Optimal. Leaf size=77 \[ \frac {b \sqrt {1-c x}}{9 x^3 \sqrt {\frac {1}{1+c x}}}+\frac {2 b c^2 \sqrt {1-c x}}{9 x \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{3 x^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 4, 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)}{3 x^3}+\frac {2 b c^2 \sqrt {1-c x}}{9 x \sqrt {\frac {1}{c x+1}}}+\frac {b \sqrt {1-c x}}{9 x^3 \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^4} \, dx &=-\frac {a+b \text {sech}^{-1}(c x)}{3 x^3}-\frac {1}{3} \left (b \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}}{9 x^3 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{3 x^3}+\frac {1}{9} \left (b \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}}{9 x^3 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{3 x^3}-\frac {1}{9} \left (2 b c^2 \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}}{9 x^3 \sqrt {\frac {1}{1+c x}}}+\frac {2 b c^2 \sqrt {1-c x}}{9 x \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 74, normalized size = 0.96 \begin {gather*} -\frac {a}{3 x^3}+b \left (\frac {2 c^3}{9}+\frac {1}{9 x^3}+\frac {c}{9 x^2}+\frac {2 c^2}{9 x}\right ) \sqrt {\frac {1-c x}{1+c x}}-\frac {b \text {sech}^{-1}(c x)}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 77, normalized size = 1.00
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {a}{3 c^{3} x^{3}}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{2} x^{2}+1\right )}{9 c^{2} x^{2}}\right )\right )\) | \(77\) |
default | \(c^{3} \left (-\frac {a}{3 c^{3} x^{3}}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{2} x^{2}+1\right )}{9 c^{2} x^{2}}\right )\right )\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 56, normalized size = 0.73 \begin {gather*} \frac {1}{9} \, b {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {3 \, \operatorname {arsech}\left (c x\right )}{x^{3}}\right )} - \frac {a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 79, normalized size = 1.03 \begin {gather*} -\frac {3 \, b \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (2 \, b c^{3} x^{3} + b c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 3 \, a}{9 \, x^{3}} \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^{4}}\, 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^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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