Optimal. Leaf size=151 \[ -\frac {i x^2 \text {PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \text {PolyLog}\left (2,i e^{-a-b x}\right )}{2 b}-\frac {i x \text {PolyLog}\left (3,-i e^{-a-b x}\right )}{b^2}+\frac {i x \text {PolyLog}\left (3,i e^{-a-b x}\right )}{b^2}-\frac {i \text {PolyLog}\left (4,-i e^{-a-b x}\right )}{b^3}+\frac {i \text {PolyLog}\left (4,i e^{-a-b x}\right )}{b^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5252, 2611,
6744, 2320, 6724} \begin {gather*} -\frac {i \text {Li}_4\left (-i e^{-a-b x}\right )}{b^3}+\frac {i \text {Li}_4\left (i e^{-a-b x}\right )}{b^3}-\frac {i x \text {Li}_3\left (-i e^{-a-b x}\right )}{b^2}+\frac {i x \text {Li}_3\left (i e^{-a-b x}\right )}{b^2}-\frac {i x^2 \text {Li}_2\left (-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \text {Li}_2\left (i e^{-a-b x}\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 5252
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int x^2 \cot ^{-1}\left (e^{a+b x}\right ) \, dx &=\frac {1}{2} i \int x^2 \log \left (1-i e^{-a-b x}\right ) \, dx-\frac {1}{2} i \int x^2 \log \left (1+i e^{-a-b x}\right ) \, dx\\ &=-\frac {i x^2 \text {Li}_2\left (-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \text {Li}_2\left (i e^{-a-b x}\right )}{2 b}+\frac {i \int x \text {Li}_2\left (-i e^{-a-b x}\right ) \, dx}{b}-\frac {i \int x \text {Li}_2\left (i e^{-a-b x}\right ) \, dx}{b}\\ &=-\frac {i x^2 \text {Li}_2\left (-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \text {Li}_2\left (i e^{-a-b x}\right )}{2 b}-\frac {i x \text {Li}_3\left (-i e^{-a-b x}\right )}{b^2}+\frac {i x \text {Li}_3\left (i e^{-a-b x}\right )}{b^2}+\frac {i \int \text {Li}_3\left (-i e^{-a-b x}\right ) \, dx}{b^2}-\frac {i \int \text {Li}_3\left (i e^{-a-b x}\right ) \, dx}{b^2}\\ &=-\frac {i x^2 \text {Li}_2\left (-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \text {Li}_2\left (i e^{-a-b x}\right )}{2 b}-\frac {i x \text {Li}_3\left (-i e^{-a-b x}\right )}{b^2}+\frac {i x \text {Li}_3\left (i e^{-a-b x}\right )}{b^2}-\frac {i \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{-a-b x}\right )}{b^3}+\frac {i \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{-a-b x}\right )}{b^3}\\ &=-\frac {i x^2 \text {Li}_2\left (-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \text {Li}_2\left (i e^{-a-b x}\right )}{2 b}-\frac {i x \text {Li}_3\left (-i e^{-a-b x}\right )}{b^2}+\frac {i x \text {Li}_3\left (i e^{-a-b x}\right )}{b^2}-\frac {i \text {Li}_4\left (-i e^{-a-b x}\right )}{b^3}+\frac {i \text {Li}_4\left (i e^{-a-b x}\right )}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 151, normalized size = 1.00 \begin {gather*} -\frac {i x^2 \text {PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \text {PolyLog}\left (2,i e^{-a-b x}\right )}{2 b}-\frac {i x \text {PolyLog}\left (3,-i e^{-a-b x}\right )}{b^2}+\frac {i x \text {PolyLog}\left (3,i e^{-a-b x}\right )}{b^2}-\frac {i \text {PolyLog}\left (4,-i e^{-a-b x}\right )}{b^3}+\frac {i \text {PolyLog}\left (4,i e^{-a-b x}\right )}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 412 vs. \(2 (129 ) = 258\).
time = 0.28, size = 413, normalized size = 2.74
method | result | size |
risch | \(\frac {\pi \,x^{3}}{6}+\frac {i a^{3} \ln \left (1+i {\mathrm e}^{b x +a}\right )}{2 b^{3}}+\frac {i \dilog \left (-i {\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}-\frac {i a^{3} \ln \left (1-i {\mathrm e}^{b x +a}\right )}{2 b^{3}}-\frac {i \ln \left (-i \left (-{\mathrm e}^{b x +a}+i\right )\right ) x \,a^{2}}{2 b^{2}}+\frac {i \ln \left (-i {\mathrm e}^{b x +a}\right ) \ln \left (-i \left (-{\mathrm e}^{b x +a}+i\right )\right ) a^{2}}{2 b^{3}}+\frac {i \dilog \left (-i \left ({\mathrm e}^{b x +a}+i\right )\right ) a^{2}}{2 b^{3}}+\frac {i \polylog \left (4, i {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {i \ln \left (1+i {\mathrm e}^{b x +a}\right ) x \,a^{2}}{2 b^{2}}+\frac {i \polylog \left (3, -i {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {i \polylog \left (4, -i {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {i \polylog \left (3, i {\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {i \polylog \left (2, i {\mathrm e}^{b x +a}\right ) x^{2}}{2 b}-\frac {i \ln \left (1-i {\mathrm e}^{b x +a}\right ) x \,a^{2}}{2 b^{2}}+\frac {i \ln \left (-i \left ({\mathrm e}^{b x +a}+i\right )\right ) x \,a^{2}}{2 b^{2}}-\frac {i \polylog \left (2, -i {\mathrm e}^{b x +a}\right ) x^{2}}{2 b}-\frac {i \polylog \left (2, i {\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}+\frac {i \ln \left (-i \left ({\mathrm e}^{b x +a}+i\right )\right ) a^{3}}{2 b^{3}}+\frac {i \polylog \left (2, -i {\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}-\frac {i \ln \left (-i \left (-{\mathrm e}^{b x +a}+i\right )\right ) a^{3}}{2 b^{3}}\) | \(413\) |
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 [A]
time = 4.37, size = 187, normalized size = 1.24 \begin {gather*} \frac {2 \, b^{3} x^{3} \operatorname {arccot}\left (e^{\left (b x + a\right )}\right ) + 3 i \, b^{2} x^{2} {\rm Li}_2\left (i \, e^{\left (b x + a\right )}\right ) - 3 i \, b^{2} x^{2} {\rm Li}_2\left (-i \, e^{\left (b x + a\right )}\right ) - i \, a^{3} \log \left (e^{\left (b x + a\right )} + i\right ) + i \, a^{3} \log \left (e^{\left (b x + a\right )} - i\right ) - 6 i \, b x {\rm polylog}\left (3, i \, e^{\left (b x + a\right )}\right ) + 6 i \, b x {\rm polylog}\left (3, -i \, e^{\left (b x + a\right )}\right ) + {\left (-i \, b^{3} x^{3} - i \, a^{3}\right )} \log \left (i \, e^{\left (b x + a\right )} + 1\right ) + {\left (i \, b^{3} x^{3} + i \, a^{3}\right )} \log \left (-i \, e^{\left (b x + a\right )} + 1\right ) + 6 i \, {\rm polylog}\left (4, i \, e^{\left (b x + a\right )}\right ) - 6 i \, {\rm polylog}\left (4, -i \, e^{\left (b x + a\right )}\right )}{6 \, b^{3}} \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^{2} \operatorname {acot}{\left (e^{a} e^{b x} \right )}\, 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^2\,\mathrm {acot}\left ({\mathrm {e}}^{a+b\,x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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