Optimal. Leaf size=105 \[ -\frac {\sqrt {e} \sqrt {d+e x^2}}{30 d x^5}+\frac {2 e^{3/2} \sqrt {d+e x^2}}{45 d^2 x^3}-\frac {4 e^{5/2} \sqrt {d+e x^2}}{45 d^3 x}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^6} \]
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Rubi [A]
time = 0.03, 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 = {6356, 277, 270}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rule 6356
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 = 0.03, size = 74, normalized size = 0.70 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 110, normalized size = 1.05
method | result | size |
default | \(-\frac {\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{6 x^{6}}-\frac {e^{\frac {3}{2}} \left (-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}\right )}{6 d}+\frac {\sqrt {e}\, \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 d \,x^{5}}+\frac {2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 d^{2} x^{3}}\right )}{6 d}\) | \(110\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 102, normalized size = 0.97 \begin {gather*} -\frac {{\left (2 \, x^{4} e^{2} + d x^{2} e - d^{2}\right )} e^{\frac {3}{2}}}{18 \, \sqrt {x^{2} e + d} d^{3} x^{3}} - \frac {\operatorname {artanh}\left (\frac {x e^{\frac {1}{2}}}{\sqrt {x^{2} e + d}}\right )}{6 \, x^{6}} + \frac {{\left (2 \, x^{4} e^{2} - d x^{2} e - 3 \, d^{2}\right )} \sqrt {x^{2} e + d} e^{\frac {1}{2}}}{90 \, d^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 246 vs.
\(2 (81) = 162\).
time = 0.36, size = 246, normalized size = 2.34 \begin {gather*} -\frac {15 \, d^{3} \log \left (\frac {2 \, x^{2} \cosh \left (\frac {1}{2}\right )^{2} + 4 \, x^{2} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + 2 \, x^{2} \sinh \left (\frac {1}{2}\right )^{2} + 2 \, {\left (x \cosh \left (\frac {1}{2}\right ) + x \sinh \left (\frac {1}{2}\right )\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}} + d}{d}\right ) + 2 \, {\left (8 \, x^{5} \cosh \left (\frac {1}{2}\right )^{5} + 40 \, x^{5} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{4} + 8 \, x^{5} \sinh \left (\frac {1}{2}\right )^{5} - 4 \, d x^{3} \cosh \left (\frac {1}{2}\right )^{3} + 3 \, d^{2} x \cosh \left (\frac {1}{2}\right ) + 4 \, {\left (20 \, x^{5} \cosh \left (\frac {1}{2}\right )^{2} - d x^{3}\right )} \sinh \left (\frac {1}{2}\right )^{3} + 4 \, {\left (20 \, x^{5} \cosh \left (\frac {1}{2}\right )^{3} - 3 \, d x^{3} \cosh \left (\frac {1}{2}\right )\right )} \sinh \left (\frac {1}{2}\right )^{2} + {\left (40 \, x^{5} \cosh \left (\frac {1}{2}\right )^{4} - 12 \, d x^{3} \cosh \left (\frac {1}{2}\right )^{2} + 3 \, d^{2} x\right )} \sinh \left (\frac {1}{2}\right )\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}}}{180 \, d^{3} x^{6}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{x^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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