Optimal. Leaf size=132 \[ \frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d}+\frac {3 b^3 \text {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d} \]
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Rubi [A]
time = 0.16, 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 = {6238, 6021,
6131, 6055, 6095, 6205, 6745} \begin {gather*} -\frac {3 b^2 \text {Li}_2\left (1-\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{d}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-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 6021
Rule 6055
Rule 6095
Rule 6131
Rule 6205
Rule 6238
Rule 6745
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \tanh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{d}-\frac {(3 b) \text {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{d}-\frac {(3 b) \text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{1-x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \tanh ^{-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 \tanh ^{-1}(x)\right ) \log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {3 b^2 \left (a+b \tanh ^{-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 \tanh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {3 b^2 \left (a+b \tanh ^{-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 [A]
time = 0.18, size = 205, normalized size = 1.55 \begin {gather*} \frac {2 a^3 d x+6 a^2 b d x \tanh ^{-1}(c+d x)-3 a^2 b (-1+c) \log (1-c-d x)+3 a^2 b (1+c) \log (1+c+d x)+6 a b^2 \left (\tanh ^{-1}(c+d x) \left ((-1+c+d x) \tanh ^{-1}(c+d x)-2 \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )\right )+\text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )+2 b^3 \left (\tanh ^{-1}(c+d x)^2 \left ((-1+c+d x) \tanh ^{-1}(c+d x)-3 \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )\right )+3 \tanh ^{-1}(c+d x) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )+\frac {3}{2} \text {PolyLog}\left (3,-e^{-2 \tanh ^{-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. \(283\) vs.
\(2(130)=260\).
time = 0.50, size = 284, normalized size = 2.15
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{3}+\left (d x +c \right ) b^{3} \arctanh \left (d x +c \right )^{3}+b^{3} \arctanh \left (d x +c \right )^{3}-3 b^{3} \arctanh \left (d x +c \right )^{2} \ln \left (1+\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )-3 b^{3} \arctanh \left (d x +c \right ) \polylog \left (2, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )+\frac {3 b^{3} \polylog \left (3, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )}{2}+3 \left (d x +c \right ) a \,b^{2} \arctanh \left (d x +c \right )^{2}+3 a \,b^{2} \arctanh \left (d x +c \right )^{2}-6 \arctanh \left (d x +c \right ) \ln \left (1+\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right ) a \,b^{2}-3 \polylog \left (2, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right ) a \,b^{2}+3 \left (d x +c \right ) a^{2} b \arctanh \left (d x +c \right )+\frac {3 a^{2} b \ln \left (1-\left (d x +c \right )^{2}\right )}{2}}{d}\) | \(284\) |
default | \(\frac {\left (d x +c \right ) a^{3}+\left (d x +c \right ) b^{3} \arctanh \left (d x +c \right )^{3}+b^{3} \arctanh \left (d x +c \right )^{3}-3 b^{3} \arctanh \left (d x +c \right )^{2} \ln \left (1+\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )-3 b^{3} \arctanh \left (d x +c \right ) \polylog \left (2, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )+\frac {3 b^{3} \polylog \left (3, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )}{2}+3 \left (d x +c \right ) a \,b^{2} \arctanh \left (d x +c \right )^{2}+3 a \,b^{2} \arctanh \left (d x +c \right )^{2}-6 \arctanh \left (d x +c \right ) \ln \left (1+\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right ) a \,b^{2}-3 \polylog \left (2, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right ) a \,b^{2}+3 \left (d x +c \right ) a^{2} b \arctanh \left (d x +c \right )+\frac {3 a^{2} b \ln \left (1-\left (d x +c \right )^{2}\right )}{2}}{d}\) | \(284\) |
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 {atanh}{\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 {atanh}\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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