Optimal. Leaf size=145 \[ -\frac {1}{12} i b x^4+\frac {1}{3} x^3 \cot ^{-1}(c-(i-c) \coth (a+b x))+\frac {1}{6} i x^3 \log \left (1+i c e^{2 a+2 b x}\right )+\frac {i x^2 \text {PolyLog}\left (2,-i c e^{2 a+2 b x}\right )}{4 b}-\frac {i x \text {PolyLog}\left (3,-i c e^{2 a+2 b x}\right )}{4 b^2}+\frac {i \text {PolyLog}\left (4,-i c e^{2 a+2 b x}\right )}{8 b^3} \]
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Rubi [A]
time = 0.17, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5306, 2215,
2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {i \text {Li}_4\left (-i c e^{2 a+2 b x}\right )}{8 b^3}-\frac {i x \text {Li}_3\left (-i c e^{2 a+2 b x}\right )}{4 b^2}+\frac {i x^2 \text {Li}_2\left (-i c e^{2 a+2 b x}\right )}{4 b}+\frac {1}{6} i x^3 \log \left (1+i c e^{2 a+2 b x}\right )+\frac {1}{3} x^3 \cot ^{-1}(c-(-c+i) \coth (a+b x))-\frac {1}{12} i b x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 2215
Rule 2221
Rule 2320
Rule 2611
Rule 5306
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int x^2 \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx &=\frac {1}{3} x^3 \cot ^{-1}(c-(i-c) \coth (a+b x))+\frac {1}{3} b \int \frac {x^3}{i-c e^{2 a+2 b x}} \, dx\\ &=-\frac {1}{12} i b x^4+\frac {1}{3} x^3 \cot ^{-1}(c-(i-c) \coth (a+b x))-\frac {1}{3} (i b c) \int \frac {e^{2 a+2 b x} x^3}{i-c e^{2 a+2 b x}} \, dx\\ &=-\frac {1}{12} i b x^4+\frac {1}{3} x^3 \cot ^{-1}(c-(i-c) \coth (a+b x))+\frac {1}{6} i x^3 \log \left (1+i c e^{2 a+2 b x}\right )-\frac {1}{2} i \int x^2 \log \left (1+i c e^{2 a+2 b x}\right ) \, dx\\ &=-\frac {1}{12} i b x^4+\frac {1}{3} x^3 \cot ^{-1}(c-(i-c) \coth (a+b x))+\frac {1}{6} i x^3 \log \left (1+i c e^{2 a+2 b x}\right )+\frac {i x^2 \text {Li}_2\left (-i c e^{2 a+2 b x}\right )}{4 b}-\frac {i \int x \text {Li}_2\left (-i c e^{2 a+2 b x}\right ) \, dx}{2 b}\\ &=-\frac {1}{12} i b x^4+\frac {1}{3} x^3 \cot ^{-1}(c-(i-c) \coth (a+b x))+\frac {1}{6} i x^3 \log \left (1+i c e^{2 a+2 b x}\right )+\frac {i x^2 \text {Li}_2\left (-i c e^{2 a+2 b x}\right )}{4 b}-\frac {i x \text {Li}_3\left (-i c e^{2 a+2 b x}\right )}{4 b^2}+\frac {i \int \text {Li}_3\left (-i c e^{2 a+2 b x}\right ) \, dx}{4 b^2}\\ &=-\frac {1}{12} i b x^4+\frac {1}{3} x^3 \cot ^{-1}(c-(i-c) \coth (a+b x))+\frac {1}{6} i x^3 \log \left (1+i c e^{2 a+2 b x}\right )+\frac {i x^2 \text {Li}_2\left (-i c e^{2 a+2 b x}\right )}{4 b}-\frac {i x \text {Li}_3\left (-i c e^{2 a+2 b x}\right )}{4 b^2}+\frac {i \text {Subst}\left (\int \frac {\text {Li}_3(-i c x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b^3}\\ &=-\frac {1}{12} i b x^4+\frac {1}{3} x^3 \cot ^{-1}(c-(i-c) \coth (a+b x))+\frac {1}{6} i x^3 \log \left (1+i c e^{2 a+2 b x}\right )+\frac {i x^2 \text {Li}_2\left (-i c e^{2 a+2 b x}\right )}{4 b}-\frac {i x \text {Li}_3\left (-i c e^{2 a+2 b x}\right )}{4 b^2}+\frac {i \text {Li}_4\left (-i c e^{2 a+2 b x}\right )}{8 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 128, normalized size = 0.88 \begin {gather*} \frac {1}{3} x^3 \cot ^{-1}(c+(-i+c) \coth (a+b x))+\frac {i \left (4 b^3 x^3 \log \left (1-\frac {i e^{-2 (a+b x)}}{c}\right )-6 b^2 x^2 \text {PolyLog}\left (2,\frac {i e^{-2 (a+b x)}}{c}\right )-6 b x \text {PolyLog}\left (3,\frac {i e^{-2 (a+b x)}}{c}\right )-3 \text {PolyLog}\left (4,\frac {i e^{-2 (a+b x)}}{c}\right )\right )}{24 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.92, size = 1487, normalized size = 10.26
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1487\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.23, size = 129, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {arccot}\left ({\left (c - i\right )} \coth \left (b x + a\right ) + c\right ) + \frac {4}{9} \, {\left (\frac {3 \, x^{4}}{4 i \, c + 4} - \frac {4 \, b^{3} x^{3} \log \left (i \, c e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 6 \, b^{2} x^{2} {\rm Li}_2\left (-i \, c e^{\left (2 \, b x + 2 \, a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-i \, c e^{\left (2 \, b x + 2 \, a\right )}) + 3 \, {\rm Li}_{4}(-i \, c e^{\left (2 \, b x + 2 \, a\right )})}{-2 \, b^{4} {\left (-i \, c - 1\right )}}\right )} b {\left (c - i\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 292 vs. \(2 (105) = 210\).
time = 2.51, size = 292, normalized size = 2.01 \begin {gather*} \frac {-i \, b^{4} x^{4} + 2 i \, b^{3} x^{3} \log \left (\frac {{\left (c - i\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c e^{\left (2 \, b x + 2 \, a\right )} - i}\right ) + 6 i \, b^{2} x^{2} {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )}\right ) + 6 i \, b^{2} x^{2} {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )}\right ) + i \, a^{4} - 2 i \, a^{3} \log \left (\frac {2 \, c e^{\left (b x + a\right )} + i \, \sqrt {-4 i \, c}}{2 \, c}\right ) - 2 i \, a^{3} \log \left (\frac {2 \, c e^{\left (b x + a\right )} - i \, \sqrt {-4 i \, c}}{2 \, c}\right ) - 12 i \, b x {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )}\right ) - 12 i \, b x {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )}\right ) - 2 \, {\left (-i \, b^{3} x^{3} - i \, a^{3}\right )} \log \left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )} + 1\right ) - 2 \, {\left (-i \, b^{3} x^{3} - i \, a^{3}\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )} + 1\right ) + 12 i \, {\rm polylog}\left (4, \frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )}\right ) + 12 i \, {\rm polylog}\left (4, -\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )}\right )}{12 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: CoercionFailed} \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 (c+\mathrm {coth}\left (a+b\,x\right )\,\left (c-\mathrm {i}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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