Optimal. Leaf size=109 \[ \frac {b \sqrt {1-c x}}{25 x^5 \sqrt {\frac {1}{1+c x}}}+\frac {4 b c^2 \sqrt {1-c x}}{75 x^3 \sqrt {\frac {1}{1+c x}}}+\frac {8 b c^4 \sqrt {1-c x}}{75 x \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5} \]
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Rubi [A]
time = 0.03, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6418, 105, 12,
97} \begin {gather*} -\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}+\frac {8 b c^4 \sqrt {1-c x}}{75 x \sqrt {\frac {1}{c x+1}}}+\frac {4 b c^2 \sqrt {1-c x}}{75 x^3 \sqrt {\frac {1}{c x+1}}}+\frac {b \sqrt {1-c x}}{25 x^5 \sqrt {\frac {1}{c x+1}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 97
Rule 105
Rule 6418
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{x^6} \, dx &=-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}-\frac {1}{5} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^6 \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{25 x^5 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}+\frac {1}{25} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int -\frac {4 c^2}{x^4 \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{25 x^5 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}-\frac {1}{25} \left (4 b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^4 \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{25 x^5 \sqrt {\frac {1}{1+c x}}}+\frac {4 b c^2 \sqrt {1-c x}}{75 x^3 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}+\frac {1}{75} \left (4 b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int -\frac {2 c^2}{x^2 \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{25 x^5 \sqrt {\frac {1}{1+c x}}}+\frac {4 b c^2 \sqrt {1-c x}}{75 x^3 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}-\frac {1}{75} \left (8 b c^4 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{25 x^5 \sqrt {\frac {1}{1+c x}}}+\frac {4 b c^2 \sqrt {1-c x}}{75 x^3 \sqrt {\frac {1}{1+c x}}}+\frac {8 b c^4 \sqrt {1-c x}}{75 x \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 94, normalized size = 0.86 \begin {gather*} -\frac {a}{5 x^5}+b \left (\frac {8 c^5}{75}+\frac {1}{25 x^5}+\frac {c}{25 x^4}+\frac {4 c^2}{75 x^3}+\frac {4 c^3}{75 x^2}+\frac {8 c^4}{75 x}\right ) \sqrt {\frac {1-c x}{1+c x}}-\frac {b \text {sech}^{-1}(c x)}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 85, normalized size = 0.78
method | result | size |
derivativedivides | \(c^{5} \left (-\frac {a}{5 c^{5} x^{5}}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{5 c^{5} x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (8 c^{4} x^{4}+4 c^{2} x^{2}+3\right )}{75 c^{4} x^{4}}\right )\right )\) | \(85\) |
default | \(c^{5} \left (-\frac {a}{5 c^{5} x^{5}}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{5 c^{5} x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (8 c^{4} x^{4}+4 c^{2} x^{2}+3\right )}{75 c^{4} x^{4}}\right )\right )\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 73, normalized size = 0.67 \begin {gather*} \frac {1}{75} \, b {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {15 \, \operatorname {arsech}\left (c x\right )}{x^{5}}\right )} - \frac {a}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 89, normalized size = 0.82 \begin {gather*} -\frac {15 \, b \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (8 \, b c^{5} x^{5} + 4 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 15 \, a}{75 \, x^{5}} \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 {a + b \operatorname {asech}{\left (c x \right )}}{x^{6}}\, dx \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 {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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