Optimal. Leaf size=109 \[ -\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}+\frac {8 b c^4 \sqrt {1-c x}}{75 x \sqrt {\frac {1}{c x+1}}}+\frac {4 b c^2 \sqrt {1-c x}}{75 x^3 \sqrt {\frac {1}{c x+1}}}+\frac {b \sqrt {1-c x}}{25 x^5 \sqrt {\frac {1}{c x+1}}} \]
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Rubi [A] time = 0.05, 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 = {6283, 103, 12, 95} \[ -\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}+\frac {4 b c^2 \sqrt {1-c x}}{75 x^3 \sqrt {\frac {1}{c x+1}}}+\frac {8 b c^4 \sqrt {1-c x}}{75 x \sqrt {\frac {1}{c x+1}}}+\frac {b \sqrt {1-c x}}{25 x^5 \sqrt {\frac {1}{c x+1}}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 103
Rule 6283
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 = 0.09, size = 94, normalized size = 0.86 \[ -\frac {a}{5 x^5}+b \left (\frac {8 c^5}{75}+\frac {8 c^4}{75 x}+\frac {4 c^3}{75 x^2}+\frac {4 c^2}{75 x^3}+\frac {c}{25 x^4}+\frac {1}{25 x^5}\right ) \sqrt {\frac {1-c x}{c x+1}}-\frac {b \text {sech}^{-1}(c x)}{5 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 89, normalized size = 0.82 \[ -\frac {15 \, b \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (8 \, b c^{5} x^{5} + 4 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 15 \, a}{75 \, x^{5}} \]
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 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 85, normalized size = 0.78 \[ c^{5} \left (-\frac {a}{5 c^{5} x^{5}}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{5 c^{5} x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (8 c^{4} x^{4}+4 c^{2} x^{2}+3\right )}{75 c^{4} x^{4}}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 73, normalized size = 0.67 \[ \frac {1}{75} \, b {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {15 \, \operatorname {arsech}\left (c x\right )}{x^{5}}\right )} - \frac {a}{5 \, x^{5}} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^6} \,d x \]
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 \[ \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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