Optimal. Leaf size=223 \[ \frac {b^2 \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c^4}-\frac {b^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 c^2}-\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}-\frac {b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}-\frac {b x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 c^2}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {b^3 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right )}{2 c^4}+\frac {b^3 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{4 c^4} \]
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Rubi [A] time = 0.24, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6285, 5451, 4186, 3767, 8, 4184, 3718, 2190, 2279, 2391} \[ \frac {b^3 \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right )}{2 c^4}-\frac {b^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 c^2}+\frac {b^2 \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c^4}-\frac {b x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 c^2}-\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}-\frac {b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {b^3 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{4 c^4} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 3767
Rule 4184
Rule 4186
Rule 5451
Rule 6285
Rubi steps
\begin {align*} \int x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^3 \text {sech}^4(x) \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^4}\\ &=\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {(3 b) \operatorname {Subst}\left (\int (a+b x)^2 \text {sech}^4(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{4 c^4}\\ &=-\frac {b^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 c^2}-\frac {b x^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 c^2}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {b \operatorname {Subst}\left (\int (a+b x)^2 \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{2 c^4}+\frac {b^3 \operatorname {Subst}\left (\int \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{4 c^4}\\ &=-\frac {b^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 c^2}-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}-\frac {b x^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 c^2}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {b^2 \operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^4}+\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int 1 \, dx,x,-i \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{4 c^4}\\ &=\frac {b^3 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 c^4}-\frac {b^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 c^2}-\frac {b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}-\frac {b x^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 c^2}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {\left (2 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^4}\\ &=\frac {b^3 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 c^4}-\frac {b^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 c^2}-\frac {b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}-\frac {b x^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 c^2}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{c^4}-\frac {b^3 \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^4}\\ &=\frac {b^3 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 c^4}-\frac {b^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 c^2}-\frac {b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}-\frac {b x^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 c^2}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{c^4}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {sech}^{-1}(c x)}\right )}{2 c^4}\\ &=\frac {b^3 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 c^4}-\frac {b^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 c^2}-\frac {b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^4}-\frac {b x^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 c^2}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{c^4}+\frac {b^3 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right )}{2 c^4}\\ \end {align*}
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Mathematica [A] time = 2.15, size = 337, normalized size = 1.51 \[ \frac {1}{4} \left (a^3 x^4+a^2 b \left (3 x^4 \text {sech}^{-1}(c x)-\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (c^2 x^2+2\right )}{c^4}\right )+\frac {a b^2 \left (3 c^4 x^4 \text {sech}^{-1}(c x)^2-c^2 x^2-2 \sqrt {\frac {1-c x}{c x+1}} \left (c^3 x^3+c^2 x^2+2 c x+2\right ) \text {sech}^{-1}(c x)+4 \log \left (\frac {1}{c x}\right )\right )}{c^4}-\frac {b^3 \left (\text {sech}^{-1}(c x) \left (c^2 x^2-4 \log \left (e^{-2 \text {sech}^{-1}(c x)}+1\right )\right )+\left (c^3 x^3 \sqrt {\frac {1-c x}{c x+1}}+c^2 x^2 \sqrt {\frac {1-c x}{c x+1}}+2 c x \sqrt {\frac {1-c x}{c x+1}}+2 \sqrt {\frac {1-c x}{c x+1}}-2\right ) \text {sech}^{-1}(c x)^2+2 \text {Li}_2\left (-e^{-2 \text {sech}^{-1}(c x)}\right )-\sqrt {\frac {1-c x}{c x+1}} (c x+1)\right )}{c^4}+b^3 x^4 \text {sech}^{-1}(c x)^3\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{3} \operatorname {arsech}\left (c x\right )^{3} + 3 \, a b^{2} x^{3} \operatorname {arsech}\left (c x\right )^{2} + 3 \, a^{2} b x^{3} \operatorname {arsech}\left (c x\right ) + a^{3} x^{3}, x\right ) \]
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 \[ \int {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.96, size = 546, normalized size = 2.45 \[ \frac {x^{4} a^{3}}{4}+\frac {b^{3} \mathrm {arcsech}\left (c x \right )^{3} x^{4}}{4}-\frac {b^{3} \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x^{3}}{4 c}-\frac {b^{3} \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x}{2 c^{3}}-\frac {b^{3} \mathrm {arcsech}\left (c x \right ) x^{2}}{4 c^{2}}+\frac {b^{3} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x}{4 c^{3}}-\frac {b^{3} \mathrm {arcsech}\left (c x \right )^{2}}{2 c^{4}}-\frac {b^{3}}{4 c^{4}}+\frac {b^{3} \mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{4}}+\frac {b^{3} \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2 c^{4}}-\frac {a \,b^{2} \mathrm {arcsech}\left (c x \right )}{c^{4}}+\frac {3 a \,b^{2} \mathrm {arcsech}\left (c x \right )^{2} x^{4}}{4}-\frac {a \,b^{2} \mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x^{3}}{2 c}-\frac {a \,b^{2} \mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x}{c^{3}}-\frac {x^{2} a \,b^{2}}{4 c^{2}}+\frac {a \,b^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{4}}+\frac {3 a^{2} b \,x^{4} \mathrm {arcsech}\left (c x \right )}{4}-\frac {a^{2} b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x^{3}}{4 c}-\frac {a^{2} b \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}}{2 c^{3}} \]
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 \[ \frac {1}{4} \, a^{3} x^{4} + \frac {1}{4} \, {\left (3 \, x^{4} \operatorname {arsech}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} a^{2} b + \int b^{3} x^{3} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{3} + 3 \, a b^{2} x^{3} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]
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 \[ \int x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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