Optimal. Leaf size=303 \[ \frac {1}{2} x^2 \text {ArcTan}(c+d \cot (a+b x))+\frac {1}{4} i x^2 \log \left (1-\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )-\frac {1}{4} i x^2 \log \left (1-\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )+\frac {x \text {PolyLog}\left (2,\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )}{4 b}-\frac {x \text {PolyLog}\left (2,\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{4 b}+\frac {i \text {PolyLog}\left (3,\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )}{8 b^2}-\frac {i \text {PolyLog}\left (3,\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{8 b^2} \]
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Rubi [A]
time = 0.31, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5285, 2221,
2611, 2320, 6724} \begin {gather*} \frac {1}{2} x^2 \text {ArcTan}(d \cot (a+b x)+c)+\frac {i \text {Li}_3\left (\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )}{8 b^2}-\frac {i \text {Li}_3\left (\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{8 b^2}+\frac {x \text {Li}_2\left (\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )}{4 b}-\frac {x \text {Li}_2\left (\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{4 b}+\frac {1}{4} i x^2 \log \left (1-\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )-\frac {1}{4} i x^2 \log \left (1-\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 5285
Rule 6724
Rubi steps
\begin {align*} \int x \tan ^{-1}(c+d \cot (a+b x)) \, dx &=\frac {1}{2} x^2 \tan ^{-1}(c+d \cot (a+b x))+\frac {1}{2} (b (1+i c-d)) \int \frac {e^{2 i a+2 i b x} x^2}{1+i c+d+(-1-i c+d) e^{2 i a+2 i b x}} \, dx-\frac {1}{2} (b (1-i c+d)) \int \frac {e^{2 i a+2 i b x} x^2}{1-i c-d+(-1+i c-d) e^{2 i a+2 i b x}} \, dx\\ &=\frac {1}{2} x^2 \tan ^{-1}(c+d \cot (a+b x))+\frac {1}{4} i x^2 \log \left (1-\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )-\frac {1}{4} i x^2 \log \left (1-\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )+\frac {1}{2} i \int x \log \left (1+\frac {(-1+i c-d) e^{2 i a+2 i b x}}{1-i c-d}\right ) \, dx-\frac {1}{2} i \int x \log \left (1+\frac {(-1-i c+d) e^{2 i a+2 i b x}}{1+i c+d}\right ) \, dx\\ &=\frac {1}{2} x^2 \tan ^{-1}(c+d \cot (a+b x))+\frac {1}{4} i x^2 \log \left (1-\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )-\frac {1}{4} i x^2 \log \left (1-\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )+\frac {x \text {Li}_2\left (\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )}{4 b}-\frac {x \text {Li}_2\left (\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{4 b}+\frac {\int \text {Li}_2\left (-\frac {(-1+i c-d) e^{2 i a+2 i b x}}{1-i c-d}\right ) \, dx}{4 b}-\frac {\int \text {Li}_2\left (-\frac {(-1-i c+d) e^{2 i a+2 i b x}}{1+i c+d}\right ) \, dx}{4 b}\\ &=\frac {1}{2} x^2 \tan ^{-1}(c+d \cot (a+b x))+\frac {1}{4} i x^2 \log \left (1-\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )-\frac {1}{4} i x^2 \log \left (1-\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )+\frac {x \text {Li}_2\left (\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )}{4 b}-\frac {x \text {Li}_2\left (\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{4 b}+\frac {i \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {(-1-i c+d) x}{1+i c+d}\right )}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^2}-\frac {i \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {(c+i (1+d)) x}{c-i (-1+d)}\right )}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^2}\\ &=\frac {1}{2} x^2 \tan ^{-1}(c+d \cot (a+b x))+\frac {1}{4} i x^2 \log \left (1-\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )-\frac {1}{4} i x^2 \log \left (1-\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )+\frac {x \text {Li}_2\left (\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )}{4 b}-\frac {x \text {Li}_2\left (\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{4 b}+\frac {i \text {Li}_3\left (\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )}{8 b^2}-\frac {i \text {Li}_3\left (\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{8 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 270, normalized size = 0.89 \begin {gather*} \frac {1}{2} x^2 \text {ArcTan}(c+d \cot (a+b x))+\frac {i \left (2 b^2 x^2 \log \left (1-\frac {(c+i (-1+d)) e^{2 i (a+b x)}}{c-i (1+d)}\right )-2 b^2 x^2 \log \left (1-\frac {(c+i (1+d)) e^{2 i (a+b x)}}{i+c-i d}\right )-2 i b x \text {PolyLog}\left (2,\frac {(c+i (-1+d)) e^{2 i (a+b x)}}{c-i (1+d)}\right )+2 i b x \text {PolyLog}\left (2,\frac {(c+i (1+d)) e^{2 i (a+b x)}}{i+c-i d}\right )+\text {PolyLog}\left (3,\frac {(c+i (-1+d)) e^{2 i (a+b x)}}{c-i (1+d)}\right )-\text {PolyLog}\left (3,\frac {(c+i (1+d)) e^{2 i (a+b x)}}{i+c-i d}\right )\right )}{8 b^2} \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 = 1.54, size = 7556, normalized size = 24.94
method | result | size |
risch | \(\text {Expression too large to display}\) | \(7556\) |
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 [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1289 vs. \(2 (213) = 426\).
time = 1.56, size = 1289, normalized size = 4.25 \begin {gather*} \frac {8 \, b^{2} x^{2} \arctan \left (d \cot \left (b x + a\right ) + c\right ) + 2 \, b x {\rm Li}_2\left (-\frac {c^{2} + d^{2} - {\left (c^{2} + 2 i \, c d - d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (-i \, c^{2} + 2 \, c d + i \, d^{2} - i\right )} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, d + 1}{c^{2} + d^{2} + 2 \, d + 1} + 1\right ) + 2 \, b x {\rm Li}_2\left (-\frac {c^{2} + d^{2} - {\left (c^{2} - 2 i \, c d - d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (i \, c^{2} + 2 \, c d - i \, d^{2} + i\right )} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, d + 1}{c^{2} + d^{2} + 2 \, d + 1} + 1\right ) - 2 \, b x {\rm Li}_2\left (-\frac {c^{2} + d^{2} - {\left (c^{2} + 2 i \, c d - d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (-i \, c^{2} + 2 \, c d + i \, d^{2} - i\right )} \sin \left (2 \, b x + 2 \, a\right ) - 2 \, d + 1}{c^{2} + d^{2} - 2 \, d + 1} + 1\right ) - 2 \, b x {\rm Li}_2\left (-\frac {c^{2} + d^{2} - {\left (c^{2} - 2 i \, c d - d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (i \, c^{2} + 2 \, c d - i \, d^{2} + i\right )} \sin \left (2 \, b x + 2 \, a\right ) - 2 \, d + 1}{c^{2} + d^{2} - 2 \, d + 1} + 1\right ) + 2 i \, a^{2} \log \left (\frac {1}{2} \, c^{2} + i \, c d - \frac {1}{2} \, d^{2} - \frac {1}{2} \, {\left (c^{2} + d^{2} + 2 \, d + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} \, {\left (i \, c^{2} + i \, d^{2} + 2 i \, d + i\right )} \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) - 2 i \, a^{2} \log \left (\frac {1}{2} \, c^{2} + i \, c d - \frac {1}{2} \, d^{2} - \frac {1}{2} \, {\left (c^{2} + d^{2} - 2 \, d + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} \, {\left (i \, c^{2} + i \, d^{2} - 2 i \, d + i\right )} \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) - 2 i \, a^{2} \log \left (-\frac {1}{2} \, c^{2} + i \, c d + \frac {1}{2} \, d^{2} + \frac {1}{2} \, {\left (c^{2} + d^{2} + 2 \, d + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} \, {\left (i \, c^{2} + i \, d^{2} + 2 i \, d + i\right )} \sin \left (2 \, b x + 2 \, a\right ) - \frac {1}{2}\right ) + 2 i \, a^{2} \log \left (-\frac {1}{2} \, c^{2} + i \, c d + \frac {1}{2} \, d^{2} + \frac {1}{2} \, {\left (c^{2} + d^{2} - 2 \, d + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} \, {\left (i \, c^{2} + i \, d^{2} - 2 i \, d + i\right )} \sin \left (2 \, b x + 2 \, a\right ) - \frac {1}{2}\right ) - 2 \, {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \log \left (\frac {c^{2} + d^{2} - {\left (c^{2} + 2 i \, c d - d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (-i \, c^{2} + 2 \, c d + i \, d^{2} - i\right )} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, d + 1}{c^{2} + d^{2} + 2 \, d + 1}\right ) - 2 \, {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \log \left (\frac {c^{2} + d^{2} - {\left (c^{2} - 2 i \, c d - d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (i \, c^{2} + 2 \, c d - i \, d^{2} + i\right )} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, d + 1}{c^{2} + d^{2} + 2 \, d + 1}\right ) - 2 \, {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \log \left (\frac {c^{2} + d^{2} - {\left (c^{2} + 2 i \, c d - d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (-i \, c^{2} + 2 \, c d + i \, d^{2} - i\right )} \sin \left (2 \, b x + 2 \, a\right ) - 2 \, d + 1}{c^{2} + d^{2} - 2 \, d + 1}\right ) - 2 \, {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \log \left (\frac {c^{2} + d^{2} - {\left (c^{2} - 2 i \, c d - d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (i \, c^{2} + 2 \, c d - i \, d^{2} + i\right )} \sin \left (2 \, b x + 2 \, a\right ) - 2 \, d + 1}{c^{2} + d^{2} - 2 \, d + 1}\right ) + i \, {\rm polylog}\left (3, \frac {{\left (c^{2} + 2 i \, c d - d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (i \, c^{2} - 2 \, c d - i \, d^{2} + i\right )} \sin \left (2 \, b x + 2 \, a\right )}{c^{2} + d^{2} + 2 \, d + 1}\right ) - i \, {\rm polylog}\left (3, \frac {{\left (c^{2} + 2 i \, c d - d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (i \, c^{2} - 2 \, c d - i \, d^{2} + i\right )} \sin \left (2 \, b x + 2 \, a\right )}{c^{2} + d^{2} - 2 \, d + 1}\right ) - i \, {\rm polylog}\left (3, \frac {{\left (c^{2} - 2 i \, c d - d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (-i \, c^{2} - 2 \, c d + i \, d^{2} - i\right )} \sin \left (2 \, b x + 2 \, a\right )}{c^{2} + d^{2} + 2 \, d + 1}\right ) + i \, {\rm polylog}\left (3, \frac {{\left (c^{2} - 2 i \, c d - d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (-i \, c^{2} - 2 \, c d + i \, d^{2} - i\right )} \sin \left (2 \, b x + 2 \, a\right )}{c^{2} + d^{2} - 2 \, d + 1}\right )}{16 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int x\,\mathrm {atan}\left (c+d\,\mathrm {cot}\left (a+b\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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