Optimal. Leaf size=97 \[ \frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d}-\frac {2 b \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d}-\frac {b^2 \text {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right )}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6103, 5910, 5984, 5918, 2402, 2315} \[ -\frac {b^2 \text {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{d}+\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d}-\frac {2 b \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 5910
Rule 5918
Rule 5984
Rule 6103
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \tanh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d}-\frac {2 b \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {\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 )^2}{d}+\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d}-\frac {2 b \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c-d x}\right )}{d}\\ &=\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d}-\frac {2 b \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 107, normalized size = 1.10 \[ \frac {a (a d x+(b-b c) \log (-c-d x+1)+b (c+1) \log (c+d x+1))+2 b \tanh ^{-1}(c+d x) \left (a d x-b \log \left (e^{-2 \tanh ^{-1}(c+d x)}+1\right )\right )+b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c+d x)}\right )+b^2 (c+d x-1) \tanh ^{-1}(c+d x)^2}{d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \operatorname {artanh}\left (d x + c\right )^{2} + 2 \, a b \operatorname {artanh}\left (d x + c\right ) + a^{2}, x\right ) \]
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 \[ \int {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 174, normalized size = 1.79 \[ \arctanh \left (d x +c \right )^{2} x \,b^{2}+\frac {\arctanh \left (d x +c \right )^{2} b^{2} c}{d}+2 \arctanh \left (d x +c \right ) x a b +\frac {b^{2} \arctanh \left (d x +c \right )^{2}}{d}-\frac {2 \arctanh \left (d x +c \right ) \ln \left (1+\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right ) b^{2}}{d}+\frac {2 \arctanh \left (d x +c \right ) a b c}{d}+a^{2} x +\frac {a b \ln \left (1-\left (d x +c \right )^{2}\right )}{d}-\frac {\polylog \left (2, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right ) b^{2}}{d}+\frac {a^{2} c}{d} \]
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 \[ -\frac {1}{4} \, {\left (c d {\left (\frac {{\left (c + 1\right )} \log \left (d x + c + 1\right )}{d^{2}} - \frac {{\left (c - 1\right )} \log \left (d x + c - 1\right )}{d^{2}}\right )} + d^{2} {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )} - 2 \, c d \int \frac {x \log \left (d x + c + 1\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} - 2 \, c^{2} \int \frac {\log \left (d x + c + 1\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} + d {\left (\frac {{\left (c + 1\right )} \log \left (d x + c + 1\right )}{d^{2}} - \frac {{\left (c - 1\right )} \log \left (d x + c - 1\right )}{d^{2}}\right )} - 6 \, d \int \frac {x \log \left (d x + c + 1\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} - 4 \, c \int \frac {\log \left (d x + c + 1\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} - \frac {{\left (d x + c - 1\right )} {\left (\log \left (-d x - c + 1\right )^{2} - 2 \, \log \left (-d x - c + 1\right ) + 2\right )}}{d} - \frac {d x \log \left (d x + c + 1\right )^{2} + 2 \, {\left (d x - {\left (d x + c + 1\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )}{d} - 2 \, \int \frac {\log \left (d x + c + 1\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x}\right )} b^{2} + a^{2} x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} a b}{d} \]
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 \[ \int {\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \operatorname {atanh}{\left (c + d x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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