Optimal. Leaf size=104 \[ -\frac {x^2}{12 a^2}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{3 a^4}-\frac {x^2 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{6 a^2}+\frac {1}{4} x^4 \text {sech}^{-1}(a x)^2-\frac {\log (x)}{3 a^4} \]
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Rubi [A]
time = 0.06, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6420, 5526,
4270, 4269, 3556} \begin {gather*} -\frac {\log (x)}{3 a^4}-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{3 a^4}-\frac {x^2}{12 a^2}-\frac {x^2 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{6 a^2}+\frac {1}{4} x^4 \text {sech}^{-1}(a x)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 4269
Rule 4270
Rule 5526
Rule 6420
Rubi steps
\begin {align*} \int x^3 \text {sech}^{-1}(a x)^2 \, dx &=-\frac {\text {Subst}\left (\int x^2 \text {sech}^4(x) \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^4}\\ &=\frac {1}{4} x^4 \text {sech}^{-1}(a x)^2-\frac {\text {Subst}\left (\int x \text {sech}^4(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{2 a^4}\\ &=-\frac {x^2}{12 a^2}-\frac {x^2 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{6 a^2}+\frac {1}{4} x^4 \text {sech}^{-1}(a x)^2-\frac {\text {Subst}\left (\int x \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {x^2}{12 a^2}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{3 a^4}-\frac {x^2 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{6 a^2}+\frac {1}{4} x^4 \text {sech}^{-1}(a x)^2+\frac {\text {Subst}\left (\int \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {x^2}{12 a^2}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{3 a^4}-\frac {x^2 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{6 a^2}+\frac {1}{4} x^4 \text {sech}^{-1}(a x)^2-\frac {\log (x)}{3 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 77, normalized size = 0.74 \begin {gather*} -\frac {a^2 x^2+2 \sqrt {\frac {1-a x}{1+a x}} \left (2+2 a x+a^2 x^2+a^3 x^3\right ) \text {sech}^{-1}(a x)-3 a^4 x^4 \text {sech}^{-1}(a x)^2+4 \log (x)}{12 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 150, normalized size = 1.44
method | result | size |
derivativedivides | \(\frac {-\frac {\mathrm {arcsech}\left (a x \right )}{3}+\frac {a^{4} x^{4} \mathrm {arcsech}\left (a x \right )^{2}}{4}-\frac {\mathrm {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, a^{3} x^{3}}{6}-\frac {\mathrm {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, a x}{3}-\frac {a^{2} x^{2}}{12}+\frac {\ln \left (1+\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{3}}{a^{4}}\) | \(150\) |
default | \(\frac {-\frac {\mathrm {arcsech}\left (a x \right )}{3}+\frac {a^{4} x^{4} \mathrm {arcsech}\left (a x \right )^{2}}{4}-\frac {\mathrm {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, a^{3} x^{3}}{6}-\frac {\mathrm {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, a x}{3}-\frac {a^{2} x^{2}}{12}+\frac {\ln \left (1+\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{3}}{a^{4}}\) | \(150\) |
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 = 0.37, size = 125, normalized size = 1.20 \begin {gather*} \frac {3 \, a^{4} x^{4} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} - a^{2} x^{2} - 2 \, {\left (a^{3} x^{3} + 2 \, a x\right )} \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right ) - 4 \, \log \left (x\right )}{12 \, a^{4}} \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 x^{3} \operatorname {asech}^{2}{\left (a x \right )}\, 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 x^3\,{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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