Optimal. Leaf size=105 \[ -\frac {4 e^{5/2} \sqrt {d+e x^2}}{45 d^3 x}+\frac {2 e^{3/2} \sqrt {d+e x^2}}{45 d^2 x^3}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}-\frac {\sqrt {e} \sqrt {d+e x^2}}{30 d x^5} \]
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Rubi [A] time = 0.04, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6221, 271, 264} \[ -\frac {4 e^{5/2} \sqrt {d+e x^2}}{45 d^3 x}+\frac {2 e^{3/2} \sqrt {d+e x^2}}{45 d^2 x^3}-\frac {\sqrt {e} \sqrt {d+e x^2}}{30 d x^5}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rule 6221
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}+\frac {1}{6} \sqrt {e} \int \frac {1}{x^6 \sqrt {d+e x^2}} \, dx\\ &=-\frac {\sqrt {e} \sqrt {d+e x^2}}{30 d x^5}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}-\frac {\left (2 e^{3/2}\right ) \int \frac {1}{x^4 \sqrt {d+e x^2}} \, dx}{15 d}\\ &=-\frac {\sqrt {e} \sqrt {d+e x^2}}{30 d x^5}+\frac {2 e^{3/2} \sqrt {d+e x^2}}{45 d^2 x^3}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}+\frac {\left (4 e^{5/2}\right ) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{45 d^2}\\ &=-\frac {\sqrt {e} \sqrt {d+e x^2}}{30 d x^5}+\frac {2 e^{3/2} \sqrt {d+e x^2}}{45 d^2 x^3}-\frac {4 e^{5/2} \sqrt {d+e x^2}}{45 d^3 x}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 74, normalized size = 0.70 \[ \frac {\sqrt {e} x \sqrt {d+e x^2} \left (-3 d^2+4 d e x^2-8 e^2 x^4\right )-15 d^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{90 d^3 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 78, normalized size = 0.74 \[ -\frac {15 \, d^{3} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right ) + 2 \, {\left (8 \, e^{2} x^{5} - 4 \, d e x^{3} + 3 \, d^{2} x\right )} \sqrt {e x^{2} + d} \sqrt {e}}{180 \, d^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 134, normalized size = 1.28 \[ \frac {8 \, {\left (10 \, {\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{4} d^{2} e^{2} - 5 \, {\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} d^{3} e^{2} + d^{4} e^{2}\right )} e}{45 \, {\left ({\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} - d\right )}^{5} d^{2}} - \frac {\log \left (-\frac {\frac {x e^{\frac {1}{2}}}{\sqrt {x^{2} e + d}} + 1}{\frac {x e^{\frac {1}{2}}}{\sqrt {x^{2} e + d}} - 1}\right )}{12 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 110, normalized size = 1.05 \[ -\frac {\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{6 x^{6}}-\frac {e^{\frac {3}{2}} \left (-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}\right )}{6 d}+\frac {\sqrt {e}\, \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 d \,x^{5}}+\frac {2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 d^{2} x^{3}}\right )}{6 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 102, normalized size = 0.97 \[ -\frac {{\left (2 \, e^{2} x^{4} + d e x^{2} - d^{2}\right )} e^{\frac {3}{2}}}{18 \, \sqrt {e x^{2} + d} d^{3} x^{3}} - \frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{6 \, x^{6}} + \frac {{\left (2 \, e^{2} x^{4} - d e x^{2} - 3 \, d^{2}\right )} \sqrt {e x^{2} + d} \sqrt {e}}{90 \, d^{3} x^{5}} \]
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 \[ \int \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^7} \,d x \]
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 \[ \int \frac {\operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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