Optimal. Leaf size=79 \[ -\frac {a+b \text {csch}^{-1}(c x)}{5 x^5}+\frac {1}{25} b c^5 \left (\frac {1}{c^2 x^2}+1\right )^{5/2}-\frac {2}{15} b c^5 \left (\frac {1}{c^2 x^2}+1\right )^{3/2}+\frac {1}{5} b c^5 \sqrt {\frac {1}{c^2 x^2}+1} \]
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Rubi [A] time = 0.05, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6284, 266, 43} \[ -\frac {a+b \text {csch}^{-1}(c x)}{5 x^5}+\frac {1}{25} b c^5 \left (\frac {1}{c^2 x^2}+1\right )^{5/2}-\frac {2}{15} b c^5 \left (\frac {1}{c^2 x^2}+1\right )^{3/2}+\frac {1}{5} b c^5 \sqrt {\frac {1}{c^2 x^2}+1} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 6284
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{x^6} \, dx &=-\frac {a+b \text {csch}^{-1}(c x)}{5 x^5}-\frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^7} \, dx}{5 c}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{5 x^5}+\frac {b \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{10 c}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{5 x^5}+\frac {b \operatorname {Subst}\left (\int \left (\frac {c^4}{\sqrt {1+\frac {x}{c^2}}}-2 c^4 \sqrt {1+\frac {x}{c^2}}+c^4 \left (1+\frac {x}{c^2}\right )^{3/2}\right ) \, dx,x,\frac {1}{x^2}\right )}{10 c}\\ &=\frac {1}{5} b c^5 \sqrt {1+\frac {1}{c^2 x^2}}-\frac {2}{15} b c^5 \left (1+\frac {1}{c^2 x^2}\right )^{3/2}+\frac {1}{25} b c^5 \left (1+\frac {1}{c^2 x^2}\right )^{5/2}-\frac {a+b \text {csch}^{-1}(c x)}{5 x^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 69, normalized size = 0.87 \[ -\frac {a}{5 x^5}+b \left (\frac {8 c^5}{75}-\frac {4 c^3}{75 x^2}+\frac {c}{25 x^4}\right ) \sqrt {\frac {c^2 x^2+1}{c^2 x^2}}-\frac {b \text {csch}^{-1}(c x)}{5 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 87, normalized size = 1.10 \[ -\frac {15 \, b \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (8 \, b c^{5} x^{5} - 4 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 15 \, a}{75 \, x^{5}} \]
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 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 83, normalized size = 1.05 \[ c^{5} \left (-\frac {a}{5 c^{5} x^{5}}+b \left (-\frac {\mathrm {arccsch}\left (c x \right )}{5 c^{5} x^{5}}+\frac {\left (c^{2} x^{2}+1\right ) \left (8 c^{4} x^{4}-4 c^{2} x^{2}+3\right )}{75 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 73, normalized size = 0.92 \[ \frac {1}{75} \, b {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {15 \, \operatorname {arcsch}\left (c x\right )}{x^{5}}\right )} - \frac {a}{5 \, x^{5}} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x^6} \,d x \]
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 \[ \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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