Optimal. Leaf size=140 \[ \frac {6 i b^2 \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c}-\frac {6 i b^2 \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c}+x \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {6 b \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c}-\frac {6 i b^3 \text {Li}_3\left (-i e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {6 i b^3 \text {Li}_3\left (i e^{\text {sech}^{-1}(c x)}\right )}{c} \]
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Rubi [A] time = 0.11, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6279, 5451, 4180, 2531, 2282, 6589} \[ \frac {6 i b^2 \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c}-\frac {6 i b^2 \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c}-\frac {6 i b^3 \text {PolyLog}\left (3,-i e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {6 i b^3 \text {PolyLog}\left (3,i e^{\text {sech}^{-1}(c x)}\right )}{c}+x \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {6 b \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4180
Rule 5451
Rule 6279
Rule 6589
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^3 \text {sech}(x) \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {(3 b) \operatorname {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {6 b \left (a+b \text {sech}^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {\left (6 i b^2\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{c}-\frac {\left (6 i b^2\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {6 b \left (a+b \text {sech}^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {6 i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {6 i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {\left (6 i b^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{c}+\frac {\left (6 i b^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {6 b \left (a+b \text {sech}^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {6 i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {6 i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {\left (6 i b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {\left (6 i b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{c}\\ &=x \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {6 b \left (a+b \text {sech}^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {6 i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {6 i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {6 i b^3 \text {Li}_3\left (-i e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {6 i b^3 \text {Li}_3\left (i e^{\text {sech}^{-1}(c x)}\right )}{c}\\ \end {align*}
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Mathematica [B] time = 0.44, size = 282, normalized size = 2.01 \[ a^3 x-\frac {3 a^2 b \tan ^{-1}\left (\frac {c x \sqrt {\frac {1-c x}{c x+1}}}{c x-1}\right )}{c}+3 a^2 b x \text {sech}^{-1}(c x)+\frac {3 i a b^2 \left (2 \text {Li}_2\left (-i e^{-\text {sech}^{-1}(c x)}\right )-2 \text {Li}_2\left (i e^{-\text {sech}^{-1}(c x)}\right )+\text {sech}^{-1}(c x) \left (-i c x \text {sech}^{-1}(c x)+2 \log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )-2 \log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )\right )\right )}{c}+\frac {b^3 \left (c x \text {sech}^{-1}(c x)^3-3 i \left (-2 \text {sech}^{-1}(c x) \left (\text {Li}_2\left (-i e^{-\text {sech}^{-1}(c x)}\right )-\text {Li}_2\left (i e^{-\text {sech}^{-1}(c x)}\right )\right )-2 \left (\text {Li}_3\left (-i e^{-\text {sech}^{-1}(c x)}\right )-\text {Li}_3\left (i e^{-\text {sech}^{-1}(c x)}\right )\right )-\left (\text {sech}^{-1}(c x)^2 \left (\log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )-\log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )\right )\right )\right )\right )}{c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} \operatorname {arsech}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {arsech}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname {arsech}\left (c x\right ) + a^{3}, x\right ) \]
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 \[ \int {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )^{3}\, dx \]
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 \[ b^{3} x \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )^{3} + a^{3} x + \frac {3 \, {\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} a^{2} b}{c} - \int -\frac {b^{3} \log \relax (c)^{3} - 3 \, a b^{2} \log \relax (c)^{2} - {\left (b^{3} c^{2} x^{2} - b^{3}\right )} \log \relax (x)^{3} - {\left (b^{3} c^{2} \log \relax (c)^{3} - 3 \, a b^{2} c^{2} \log \relax (c)^{2}\right )} x^{2} + 3 \, {\left (b^{3} \log \relax (c) - a b^{2} - {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2} + {\left (b^{3} \log \relax (c) - a b^{2} - {\left (b^{3} c^{2} {\left (\log \relax (c) + 1\right )} - a b^{2} c^{2}\right )} x^{2} - {\left (b^{3} c^{2} x^{2} - b^{3}\right )} \log \relax (x)\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - {\left (b^{3} c^{2} x^{2} - b^{3}\right )} \log \relax (x)\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )^{2} + 3 \, {\left (b^{3} \log \relax (c) - a b^{2} - {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x)^{2} + {\left (b^{3} \log \relax (c)^{3} - 3 \, a b^{2} \log \relax (c)^{2} - {\left (b^{3} c^{2} x^{2} - b^{3}\right )} \log \relax (x)^{3} - {\left (b^{3} c^{2} \log \relax (c)^{3} - 3 \, a b^{2} c^{2} \log \relax (c)^{2}\right )} x^{2} + 3 \, {\left (b^{3} \log \relax (c) - a b^{2} - {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x)^{2} + 3 \, {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) - {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c)\right )} x^{2}\right )} \log \relax (x)\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - 3 \, {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) - {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c)\right )} x^{2} - {\left (b^{3} c^{2} x^{2} - b^{3}\right )} \log \relax (x)^{2} + {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) - {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c)\right )} x^{2} - {\left (b^{3} c^{2} x^{2} - b^{3}\right )} \log \relax (x)^{2} + 2 \, {\left (b^{3} \log \relax (c) - a b^{2} - {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x)\right )} \sqrt {c x + 1} \sqrt {-c x + 1} + 2 \, {\left (b^{3} \log \relax (c) - a b^{2} - {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x)\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right ) + 3 \, {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) - {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c)\right )} x^{2}\right )} \log \relax (x)}{c^{2} x^{2} + {\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - 1}\,{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 \[ \int {\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]
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 \[ \int \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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