Optimal. Leaf size=126 \[ -\frac {3 b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3 b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{c^2}+\frac {3 b^3 \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right )}{2 c^2} \]
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Rubi [A]
time = 0.11, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6420, 5559,
4269, 3799, 2221, 2317, 2438} \begin {gather*} \frac {3 b^2 \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c^2}-\frac {3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3 b^3 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right )}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 4269
Rule 5559
Rule 6420
Rubi steps
\begin {align*} \int x \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx &=-\frac {\text {Subst}\left (\int (a+b x)^3 \text {sech}^2(x) \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^2}\\ &=\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {(3 b) \text {Subst}\left (\int (a+b x)^2 \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{2 c^2}\\ &=-\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {\left (3 b^2\right ) \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^2}\\ &=-\frac {3 b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {\left (6 b^2\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {sech}^{-1}(c x)\right )}{c^2}\\ &=-\frac {3 b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3 b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^2}\\ &=-\frac {3 b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3 b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {sech}^{-1}(c x)}\right )}{2 c^2}\\ &=-\frac {3 b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3 b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{c^2}+\frac {3 b^3 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right )}{2 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 219, normalized size = 1.74 \begin {gather*} \frac {-3 b^2 \left (-a c^2 x^2+b \left (-1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )\right ) \text {sech}^{-1}(c x)^2+b^3 c^2 x^2 \text {sech}^{-1}(c x)^3+3 b \text {sech}^{-1}(c x) \left (a \left (a c^2 x^2-2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )+2 b^2 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )\right )+a \left (a \left (a c^2 x^2-3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )+6 b^2 \log \left (\frac {1}{c x}\right )\right )-3 b^3 \text {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(336\) vs.
\(2(166)=332\).
time = 0.50, size = 337, normalized size = 2.67
method | result | size |
derivativedivides | \(\frac {\frac {c^{2} x^{2} a^{3}}{2}+\frac {b^{3} \mathrm {arcsech}\left (c x \right )^{3} c^{2} x^{2}}{2}-\frac {3 b^{3} \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{2}-\frac {3 b^{3} \mathrm {arcsech}\left (c x \right )^{2}}{2}+3 b^{3} \mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+\frac {3 b^{3} \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}-3 a \,b^{2} \mathrm {arcsech}\left (c x \right )+\frac {3 a \,b^{2} \mathrm {arcsech}\left (c x \right )^{2} c^{2} x^{2}}{2}-3 a \,b^{2} \mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +3 a \,b^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+3 a^{2} b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{2} x^{2}}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{2}\right )}{c^{2}}\) | \(337\) |
default | \(\frac {\frac {c^{2} x^{2} a^{3}}{2}+\frac {b^{3} \mathrm {arcsech}\left (c x \right )^{3} c^{2} x^{2}}{2}-\frac {3 b^{3} \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{2}-\frac {3 b^{3} \mathrm {arcsech}\left (c x \right )^{2}}{2}+3 b^{3} \mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+\frac {3 b^{3} \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}-3 a \,b^{2} \mathrm {arcsech}\left (c x \right )+\frac {3 a \,b^{2} \mathrm {arcsech}\left (c x \right )^{2} c^{2} x^{2}}{2}-3 a \,b^{2} \mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +3 a \,b^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+3 a^{2} b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{2} x^{2}}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{2}\right )}{c^{2}}\) | \(337\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}\, 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\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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