Optimal. Leaf size=126 \[ \frac {3 b^2 \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c^2}-\frac {3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3 b^3 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right )}{2 c^2} \]
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Rubi [A] time = 0.16, 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 = {6285, 5451, 4184, 3718, 2190, 2279, 2391} \[ \frac {3 b^3 \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right )}{2 c^2}+\frac {3 b^2 \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c^2}-\frac {3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3 \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 4184
Rule 5451
Rule 6285
Rubi steps
\begin {align*} \int x \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx &=-\frac {\operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.95, size = 219, normalized size = 1.74 \[ \frac {a \left (a \left (a c^2 x^2-3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1)\right )+6 b^2 \log \left (\frac {1}{c x}\right )\right )-3 b^2 \text {sech}^{-1}(c x)^2 \left (b \left (c x \sqrt {\frac {1-c x}{c x+1}}+\sqrt {\frac {1-c x}{c x+1}}-1\right )-a c^2 x^2\right )+3 b \text {sech}^{-1}(c x) \left (a \left (a c^2 x^2-2 b \sqrt {\frac {1-c x}{c x+1}} (c x+1)\right )+2 b^2 \log \left (e^{-2 \text {sech}^{-1}(c x)}+1\right )\right )+b^3 c^2 x^2 \text {sech}^{-1}(c x)^3-3 b^3 \text {Li}_2\left (-e^{-2 \text {sech}^{-1}(c x)}\right )}{2 c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x \operatorname {arsech}\left (c x\right )^{3} + 3 \, a b^{2} x \operatorname {arsech}\left (c x\right )^{2} + 3 \, a^{2} b x \operatorname {arsech}\left (c x\right ) + a^{3} x, x\right ) \]
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 \[ \int {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.72, size = 343, normalized size = 2.72 \[ \frac {x^{2} a^{3}}{2}+\frac {x^{2} b^{3} \mathrm {arcsech}\left (c x \right )^{3}}{2}-\frac {3 b^{3} \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x}{2 c}-\frac {3 b^{3} \mathrm {arcsech}\left (c x \right )^{2}}{2 c^{2}}+\frac {3 b^{3} \mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}+\frac {3 b^{3} \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2 c^{2}}-\frac {3 a \,b^{2} \mathrm {arcsech}\left (c x \right )}{c^{2}}+\frac {3 x^{2} a \,b^{2} \mathrm {arcsech}\left (c x \right )^{2}}{2}-\frac {3 a \,b^{2} \mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x}{c}+\frac {3 a \,b^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}+\frac {3 x^{2} a^{2} b \,\mathrm {arcsech}\left (c x \right )}{2}-\frac {3 a^{2} b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x}{2 c} \]
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 \[ \frac {3}{2} \, a b^{2} x^{2} \operatorname {arsech}\left (c x\right )^{2} + \frac {1}{2} \, a^{3} x^{2} + \frac {3}{2} \, {\left (x^{2} \operatorname {arsech}\left (c x\right ) - \frac {x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c}\right )} a^{2} b - 3 \, {\left (\frac {x \sqrt {\frac {1}{c^{2} x^{2}} - 1} \operatorname {arsech}\left (c x\right )}{c} + \frac {\log \relax (x)}{c^{2}}\right )} a b^{2} + b^{3} \int x \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{3}\,{d x} \]
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 \[ \int x\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]
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 \[ \int x \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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