Optimal. Leaf size=91 \[ -\frac {(-e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{6 d^{3/2}}-\frac {\sqrt {-e} \sqrt {d+e x^2}}{6 d x^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{3 x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ -\frac {(-e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{6 d^{3/2}}-\frac {\sqrt {-e} \sqrt {d+e x^2}}{6 d x^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{3 x^3} \]
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^4} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{3 x^3}+\frac {1}{3} \sqrt {-e} \int \frac {1}{x^3 \sqrt {d+e x^2}} \, dx\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{3 x^3}+\frac {1}{6} \sqrt {-e} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{6 d x^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{3 x^3}+\frac {(-e)^{3/2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{12 d}\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{6 d x^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{3 x^3}-\frac {\sqrt {-e} \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{6 d}\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{6 d x^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{3 x^3}-\frac {(-e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{6 d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 101, normalized size = 1.11 \[ \frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {-e}}{\sqrt {e} \sqrt {d+e x^2}}\right )}{6 d^{3/2}}-\frac {\sqrt {-e} \sqrt {d+e x^2}}{6 d x^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 198, normalized size = 2.18 \[ \left [\frac {e x^{3} \sqrt {-\frac {e}{d}} \log \left (-\frac {e^{2} x^{2} - 2 \, \sqrt {e x^{2} + d} d \sqrt {-e} \sqrt {-\frac {e}{d}} + 2 \, d e}{x^{2}}\right ) - 2 \, \sqrt {e x^{2} + d} \sqrt {-e} x - 4 \, d \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{12 \, d x^{3}}, \frac {e x^{3} \sqrt {\frac {e}{d}} \arctan \left (\frac {\sqrt {e x^{2} + d} d \sqrt {-e} \sqrt {\frac {e}{d}}}{e^{2} x^{2} + d e}\right ) - \sqrt {e x^{2} + d} \sqrt {-e} x - 2 \, d \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{6 \, d x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 83, normalized size = 0.91 \[ \frac {1}{6} \, {\left (\frac {\arctan \left (\frac {\sqrt {-x^{2} e^{2} - d e} e^{\left (-\frac {1}{2}\right )}}{\sqrt {d}}\right ) e^{\frac {5}{2}}}{d^{\frac {3}{2}}} - \frac {\sqrt {-x^{2} e^{2} - d e} e}{d x^{2}}\right )} e^{\left (-1\right )} - \frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 100, normalized size = 1.10 \[ -\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{3 x^{3}}+\frac {\sqrt {-e}\, e \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )}{6 d^{\frac {3}{2}}}-\frac {\sqrt {-e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{6 d^{2} x^{2}}+\frac {\sqrt {-e}\, e \sqrt {e \,x^{2}+d}}{6 d^{2}} \]
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 \[ \int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.58, size = 82, normalized size = 0.90 \[ - \frac {\operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{3 x^{3}} - \frac {\sqrt {e} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{6 d x} + \frac {e \sqrt {- e} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} x} \right )}}{6 d^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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