Optimal. Leaf size=204 \[ -\frac {\left (3 a^2+1\right ) \text {Li}_2\left (-\frac {a+b x+1}{-a-b x+1}\right )}{3 b^3}+\frac {a \left (a^2+3\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {\left (3 a^2+1\right ) \coth ^{-1}(a+b x)^2}{3 b^3}-\frac {2 \left (3 a^2+1\right ) \log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{3 b^3}-\frac {a \log \left (1-(a+b x)^2\right )}{b^3}-\frac {\tanh ^{-1}(a+b x)}{3 b^3}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{3 b^3}-\frac {2 a (a+b x) \coth ^{-1}(a+b x)}{b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2+\frac {x}{3 b^2} \]
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Rubi [A] time = 0.28, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6112, 5929, 5911, 260, 5917, 321, 206, 6049, 5949, 5985, 5919, 2402, 2315} \[ -\frac {\left (3 a^2+1\right ) \text {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )}{3 b^3}+\frac {a \left (a^2+3\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {\left (3 a^2+1\right ) \coth ^{-1}(a+b x)^2}{3 b^3}-\frac {2 \left (3 a^2+1\right ) \log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{3 b^3}-\frac {a \log \left (1-(a+b x)^2\right )}{b^3}-\frac {\tanh ^{-1}(a+b x)}{3 b^3}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{3 b^3}-\frac {2 a (a+b x) \coth ^{-1}(a+b x)}{b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2+\frac {x}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 260
Rule 321
Rule 2315
Rule 2402
Rule 5911
Rule 5917
Rule 5919
Rule 5929
Rule 5949
Rule 5985
Rule 6049
Rule 6112
Rubi steps
\begin {align*} \int x^2 \coth ^{-1}(a+b x)^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \coth ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2-\frac {2}{3} \operatorname {Subst}\left (\int \left (\frac {3 a \coth ^{-1}(x)}{b^3}-\frac {x \coth ^{-1}(x)}{b^3}-\frac {\left (a \left (3+a^2\right )-\left (1+3 a^2\right ) x\right ) \coth ^{-1}(x)}{b^3 \left (1-x^2\right )}\right ) \, dx,x,a+b x\right )\\ &=\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2+\frac {2 \operatorname {Subst}\left (\int x \coth ^{-1}(x) \, dx,x,a+b x\right )}{3 b^3}+\frac {2 \operatorname {Subst}\left (\int \frac {\left (a \left (3+a^2\right )-\left (1+3 a^2\right ) x\right ) \coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{3 b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \coth ^{-1}(x) \, dx,x,a+b x\right )}{b^3}\\ &=-\frac {2 a (a+b x) \coth ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2-\frac {\operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,a+b x\right )}{3 b^3}+\frac {2 \operatorname {Subst}\left (\int \left (\frac {a \left (3+a^2\right ) \coth ^{-1}(x)}{1-x^2}-\frac {\left (1+3 a^2\right ) x \coth ^{-1}(x)}{1-x^2}\right ) \, dx,x,a+b x\right )}{3 b^3}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,a+b x\right )}{b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \coth ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2-\frac {a \log \left (1-(a+b x)^2\right )}{b^3}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,a+b x\right )}{3 b^3}+\frac {\left (2 a \left (3+a^2\right )\right ) \operatorname {Subst}\left (\int \frac {\coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{3 b^3}-\frac {\left (2 \left (1+3 a^2\right )\right ) \operatorname {Subst}\left (\int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{3 b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \coth ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{3 b^3}+\frac {a \left (3+a^2\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {\left (1+3 a^2\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2-\frac {\tanh ^{-1}(a+b x)}{3 b^3}-\frac {a \log \left (1-(a+b x)^2\right )}{b^3}-\frac {\left (2 \left (1+3 a^2\right )\right ) \operatorname {Subst}\left (\int \frac {\coth ^{-1}(x)}{1-x} \, dx,x,a+b x\right )}{3 b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \coth ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{3 b^3}+\frac {a \left (3+a^2\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {\left (1+3 a^2\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2-\frac {\tanh ^{-1}(a+b x)}{3 b^3}-\frac {2 \left (1+3 a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{3 b^3}-\frac {a \log \left (1-(a+b x)^2\right )}{b^3}+\frac {\left (2 \left (1+3 a^2\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,a+b x\right )}{3 b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \coth ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{3 b^3}+\frac {a \left (3+a^2\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {\left (1+3 a^2\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2-\frac {\tanh ^{-1}(a+b x)}{3 b^3}-\frac {2 \left (1+3 a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{3 b^3}-\frac {a \log \left (1-(a+b x)^2\right )}{b^3}-\frac {\left (2 \left (1+3 a^2\right )\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a-b x}\right )}{3 b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \coth ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{3 b^3}+\frac {a \left (3+a^2\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {\left (1+3 a^2\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2-\frac {\tanh ^{-1}(a+b x)}{3 b^3}-\frac {2 \left (1+3 a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{3 b^3}-\frac {a \log \left (1-(a+b x)^2\right )}{b^3}-\frac {\left (1+3 a^2\right ) \text {Li}_2\left (1-\frac {2}{1-a-b x}\right )}{3 b^3}\\ \end {align*}
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Mathematica [B] time = 4.54, size = 607, normalized size = 2.98 \[ -\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \left (1-(a+b x)^2\right ) \left (\frac {4 \left (3 a^2+1\right ) \text {Li}_2\left (e^{-2 \coth ^{-1}(a+b x)}\right )}{(a+b x)^3 \left (1-\frac {1}{(a+b x)^2}\right )^{3/2}}+\frac {9 a^2 \coth ^{-1}(a+b x)^2}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {-3 \left (a^2-1\right ) \coth ^{-1}(a+b x)^2+6 a \coth ^{-1}(a+b x)-1}{\sqrt {1-\frac {1}{(a+b x)^2}}}+3 a^2 \coth ^{-1}(a+b x)^2 \cosh \left (3 \coth ^{-1}(a+b x)\right )+\frac {18 a^2 \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}-3 a^2 \coth ^{-1}(a+b x)^2 \sinh \left (3 \coth ^{-1}(a+b x)\right )-6 a^2 \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right ) \sinh \left (3 \coth ^{-1}(a+b x)\right )-\frac {18 a \log \left (\frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}-\frac {12 a \coth ^{-1}(a+b x)^2}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {3 \coth ^{-1}(a+b x)^2}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {4 \coth ^{-1}(a+b x)}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}-6 a \coth ^{-1}(a+b x) \cosh \left (3 \coth ^{-1}(a+b x)\right )+\coth ^{-1}(a+b x)^2 \cosh \left (3 \coth ^{-1}(a+b x)\right )+\cosh \left (3 \coth ^{-1}(a+b x)\right )+\frac {6 \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}-\coth ^{-1}(a+b x)^2 \sinh \left (3 \coth ^{-1}(a+b x)\right )+6 a \log \left (\frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right ) \sinh \left (3 \coth ^{-1}(a+b x)\right )-2 \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right ) \sinh \left (3 \coth ^{-1}(a+b x)\right )\right )}{12 b^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {arcoth}\left (b x + a\right )^{2}, x\right ) \]
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 \[ \int x^{2} \operatorname {arcoth}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 729, normalized size = 3.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 259, normalized size = 1.27 \[ \frac {1}{3} \, x^{3} \operatorname {arcoth}\left (b x + a\right )^{2} - \frac {1}{12} \, b^{2} {\left (\frac {4 \, {\left (3 \, a^{2} + 1\right )} {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )}}{b^{5}} + \frac {2 \, {\left (5 \, a^{2} + 6 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{5}} + \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 2 \, {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )^{2} - 4 \, b x - 2 \, {\left (5 \, a^{2} - 6 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{5}}\right )} + \frac {1}{3} \, b {\left (\frac {b x^{2} - 4 \, a x}{b^{3}} + \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{4}} - \frac {{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )}{b^{4}}\right )} \operatorname {arcoth}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\mathrm {acoth}\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int x^{2} \operatorname {acoth}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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