Optimal. Leaf size=155 \[ -\frac {b x^4}{12}+\frac {1}{3} x^3 \text {ArcTan}(c-(1+i c) \cot (a+b x))-\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x^2 \text {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i x \text {PolyLog}\left (3,-i c e^{2 i a+2 i b x}\right )}{4 b^2}+\frac {\text {PolyLog}\left (4,-i c e^{2 i a+2 i b x}\right )}{8 b^3} \]
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Rubi [A]
time = 0.18, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5281, 2215,
2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {1}{3} x^3 \text {ArcTan}(c-(1+i c) \cot (a+b x))+\frac {\text {Li}_4\left (-i c e^{2 i a+2 i b x}\right )}{8 b^3}-\frac {i x \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac {x^2 \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {b x^4}{12} \end {gather*}
Antiderivative was successfully verified.
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Rule 2215
Rule 2221
Rule 2320
Rule 2611
Rule 5281
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int x^2 \tan ^{-1}(c+(-1-i c) \cot (a+b x)) \, dx &=\frac {1}{3} x^3 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{3} (i b) \int \frac {x^3}{-i (-1-i c)+c-c e^{2 i a+2 i b x}} \, dx\\ &=-\frac {b x^4}{12}+\frac {1}{3} x^3 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{3} (b c) \int \frac {e^{2 i a+2 i b x} x^3}{-i (-1-i c)+c-c e^{2 i a+2 i b x}} \, dx\\ &=-\frac {b x^4}{12}+\frac {1}{3} x^3 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {1}{2} i \int x^2 \log \left (1-\frac {c e^{2 i a+2 i b x}}{-i (-1-i c)+c}\right ) \, dx\\ &=-\frac {b x^4}{12}+\frac {1}{3} x^3 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x^2 \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {\int x \text {Li}_2\left (\frac {c e^{2 i a+2 i b x}}{-i (-1-i c)+c}\right ) \, dx}{2 b}\\ &=-\frac {b x^4}{12}+\frac {1}{3} x^3 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x^2 \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i x \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{4 b^2}+\frac {i \int \text {Li}_3\left (\frac {c e^{2 i a+2 i b x}}{-i (-1-i c)+c}\right ) \, dx}{4 b^2}\\ &=-\frac {b x^4}{12}+\frac {1}{3} x^3 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x^2 \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i x \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{4 b^2}+\frac {\text {Subst}\left (\int \frac {\text {Li}_3(-i c x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^3}\\ &=-\frac {b x^4}{12}+\frac {1}{3} x^3 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x^2 \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i x \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{4 b^2}+\frac {\text {Li}_4\left (-i c e^{2 i a+2 i b x}\right )}{8 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 140, normalized size = 0.90 \begin {gather*} \frac {1}{3} x^3 \text {ArcTan}(c+(-1-i c) \cot (a+b x))-\frac {4 i b^3 x^3 \log \left (1-\frac {i e^{-2 i (a+b x)}}{c}\right )-6 b^2 x^2 \text {PolyLog}\left (2,\frac {i e^{-2 i (a+b x)}}{c}\right )+6 i b x \text {PolyLog}\left (3,\frac {i e^{-2 i (a+b x)}}{c}\right )+3 \text {PolyLog}\left (4,\frac {i e^{-2 i (a+b x)}}{c}\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 = 0.76, size = 1533, normalized size = 9.89
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1533\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 311 vs. \(2 (109) = 218\).
time = 0.30, size = 311, normalized size = 2.01 \begin {gather*} \frac {\frac {4 \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2}\right )} \arctan \left ({\left (-i \, c - 1\right )} \cot \left (b x + a\right ) + c\right )}{b^{2}} - \frac {{\left (-3 i \, {\left (b x + a\right )}^{4} + 12 i \, {\left (b x + a\right )}^{3} a - 18 i \, {\left (b x + a\right )}^{2} a^{2} - 2 \, {\left (-4 i \, {\left (b x + a\right )}^{3} + 9 i \, {\left (b x + a\right )}^{2} a - 9 i \, {\left (b x + a\right )} a^{2}\right )} \arctan \left (c \cos \left (2 \, b x + 2 \, a\right ), -c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) - 3 \, {\left (4 i \, {\left (b x + a\right )}^{2} - 6 i \, {\left (b x + a\right )} a + 3 i \, a^{2}\right )} {\rm Li}_2\left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + {\left (4 \, {\left (b x + a\right )}^{3} - 9 \, {\left (b x + a\right )}^{2} a + 9 \, {\left (b x + a\right )} a^{2}\right )} \log \left (c^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + c^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\left (4 \, b x + a\right )} {\rm Li}_{3}(-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}) + 6 i \, {\rm Li}_{4}(-i \, c e^{\left (2 i \, b x + 2 i \, a\right )})\right )} {\left (i \, c + 1\right )}}{b^{2} {\left (c - i\right )}}}{12 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.29, size = 166, normalized size = 1.07 \begin {gather*} -\frac {2 \, b^{4} x^{4} - 4 i \, b^{3} x^{3} \log \left (-\frac {{\left (c e^{\left (2 i \, b x + 2 i \, a\right )} - i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{c - i}\right ) + 6 \, b^{2} x^{2} {\rm Li}_2\left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 2 \, a^{4} - 4 i \, a^{3} \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} - i}{c}\right ) + 6 i \, b x {\rm polylog}\left (3, -i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 4 \, {\left (i \, b^{3} x^{3} + i \, a^{3}\right )} \log \left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 3 \, {\rm polylog}\left (4, -i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{24 \, 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 {atan}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (1+c\,1{}\mathrm {i}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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