Optimal. Leaf size=132 \[ \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d}+\frac {3 b^3 \text {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d} \]
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Rubi [A]
time = 0.15, 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 = {6239, 6022,
6132, 6056, 6096, 6206, 6745} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 6022
Rule 6056
Rule 6096
Rule 6132
Rule 6206
Rule 6239
Rule 6745
Rubi steps
\begin {align*} \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx &=\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 ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d}+\frac {\left (3 b^3\right ) \text {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] Result contains complex when optimal does not.
time = 0.20, size = 208, normalized size = 1.58 \begin {gather*} \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 )+\text {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) \text {PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )+\frac {3}{2} \text {PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs.
\(2(130)=260\).
time = 0.65, size = 408, normalized size = 3.09
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{3}+\mathrm {arccoth}\left (d x +c \right )^{3} b^{3} \left (d x +c \right )+b^{3} \mathrm {arccoth}\left (d x +c \right )^{3}-3 \mathrm {arccoth}\left (d x +c \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}-3 \mathrm {arccoth}\left (d x +c \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}-6 \,\mathrm {arccoth}\left (d x +c \right ) \polylog \left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}-6 \,\mathrm {arccoth}\left (d x +c \right ) \polylog \left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}+6 \polylog \left (3, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}+6 \polylog \left (3, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}+3 \mathrm {arccoth}\left (d x +c \right )^{2} a \,b^{2} \left (d x +c \right )+3 a \,b^{2} \mathrm {arccoth}\left (d x +c \right )^{2}-6 \,\mathrm {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) a \,b^{2}-6 \,\mathrm {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) a \,b^{2}-6 \polylog \left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) a \,b^{2}-6 \polylog \left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) a \,b^{2}+3 a^{2} b \left (d x +c \right ) \mathrm {arccoth}\left (d x +c \right )+\frac {3 a^{2} b \ln \left (\left (d x +c \right )^{2}-1\right )}{2}}{d}\) | \(408\) |
default | \(\frac {\left (d x +c \right ) a^{3}+\mathrm {arccoth}\left (d x +c \right )^{3} b^{3} \left (d x +c \right )+b^{3} \mathrm {arccoth}\left (d x +c \right )^{3}-3 \mathrm {arccoth}\left (d x +c \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}-3 \mathrm {arccoth}\left (d x +c \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}-6 \,\mathrm {arccoth}\left (d x +c \right ) \polylog \left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}-6 \,\mathrm {arccoth}\left (d x +c \right ) \polylog \left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}+6 \polylog \left (3, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}+6 \polylog \left (3, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) b^{3}+3 \mathrm {arccoth}\left (d x +c \right )^{2} a \,b^{2} \left (d x +c \right )+3 a \,b^{2} \mathrm {arccoth}\left (d x +c \right )^{2}-6 \,\mathrm {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) a \,b^{2}-6 \,\mathrm {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) a \,b^{2}-6 \polylog \left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) a \,b^{2}-6 \polylog \left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right ) a \,b^{2}+3 a^{2} b \left (d x +c \right ) \mathrm {arccoth}\left (d x +c \right )+\frac {3 a^{2} b \ln \left (\left (d x +c \right )^{2}-1\right )}{2}}{d}\) | \(408\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{3}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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