x \sqrt {\frac {d}{e x^{2}} + 1}} - \frac {3 e^{2} \sqrt {- e} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} x} \right )}}{40 d^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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