Optimal. Leaf size=111 \[ \frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {Li}_3\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {Li}_3\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}+x \text {sech}^{-1}(a x)^3-\frac {6 \text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a} \]
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Rubi [A] time = 0.09, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6279, 5418, 4180, 2531, 2282, 6589} \[ \frac {6 i \text {sech}^{-1}(a x) \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {sech}^{-1}(a x) \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {PolyLog}\left (3,-i e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {PolyLog}\left (3,i e^{\text {sech}^{-1}(a x)}\right )}{a}+x \text {sech}^{-1}(a x)^3-\frac {6 \text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4180
Rule 5418
Rule 6279
Rule 6589
Rubi steps
\begin {align*} \int \text {sech}^{-1}(a x)^3 \, dx &=-\frac {\operatorname {Subst}\left (\int x^3 \text {sech}(x) \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}\\ &=x \text {sech}^{-1}(a x)^3-\frac {3 \operatorname {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}\\ &=x \text {sech}^{-1}(a x)^3-\frac {6 \text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {(6 i) \operatorname {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}-\frac {(6 i) \operatorname {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}\\ &=x \text {sech}^{-1}(a x)^3-\frac {6 \text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {(6 i) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}+\frac {(6 i) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}\\ &=x \text {sech}^{-1}(a x)^3-\frac {6 \text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {(6 i) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {(6 i) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a x)}\right )}{a}\\ &=x \text {sech}^{-1}(a x)^3-\frac {6 \text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {Li}_3\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {Li}_3\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 128, normalized size = 1.15 \[ x \text {sech}^{-1}(a x)^3-\frac {3 i \left (-2 \text {sech}^{-1}(a x) \left (\text {Li}_2\left (-i e^{-\text {sech}^{-1}(a x)}\right )-\text {Li}_2\left (i e^{-\text {sech}^{-1}(a x)}\right )\right )-2 \left (\text {Li}_3\left (-i e^{-\text {sech}^{-1}(a x)}\right )-\text {Li}_3\left (i e^{-\text {sech}^{-1}(a x)}\right )\right )-\left (\text {sech}^{-1}(a x)^2 \left (\log \left (1-i e^{-\text {sech}^{-1}(a x)}\right )-\log \left (1+i e^{-\text {sech}^{-1}(a x)}\right )\right )\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {arsech}\left (a x\right )^{3}, x\right ) \]
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 \[ \int \operatorname {arsech}\left (a x\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \mathrm {arcsech}\left (a x \right )^{3}\, dx \]
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 \[ x \log \left (\sqrt {a x + 1} \sqrt {-a x + 1} + 1\right )^{3} - \int \frac {a^{2} x^{2} \log \relax (a)^{3} + {\left (a^{2} x^{2} - 1\right )} \log \relax (x)^{3} + 3 \, {\left (a^{2} x^{2} \log \relax (a) + {\left (a^{2} x^{2} {\left (\log \relax (a) + 1\right )} + {\left (a^{2} x^{2} - 1\right )} \log \relax (x) - \log \relax (a)\right )} \sqrt {a x + 1} \sqrt {-a x + 1} + {\left (a^{2} x^{2} - 1\right )} \log \relax (x) - \log \relax (a)\right )} \log \left (\sqrt {a x + 1} \sqrt {-a x + 1} + 1\right )^{2} - \log \relax (a)^{3} + 3 \, {\left (a^{2} x^{2} \log \relax (a) - \log \relax (a)\right )} \log \relax (x)^{2} + {\left (a^{2} x^{2} \log \relax (a)^{3} + {\left (a^{2} x^{2} - 1\right )} \log \relax (x)^{3} - \log \relax (a)^{3} + 3 \, {\left (a^{2} x^{2} \log \relax (a) - \log \relax (a)\right )} \log \relax (x)^{2} + 3 \, {\left (a^{2} x^{2} \log \relax (a)^{2} - \log \relax (a)^{2}\right )} \log \relax (x)\right )} \sqrt {a x + 1} \sqrt {-a x + 1} - 3 \, {\left (a^{2} x^{2} \log \relax (a)^{2} + {\left (a^{2} x^{2} - 1\right )} \log \relax (x)^{2} + {\left (a^{2} x^{2} \log \relax (a)^{2} + {\left (a^{2} x^{2} - 1\right )} \log \relax (x)^{2} - \log \relax (a)^{2} + 2 \, {\left (a^{2} x^{2} \log \relax (a) - \log \relax (a)\right )} \log \relax (x)\right )} \sqrt {a x + 1} \sqrt {-a x + 1} - \log \relax (a)^{2} + 2 \, {\left (a^{2} x^{2} \log \relax (a) - \log \relax (a)\right )} \log \relax (x)\right )} \log \left (\sqrt {a x + 1} \sqrt {-a x + 1} + 1\right ) + 3 \, {\left (a^{2} x^{2} \log \relax (a)^{2} - \log \relax (a)^{2}\right )} \log \relax (x)}{a^{2} x^{2} + {\left (a^{2} x^{2} - 1\right )} \sqrt {a x + 1} \sqrt {-a x + 1} - 1}\,{d x} \]
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 \[ \int {\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^3 \,d x \]
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 \[ \int \operatorname {asech}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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