Optimal. Leaf size=143 \[ \frac {3 i b^2 \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d}-\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{i (c+d x)+1}\right )}{2 d} \]
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Rubi [A] time = 0.22, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5040, 4847, 4921, 4855, 4885, 4995, 6610} \[ \frac {3 i b^2 \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d}-\frac {3 b^3 \text {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d}+\frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d} \]
Antiderivative was successfully verified.
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Rule 4847
Rule 4855
Rule 4885
Rule 4921
Rule 4995
Rule 5040
Rule 6610
Rubi steps
\begin {align*} \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \cot ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {x \left (a+b \cot ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right )^2}{i-x} \, dx,x,c+d x\right )}{d}\\ &=\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right ) \log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {3 i b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{d}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {3 i b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{d}-\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1+i (c+d x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 228, normalized size = 1.59 \[ \frac {2 a^3 (c+d x)+3 a^2 b \log \left ((c+d x)^2+1\right )+6 a^2 b (c+d x) \cot ^{-1}(c+d x)+6 a b^2 \left (i \text {Li}_2\left (e^{2 i \cot ^{-1}(c+d x)}\right )+\cot ^{-1}(c+d x) \left ((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )\right )+2 b^3 \left (-3 i \cot ^{-1}(c+d x) \text {Li}_2\left (e^{-2 i \cot ^{-1}(c+d x)}\right )-\frac {3}{2} \text {Li}_3\left (e^{-2 i \cot ^{-1}(c+d x)}\right )+(c+d x) \cot ^{-1}(c+d x)^3-i \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \cot ^{-1}(c+d x)}\right )+\frac {i \pi ^3}{8}\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} \operatorname {arccot}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arccot}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arccot}\left (d x + c\right ) + a^{3}, x\right ) \]
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 \[ \int {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.45, size = 507, normalized size = 3.55 \[ x \,a^{3}+\frac {a^{3} c}{d}+\frac {i \mathrm {arccot}\left (d x +c \right )^{3} b^{3}}{d}+\mathrm {arccot}\left (d x +c \right )^{3} x \,b^{3}+\frac {\mathrm {arccot}\left (d x +c \right )^{3} b^{3} c}{d}+\frac {3 i \mathrm {arccot}\left (d x +c \right )^{2} a \,b^{2}}{d}-\frac {3 \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) \mathrm {arccot}\left (d x +c \right )^{2} b^{3}}{d}+\frac {6 i \polylog \left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) \mathrm {arccot}\left (d x +c \right ) b^{3}}{d}-\frac {3 \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) \mathrm {arccot}\left (d x +c \right )^{2} b^{3}}{d}-\frac {6 \polylog \left (3, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) b^{3}}{d}-\frac {6 \polylog \left (3, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) b^{3}}{d}+\frac {6 i \polylog \left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) a \,b^{2}}{d}+3 \mathrm {arccot}\left (d x +c \right )^{2} x a \,b^{2}+\frac {3 \mathrm {arccot}\left (d x +c \right )^{2} a \,b^{2} c}{d}+\frac {6 i \polylog \left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) a \,b^{2}}{d}-\frac {6 \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) \mathrm {arccot}\left (d x +c \right ) a \,b^{2}}{d}+\frac {6 i \polylog \left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) \mathrm {arccot}\left (d x +c \right ) b^{3}}{d}-\frac {6 \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) \mathrm {arccot}\left (d x +c \right ) a \,b^{2}}{d}+3 \,\mathrm {arccot}\left (d x +c \right ) x \,a^{2} b +\frac {3 \,\mathrm {arccot}\left (d x +c \right ) a^{2} b c}{d}+\frac {3 a^{2} b \ln \left (1+\left (d x +c \right )^{2}\right )}{2 d} \]
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 \[ \frac {1}{8} \, b^{3} x \arctan \left (1, d x + c\right )^{3} - \frac {3}{32} \, b^{3} x \arctan \left (1, d x + c\right ) \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + a^{3} x + \frac {3 \, {\left (2 \, {\left (d x + c\right )} \operatorname {arccot}\left (d x + c\right ) + \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} a^{2} b}{2 \, d} + \int \frac {28 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 4 \, {\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2}\right )} d^{2} x^{2} + 96 \, a b^{2} \arctan \left (1, d x + c\right )^{2} + 4 \, {\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2}\right )} c^{2} + 4 \, {\left (3 \, b^{3} \arctan \left (1, d x + c\right )^{2} + 2 \, {\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2}\right )} c\right )} d x + 3 \, {\left (b^{3} d^{2} x^{2} \arctan \left (1, d x + c\right ) + b^{3} c^{2} \arctan \left (1, d x + c\right ) + b^{3} \arctan \left (1, d x + c\right ) + {\left (2 \, b^{3} c \arctan \left (1, d x + c\right ) - b^{3}\right )} d x\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + 12 \, {\left (b^{3} d^{2} x^{2} \arctan \left (1, d x + c\right ) + b^{3} c d x \arctan \left (1, d x + c\right )\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{32 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^3 \,d x \]
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 \[ \int \left (a + b \operatorname {acot}{\left (c + d x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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