Optimal. Leaf size=122 \[ \frac {x}{32 c^4 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^5}{16 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^9}{12 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {\tanh ^{-1}\left (\sqrt {1+\frac {1}{c^4 x^4}}\right )}{32 c^{12} \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \]
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Rubi [A]
time = 0.05, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {5670, 5668,
272, 43, 44, 65, 213} \begin {gather*} \frac {x}{32 c^4 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^5}{16 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {\tanh ^{-1}\left (\sqrt {\frac {1}{c^4 x^4}+1}\right )}{32 c^{12} x^3 \left (\frac {1}{c^4 x^4}+1\right )^{3/2} \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^9}{12 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 213
Rule 272
Rule 5668
Rule 5670
Rubi steps
\begin {align*} \int \frac {x^8}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx &=\frac {\text {Subst}\left (\int \frac {x^8}{\text {sech}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^9}\\ &=\frac {\text {Subst}\left (\int \left (1+\frac {1}{x^4}\right )^{3/2} x^{11} \, dx,x,c x\right )}{c^{12} \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=-\frac {\text {Subst}\left (\int \frac {(1+x)^{3/2}}{x^4} \, dx,x,\frac {1}{c^4 x^4}\right )}{4 c^{12} \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {x^9}{12 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {\text {Subst}\left (\int \frac {\sqrt {1+x}}{x^3} \, dx,x,\frac {1}{c^4 x^4}\right )}{8 c^{12} \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {x^5}{16 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^9}{12 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {\text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,\frac {1}{c^4 x^4}\right )}{32 c^{12} \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {x}{32 c^4 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^5}{16 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^9}{12 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\frac {1}{c^4 x^4}\right )}{64 c^{12} \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {x}{32 c^4 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^5}{16 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^9}{12 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\frac {1}{c^4 x^4}}\right )}{32 c^{12} \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {x}{32 c^4 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^5}{16 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^9}{12 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {\tanh ^{-1}\left (\sqrt {1+\frac {1}{c^4 x^4}}\right )}{32 c^{12} \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 98, normalized size = 0.80 \begin {gather*} \frac {c^3 x^3 \sqrt {1+c^4 x^4} \left (3+14 c^4 x^4+8 c^8 x^8\right )-3 c x \sinh ^{-1}\left (c^2 x^2\right )}{192 \sqrt {2} c^9 \sqrt {\frac {c^2 x^2}{1+c^4 x^4}} \sqrt {1+c^4 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.50, size = 121, normalized size = 0.99
method | result | size |
risch | \(\frac {x^{3} \left (8 c^{8} x^{8}+14 c^{4} x^{4}+3\right ) \sqrt {2}}{384 c^{6} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}-\frac {\ln \left (\frac {c^{4} x^{2}}{\sqrt {c^{4}}}+\sqrt {c^{4} x^{4}+1}\right ) \sqrt {2}\, x}{128 c^{6} \sqrt {c^{4}}\, \sqrt {c^{4} x^{4}+1}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}\) | \(121\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 109, normalized size = 0.89 \begin {gather*} \frac {2 \, \sqrt {2} {\left (8 \, c^{13} x^{13} + 22 \, c^{9} x^{9} + 17 \, c^{5} x^{5} + 3 \, c x\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}} + 3 \, \sqrt {2} \log \left (-2 \, c^{4} x^{4} + 2 \, {\left (c^{5} x^{5} + c x\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}} - 1\right )}{768 \, c^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {x^8}{{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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