Integrand size = 12, antiderivative size = 132 \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d} \]
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Time = 0.19 (sec) , antiderivative size = 132, 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 = {6239, 6022, 6132, 6056, 6096, 6206, 6745} \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=-\frac {3 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right )}{2 d} \]
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Rule 6022
Rule 6056
Rule 6096
Rule 6132
Rule 6206
Rule 6239
Rule 6745
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (a+b \coth ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {(3 b) \text {Subst}\left (\int \frac {x \left (a+b \coth ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {(3 b) \text {Subst}\left (\int \frac {\left (a+b \coth ^{-1}(x)\right )^2}{1-x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int \frac {\left (a+b \coth ^{-1}(x)\right ) \log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.58 \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\frac {2 a^3 (c+d x)+6 a^2 b (c+d x) \coth ^{-1}(c+d x)+3 a^2 b \log \left (1-(c+d x)^2\right )+6 a b^2 \left (\coth ^{-1}(c+d x) \left ((-1+c+d x) \coth ^{-1}(c+d x)-2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )+2 b^3 \left (-\frac {i \pi ^3}{8}+\coth ^{-1}(c+d x)^3+(c+d x) \coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )-3 \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )\right )}{2 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs. \(2(130)=260\).
Time = 0.85 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.82
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\operatorname {arccoth}\left (d x +c \right )^{3} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{3}-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+3 b \,a^{2} \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\) | \(372\) |
default | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\operatorname {arccoth}\left (d x +c \right )^{3} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{3}-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+3 b \,a^{2} \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\) | \(372\) |
parts | \(a^{3} x +\frac {b^{3} \left (\operatorname {arccoth}\left (d x +c \right )^{3} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{3}-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )}{d}+\frac {3 a \,b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )}{d}+\frac {3 b \,a^{2} \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\) | \(373\) |
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\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
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\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{3}\, dx \]
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\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
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\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
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Timed out. \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int {\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^3 \,d x \]
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