Integrand size = 12, antiderivative size = 143 \[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {3 i b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d}-\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d} \]
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Time = 0.16 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5148, 4931, 5041, 4965, 5005, 5115, 6745} \[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {3 i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d}-\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )}{2 d} \]
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Rule 4931
Rule 4965
Rule 5005
Rule 5041
Rule 5115
Rule 5148
Rule 6745
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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) \text {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) \text {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 ) \text {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 ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d}+\frac {\left (3 i b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,1-\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 ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d}-\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.59 \[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {2 a^3 (c+d x)+6 a^2 b (c+d x) \cot ^{-1}(c+d x)+3 a^2 b \log \left (1+(c+d x)^2\right )+6 a b^2 \left (\cot ^{-1}(c+d x) \left ((i+c+d x) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )+2 b^3 \left (\frac {i \pi ^3}{8}-i \cot ^{-1}(c+d x)^3+(c+d x) \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \cot ^{-1}(c+d x)}\right )-3 i \cot ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(c+d x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(c+d x)}\right )\right )}{2 d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (136 ) = 272\).
Time = 2.46 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.76
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\operatorname {arccot}\left (d x +c \right )^{3} \left (d x +c -i\right )+2 i \operatorname {arccot}\left (d x +c \right )^{3}-3 \operatorname {arccot}\left (d x +c \right )^{2} \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+6 i \operatorname {arccot}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-3 \operatorname {arccot}\left (d x +c \right )^{2} \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+6 i \operatorname {arccot}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccot}\left (d x +c \right )^{2} \left (d x +c -i\right )-2 \,\operatorname {arccot}\left (d x +c \right ) \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-2 \,\operatorname {arccot}\left (d x +c \right ) \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+2 i \operatorname {arccot}\left (d x +c \right )^{2}+2 i \operatorname {polylog}\left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+2 i \operatorname {polylog}\left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )+3 a^{2} b \left (\operatorname {arccot}\left (d x +c \right ) \left (d x +c \right )+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}\) | \(395\) |
default | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\operatorname {arccot}\left (d x +c \right )^{3} \left (d x +c -i\right )+2 i \operatorname {arccot}\left (d x +c \right )^{3}-3 \operatorname {arccot}\left (d x +c \right )^{2} \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+6 i \operatorname {arccot}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-3 \operatorname {arccot}\left (d x +c \right )^{2} \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+6 i \operatorname {arccot}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccot}\left (d x +c \right )^{2} \left (d x +c -i\right )-2 \,\operatorname {arccot}\left (d x +c \right ) \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-2 \,\operatorname {arccot}\left (d x +c \right ) \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+2 i \operatorname {arccot}\left (d x +c \right )^{2}+2 i \operatorname {polylog}\left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+2 i \operatorname {polylog}\left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )+3 a^{2} b \left (\operatorname {arccot}\left (d x +c \right ) \left (d x +c \right )+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}\) | \(395\) |
parts | \(a^{3} x +\frac {b^{3} \left (\operatorname {arccot}\left (d x +c \right )^{3} \left (d x +c -i\right )+2 i \operatorname {arccot}\left (d x +c \right )^{3}-3 \operatorname {arccot}\left (d x +c \right )^{2} \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+6 i \operatorname {arccot}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-3 \operatorname {arccot}\left (d x +c \right )^{2} \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+6 i \operatorname {arccot}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{d}+\frac {3 a \,b^{2} \left (\operatorname {arccot}\left (d x +c \right )^{2} \left (d x +c -i\right )-2 \,\operatorname {arccot}\left (d x +c \right ) \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-2 \,\operatorname {arccot}\left (d x +c \right ) \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+2 i \operatorname {arccot}\left (d x +c \right )^{2}+2 i \operatorname {polylog}\left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+2 i \operatorname {polylog}\left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{d}+\frac {3 a^{2} b \left (\operatorname {arccot}\left (d x +c \right ) \left (d x +c \right )+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}\) | \(396\) |
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\[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
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\[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {acot}{\left (c + d x \right )}\right )^{3}\, dx \]
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\[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
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\[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
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Timed out. \[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int {\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^3 \,d x \]
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