Optimal. Leaf size=85 \[ \frac {e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{6 d^{3/2}}-\frac {\sqrt {e} \sqrt {d+e x^2}}{6 d x^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6221, 266, 51, 63, 208} \[ \frac {e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{6 d^{3/2}}-\frac {\sqrt {e} \sqrt {d+e x^2}}{6 d x^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 6221
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^4} \, dx &=-\frac {\tanh ^{-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 {\tanh ^{-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 {\tanh ^{-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 {\tanh ^{-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 {\tanh ^{-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 = 92, normalized size = 1.08 \[ -\frac {\frac {\sqrt {e} x \left (\sqrt {d} \sqrt {d+e x^2}-e x^2 \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )+e x^2 \log (x)\right )}{d^{3/2}}+2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 340, normalized size = 4.00 \[ \left [\frac {e x^{3} \sqrt {\frac {e}{d}} \log \left (-\frac {e^{2} x^{2} + 2 \, \sqrt {e x^{2} + d} d \sqrt {e} \sqrt {\frac {e}{d}} + 2 \, d e}{x^{2}}\right ) - 2 \, d x^{3} \log \left (\frac {e x + \sqrt {e x^{2} + d} \sqrt {e}}{x}\right ) + 2 \, d x^{3} \log \left (\frac {e x - \sqrt {e x^{2} + d} \sqrt {e}}{x}\right ) - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + 2 \, {\left (d x^{3} - d\right )} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right )}{12 \, d x^{3}}, -\frac {e x^{3} \sqrt {-\frac {e}{d}} \arctan \left (\frac {\sqrt {e x^{2} + d} d \sqrt {e} \sqrt {-\frac {e}{d}}}{e^{2} x^{2} + d e}\right ) + d x^{3} \log \left (\frac {e x + \sqrt {e x^{2} + d} \sqrt {e}}{x}\right ) - d x^{3} \log \left (\frac {e x - \sqrt {e x^{2} + d} \sqrt {e}}{x}\right ) + \sqrt {e x^{2} + d} \sqrt {e} x - {\left (d x^{3} - d\right )} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right )}{6 \, d x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [A] time = 0.03, size = 90, normalized size = 1.06 \[ -\frac {\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{3 x^{3}}+\frac {e^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )}{6 d^{\frac {3}{2}}}-\frac {\sqrt {e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{6 d^{2} x^{2}}+\frac {e^{\frac {3}{2}} \sqrt {e \,x^{2}+d}}{6 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ d \sqrt {e} \int -\frac {\sqrt {e x^{2} + d}}{3 \, {\left (e^{2} x^{7} + d e x^{5} - {\left (e x^{5} + d x^{3}\right )} {\left (e x^{2} + d\right )}\right )}}\,{d x} - \frac {\log \left (\sqrt {e} x + \sqrt {e x^{2} + d}\right ) - \log \left (-\sqrt {e} x + \sqrt {e x^{2} + d}\right )}{6 \, x^{3}} \]
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 \[ \int \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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