Optimal. Leaf size=303 \[ -\frac {i \text {Li}_3\left (\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )}{8 b^2}+\frac {i \text {Li}_3\left (\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{8 b^2}-\frac {x \text {Li}_2\left (\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )}{4 b}+\frac {x \text {Li}_2\left (\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{4 b}-\frac {1}{4} i x^2 \log \left (1-\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )+\frac {1}{4} i x^2 \log \left (1-\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )+\frac {1}{2} x^2 \cot ^{-1}(d \cot (a+b x)+c) \]
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Rubi [A] time = 0.42, 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 = {5178, 2190, 2531, 2282, 6589} \[ -\frac {i \text {PolyLog}\left (3,\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )}{8 b^2}+\frac {i \text {PolyLog}\left (3,\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{8 b^2}-\frac {x \text {PolyLog}\left (2,\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )}{4 b}+\frac {x \text {PolyLog}\left (2,\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{4 b}-\frac {1}{4} i x^2 \log \left (1-\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )+\frac {1}{4} i x^2 \log \left (1-\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )+\frac {1}{2} x^2 \cot ^{-1}(d \cot (a+b x)+c) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 5178
Rule 6589
Rubi steps
\begin {align*} \int x \cot ^{-1}(c+d \cot (a+b x)) \, dx &=\frac {1}{2} x^2 \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-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 \operatorname {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 \operatorname {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 \cot ^{-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 = 0.60, size = 270, normalized size = 0.89 \[ \frac {1}{2} x^2 \cot ^{-1}(d \cot (a+b x)+c)-\frac {i \left (2 b^2 x^2 \log \left (1-\frac {(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right )-2 b^2 x^2 \log \left (1-\frac {(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right )-2 i b x \text {Li}_2\left (\frac {(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right )+2 i b x \text {Li}_2\left (\frac {(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right )+\text {Li}_3\left (\frac {(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right )-\text {Li}_3\left (\frac {(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right )\right )}{8 b^2} \]
Antiderivative was successfully verified.
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fricas [C] time = 1.08, size = 1285, normalized size = 4.24 \[ \text {result too large to display} \]
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 \[ \int x \operatorname {arccot}\left (d \cot \left (b x + a\right ) + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.58, size = 7532, normalized size = 24.86 \[ \text {output too large to display} \]
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 \[ \frac {1}{4} \, x^{2} \arctan \left ({\left (d + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + c \sin \left (2 \, b x + 2 \, a\right ) + d - 1, c \cos \left (2 \, b x + 2 \, a\right ) - {\left (d + 1\right )} \sin \left (2 \, b x + 2 \, a\right ) - c\right ) - \frac {1}{4} \, x^{2} \arctan \left ({\left (d - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + c \sin \left (2 \, b x + 2 \, a\right ) + d + 1, c \cos \left (2 \, b x + 2 \, a\right ) - {\left (d - 1\right )} \sin \left (2 \, b x + 2 \, a\right ) - c\right ) - 2 \, b d \int \frac {2 \, {\left (c^{2} + d^{2} + 1\right )} x^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + 2 \, c d x^{2} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, {\left (c^{2} + d^{2} + 1\right )} x^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - {\left (c^{2} - d^{2} + 1\right )} x^{2} \cos \left (2 \, b x + 2 \, a\right ) - {\left (2 \, c d x^{2} \sin \left (2 \, b x + 2 \, a\right ) + {\left (c^{2} - d^{2} + 1\right )} x^{2} \cos \left (2 \, b x + 2 \, a\right )\right )} \cos \left (4 \, b x + 4 \, a\right ) + {\left (2 \, c d x^{2} \cos \left (2 \, b x + 2 \, a\right ) - {\left (c^{2} - d^{2} + 1\right )} x^{2} \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (4 \, b x + 4 \, a\right )}{c^{4} + d^{4} + 2 \, {\left (c^{2} - 1\right )} d^{2} + {\left (c^{4} + d^{4} + 2 \, {\left (c^{2} - 1\right )} d^{2} + 2 \, c^{2} + 1\right )} \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, {\left (c^{4} + d^{4} + 2 \, {\left (c^{2} + 1\right )} d^{2} + 2 \, c^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (c^{4} + d^{4} + 2 \, {\left (c^{2} - 1\right )} d^{2} + 2 \, c^{2} + 1\right )} \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, {\left (c^{4} + d^{4} + 2 \, {\left (c^{2} + 1\right )} d^{2} + 2 \, c^{2} + 1\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, c^{2} + 2 \, {\left (c^{4} + d^{4} - 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} + 2 \, c^{2} - 2 \, {\left (c^{4} - d^{4} + 2 \, c^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 4 \, {\left (c d^{3} + {\left (c^{3} + c\right )} d\right )} \sin \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - 4 \, {\left (c^{4} - d^{4} + 2 \, c^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + 4 \, {\left (2 \, c d^{3} - 2 \, {\left (c^{3} + c\right )} d + 2 \, {\left (c d^{3} + {\left (c^{3} + c\right )} d\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (c^{4} - d^{4} + 2 \, c^{2} + 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (4 \, b x + 4 \, a\right ) + 8 \, {\left (c d^{3} + {\left (c^{3} + c\right )} d\right )} \sin \left (2 \, b x + 2 \, a\right ) + 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 = -0.00 \[ \int x\,\mathrm {acot}\left (c+d\,\mathrm {cot}\left (a+b\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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