Optimal. Leaf size=78 \[ -\frac {b x \sqrt {1-c x}}{6 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcSin}(c x)}{6 c^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 78, 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, 92, 41,
222} \begin {gather*} \frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcSin}(c x)}{6 c^3}-\frac {b x \sqrt {1-c x}}{6 c^2 \sqrt {\frac {1}{c x+1}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 92
Rule 222
Rule 6418
Rubi steps
\begin {align*} \int x^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b x \sqrt {1-c x}}{6 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{6 c^2}\\ &=-\frac {b x \sqrt {1-c x}}{6 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{6 c^2}\\ &=-\frac {b x \sqrt {1-c x}}{6 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{6 c^3}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.05, size = 103, normalized size = 1.32 \begin {gather*} \frac {a x^3}{3}+b \sqrt {\frac {1-c x}{1+c x}} \left (-\frac {x}{6 c^2}-\frac {x^2}{6 c}\right )+\frac {1}{3} b x^3 \text {sech}^{-1}(c x)+\frac {i b \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{6 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 96, normalized size = 1.23
method | result | size |
derivativedivides | \(\frac {\frac {c^{3} x^{3} a}{3}+b \left (\frac {c^{3} x^{3} \mathrm {arcsech}\left (c x \right )}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{6 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{3}}\) | \(96\) |
default | \(\frac {\frac {c^{3} x^{3} a}{3}+b \left (\frac {c^{3} x^{3} \mathrm {arcsech}\left (c x \right )}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{6 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{3}}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 73, normalized size = 0.94 \begin {gather*} \frac {1}{3} \, a x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs.
\(2 (46) = 92\).
time = 0.38, size = 162, normalized size = 2.08 \begin {gather*} \frac {2 \, a c^{3} x^{3} - b c^{2} x^{2} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, b c^{3} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 2 \, b \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) + 2 \, {\left (b c^{3} x^{3} - b c^{3}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{6 \, c^{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 x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\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^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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