Optimal. Leaf size=132 \[ -\frac {3 b^2 \text {Li}_2\left (1-\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{-c-d x+1}\right )}{2 d} \]
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Rubi [A] time = 0.23, antiderivative size = 132, 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 = {6104, 5911, 5985, 5919, 5949, 6059, 6610} \[ -\frac {3 b^2 \text {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d}+\frac {3 b^3 \text {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right )}{2 d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d} \]
Antiderivative was successfully verified.
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Rule 5911
Rule 5919
Rule 5949
Rule 5985
Rule 6059
Rule 6104
Rule 6610
Rubi steps
\begin {align*} \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx &=\frac {\operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d}+\frac {\left (3 b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (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 ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-c-d x}\right )}{2 d}\\ \end {align*}
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Mathematica [C] time = 0.33, size = 208, normalized size = 1.58 \[ \frac {2 a^3 (c+d x)+3 a^2 b \log \left (1-(c+d x)^2\right )+6 a^2 b (c+d x) \coth ^{-1}(c+d x)+6 a b^2 \left (\text {Li}_2\left (e^{-2 \coth ^{-1}(c+d x)}\right )+\coth ^{-1}(c+d x) \left ((c+d x-1) \coth ^{-1}(c+d x)-2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )\right )+2 b^3 \left (-3 \coth ^{-1}(c+d x) \text {Li}_2\left (e^{2 \coth ^{-1}(c+d x)}\right )+\frac {3}{2} \text {Li}_3\left (e^{2 \coth ^{-1}(c+d x)}\right )+(c+d x) \coth ^{-1}(c+d x)^3+\coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )-\frac {i \pi ^3}{8}\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} \operatorname {arcoth}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arcoth}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arcoth}\left (d x + c\right ) + a^{3}, x\right ) \]
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 \[ \int {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 485, normalized size = 3.67 \[ a^{3} x +\frac {a^{3} c}{d}+\mathrm {arccoth}\left (d x +c \right )^{3} x \,b^{3}+\frac {\mathrm {arccoth}\left (d x +c \right )^{3} b^{3} c}{d}+\frac {b^{3} \mathrm {arccoth}\left (d x +c \right )^{3}}{d}-\frac {3 \mathrm {arccoth}\left (d x +c \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}}{d}-\frac {3 \mathrm {arccoth}\left (d x +c \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}}{d}-\frac {6 \,\mathrm {arccoth}\left (d x +c \right ) \polylog \left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}}{d}-\frac {6 \,\mathrm {arccoth}\left (d x +c \right ) \polylog \left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}}{d}+\frac {6 \polylog \left (3, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}}{d}+\frac {6 \polylog \left (3, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}}{d}+3 \mathrm {arccoth}\left (d x +c \right )^{2} x a \,b^{2}+\frac {3 \mathrm {arccoth}\left (d x +c \right )^{2} a \,b^{2} c}{d}+\frac {3 a \,b^{2} \mathrm {arccoth}\left (d x +c \right )^{2}}{d}-\frac {6 \,\mathrm {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) a \,b^{2}}{d}-\frac {6 \,\mathrm {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) a \,b^{2}}{d}-\frac {6 \polylog \left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) a \,b^{2}}{d}-\frac {6 \polylog \left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) a \,b^{2}}{d}+3 \,\mathrm {arccoth}\left (d x +c \right ) x \,a^{2} b +\frac {3 \,\mathrm {arccoth}\left (d x +c \right ) a^{2} b c}{d}+\frac {3 a^{2} b \ln \left (\left (d x +c \right )^{2}-1\right )}{2 d} \]
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 \[ a^{3} x + \frac {3 \, {\left (2 \, {\left (d x + c\right )} \operatorname {arcoth}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} a^{2} b}{2 \, d} + \frac {{\left (b^{3} d x + b^{3} {\left (c + 1\right )}\right )} \log \left (d x + c + 1\right )^{3} + 3 \, {\left (2 \, a b^{2} d x - {\left (b^{3} d x + b^{3} {\left (c - 1\right )}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right )^{2}}{8 \, d} + \int -\frac {{\left (b^{3} d x + b^{3} {\left (c + 1\right )}\right )} \log \left (d x + c - 1\right )^{3} - 6 \, {\left (a b^{2} d x + a b^{2} {\left (c + 1\right )}\right )} \log \left (d x + c - 1\right )^{2} + 3 \, {\left (4 \, a b^{2} d x - {\left (b^{3} d x + b^{3} {\left (c + 1\right )}\right )} \log \left (d x + c - 1\right )^{2} + 2 \, {\left (2 \, a b^{2} {\left (c + 1\right )} - b^{3} {\left (c - 1\right )} + {\left (2 \, a b^{2} d - b^{3} d\right )} x\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right )}{8 \, {\left (d x + c + 1\right )}}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^3 \,d x \]
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 \[ \int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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