Optimal. Leaf size=370 \[ \frac {a b^2 \text {Li}_2\left (1-\frac {2}{a+b x+1}\right )}{\left (1-a^2\right )^2}-\frac {a b^2 \text {Li}_2\left (1-\frac {2 b x}{(1-a) (a+b x+1)}\right )}{\left (1-a^2\right )^2}+\frac {b^2 \log (x)}{\left (1-a^2\right )^2}-\frac {2 a b^2 \log \left (\frac {2}{a+b x+1}\right ) \tanh ^{-1}(a+b x)}{\left (1-a^2\right )^2}+\frac {2 a b^2 \log \left (\frac {2 b x}{(1-a) (a+b x+1)}\right ) \tanh ^{-1}(a+b x)}{\left (1-a^2\right )^2}-\frac {b \tanh ^{-1}(a+b x)}{\left (1-a^2\right ) x}+\frac {b^2 \text {Li}_2\left (-\frac {a+b x+1}{-a-b x+1}\right )}{4 (1-a)^2}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{a+b x+1}\right )}{4 (a+1)^2}-\frac {b^2 \log (-a-b x+1)}{2 (1-a)^2 (a+1)}-\frac {b^2 \log (a+b x+1)}{2 (1-a) (a+1)^2}+\frac {b^2 \log \left (\frac {2}{-a-b x+1}\right ) \tanh ^{-1}(a+b x)}{2 (1-a)^2}-\frac {b^2 \log \left (\frac {2}{a+b x+1}\right ) \tanh ^{-1}(a+b x)}{2 (a+1)^2}-\frac {\tanh ^{-1}(a+b x)^2}{2 x^2} \]
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Rubi [A] time = 0.80, antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 16, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {6109, 371, 710, 801, 6741, 6121, 6725, 5926, 706, 31, 633, 5920, 2402, 2315, 2447, 5918} \[ \frac {a b^2 \text {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )}{\left (1-a^2\right )^2}-\frac {a b^2 \text {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (a+b x+1)}\right )}{\left (1-a^2\right )^2}+\frac {b^2 \text {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )}{4 (1-a)^2}+\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )}{4 (a+1)^2}+\frac {b^2 \log (x)}{\left (1-a^2\right )^2}-\frac {2 a b^2 \log \left (\frac {2}{a+b x+1}\right ) \tanh ^{-1}(a+b x)}{\left (1-a^2\right )^2}+\frac {2 a b^2 \log \left (\frac {2 b x}{(1-a) (a+b x+1)}\right ) \tanh ^{-1}(a+b x)}{\left (1-a^2\right )^2}-\frac {b \tanh ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {b^2 \log (-a-b x+1)}{2 (1-a)^2 (a+1)}-\frac {b^2 \log (a+b x+1)}{2 (1-a) (a+1)^2}+\frac {b^2 \log \left (\frac {2}{-a-b x+1}\right ) \tanh ^{-1}(a+b x)}{2 (1-a)^2}-\frac {b^2 \log \left (\frac {2}{a+b x+1}\right ) \tanh ^{-1}(a+b x)}{2 (a+1)^2}-\frac {\tanh ^{-1}(a+b x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 371
Rule 633
Rule 706
Rule 710
Rule 801
Rule 2315
Rule 2402
Rule 2447
Rule 5918
Rule 5920
Rule 5926
Rule 6109
Rule 6121
Rule 6725
Rule 6741
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a+b x)^2}{x^3} \, dx &=-\frac {\tanh ^{-1}(a+b x)^2}{2 x^2}+b \int \frac {\tanh ^{-1}(a+b x)}{x^2 \left (1-(a+b x)^2\right )} \, dx\\ &=-\frac {\tanh ^{-1}(a+b x)^2}{2 x^2}+b \int \frac {\tanh ^{-1}(a+b x)}{x^2 \left (1-a^2-2 a b x-b^2 x^2\right )} \, dx\\ &=-\frac {\tanh ^{-1}(a+b x)^2}{2 x^2}+\operatorname {Subst}\left (\int \frac {\tanh ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \left (1-x^2\right )} \, dx,x,a+b x\right )\\ &=-\frac {\tanh ^{-1}(a+b x)^2}{2 x^2}+\operatorname {Subst}\left (\int \left (-\frac {b^2 \tanh ^{-1}(x)}{\left (-1+a^2\right ) (a-x)^2}-\frac {2 a b^2 \tanh ^{-1}(x)}{\left (-1+a^2\right )^2 (a-x)}-\frac {b^2 \tanh ^{-1}(x)}{2 (-1+a)^2 (-1+x)}+\frac {b^2 \tanh ^{-1}(x)}{2 (1+a)^2 (1+x)}\right ) \, dx,x,a+b x\right )\\ &=-\frac {\tanh ^{-1}(a+b x)^2}{2 x^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\tanh ^{-1}(x)}{-1+x} \, dx,x,a+b x\right )}{2 (1-a)^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\tanh ^{-1}(x)}{1+x} \, dx,x,a+b x\right )}{2 (1+a)^2}-\frac {\left (2 a b^2\right ) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}(x)}{a-x} \, dx,x,a+b x\right )}{\left (1-a^2\right )^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\tanh ^{-1}(x)}{(a-x)^2} \, dx,x,a+b x\right )}{1-a^2}\\ &=-\frac {b \tanh ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {\tanh ^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{2 (1-a)^2}-\frac {b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{2 (1+a)^2}-\frac {2 a b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{\left (1-a^2\right )^2}+\frac {2 a b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )}{\left (1-a^2\right )^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,a+b x\right )}{2 (1-a)^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx,x,a+b x\right )}{2 (1+a)^2}+\frac {\left (2 a b^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx,x,a+b x\right )}{\left (1-a^2\right )^2}-\frac {\left (2 a b^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 (a-x)}{(-1+a) (1+x)}\right )}{1-x^2} \, dx,x,a+b x\right )}{\left (1-a^2\right )^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{(a-x) \left (1-x^2\right )} \, dx,x,a+b x\right )}{1-a^2}\\ &=-\frac {b \tanh ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {\tanh ^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{2 (1-a)^2}-\frac {b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{2 (1+a)^2}-\frac {2 a b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{\left (1-a^2\right )^2}+\frac {2 a b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )}{\left (1-a^2\right )^2}-\frac {a b^2 \text {Li}_2\left (1-\frac {2 b x}{(1-a) (1+a+b x)}\right )}{\left (1-a^2\right )^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a-b x}\right )}{2 (1-a)^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+a+b x}\right )}{2 (1+a)^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a-x} \, dx,x,a+b x\right )}{\left (1-a^2\right )^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {-a-x}{1-x^2} \, dx,x,a+b x\right )}{\left (1-a^2\right )^2}+\frac {\left (2 a b^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+a+b x}\right )}{\left (1-a^2\right )^2}\\ &=-\frac {b \tanh ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {\tanh ^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \log (x)}{\left (1-a^2\right )^2}+\frac {b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{2 (1-a)^2}-\frac {b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{2 (1+a)^2}-\frac {2 a b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{\left (1-a^2\right )^2}+\frac {2 a b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )}{\left (1-a^2\right )^2}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-a-b x}\right )}{4 (1-a)^2}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1+a+b x}\right )}{4 (1+a)^2}+\frac {a b^2 \text {Li}_2\left (1-\frac {2}{1+a+b x}\right )}{\left (1-a^2\right )^2}-\frac {a b^2 \text {Li}_2\left (1-\frac {2 b x}{(1-a) (1+a+b x)}\right )}{\left (1-a^2\right )^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{-1-x} \, dx,x,a+b x\right )}{2 (1-a) (1+a)^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,a+b x\right )}{2 (1-a)^2 (1+a)}\\ &=-\frac {b \tanh ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {\tanh ^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \log (x)}{\left (1-a^2\right )^2}+\frac {b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{2 (1-a)^2}-\frac {b^2 \log (1-a-b x)}{2 (1-a)^2 (1+a)}-\frac {b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{2 (1+a)^2}-\frac {2 a b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{\left (1-a^2\right )^2}+\frac {2 a b^2 \tanh ^{-1}(a+b x) \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )}{\left (1-a^2\right )^2}-\frac {b^2 \log (1+a+b x)}{2 (1-a) (1+a)^2}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-a-b x}\right )}{4 (1-a)^2}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1+a+b x}\right )}{4 (1+a)^2}+\frac {a b^2 \text {Li}_2\left (1-\frac {2}{1+a+b x}\right )}{\left (1-a^2\right )^2}-\frac {a b^2 \text {Li}_2\left (1-\frac {2 b x}{(1-a) (1+a+b x)}\right )}{\left (1-a^2\right )^2}\\ \end {align*}
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Mathematica [C] time = 2.35, size = 271, normalized size = 0.73 \[ \frac {2 b x \tanh ^{-1}(a+b x) \left (a^2+a b x+i \pi a b x-2 a b x \tanh ^{-1}(a)+2 a b x \log \left (1-e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right )-1\right )-\left (a^4-a^2 \left (b^2 x^2+2\right )-b^2 x^2 \left (2 \sqrt {1-a^2} e^{\tanh ^{-1}(a)}-1\right )+1\right ) \tanh ^{-1}(a+b x)^2-2 a b^2 x^2 \text {Li}_2\left (e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right )+2 b^2 x^2 \left (i \pi a \log \left (\frac {1}{\sqrt {1-(a+b x)^2}}\right )+\log \left (-\frac {b x}{\sqrt {1-(a+b x)^2}}\right )-i \pi a \log \left (e^{2 \tanh ^{-1}(a+b x)}+1\right )-2 a \tanh ^{-1}(a) \left (\log \left (1-e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right )-\log \left (-i \sinh \left (\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right )\right )\right )\right )}{2 \left (a^2-1\right )^2 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (b x + a\right )^{2}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 467, normalized size = 1.26 \[ -\frac {\arctanh \left (b x +a \right )^{2}}{2 x^{2}}-\frac {b^{2} \arctanh \left (b x +a \right ) \ln \left (b x +a -1\right )}{2 \left (a -1\right )^{2}}+\frac {b^{2} \arctanh \left (b x +a \right ) \ln \left (b x +a +1\right )}{2 \left (1+a \right )^{2}}+\frac {b \arctanh \left (b x +a \right )}{\left (a -1\right ) \left (1+a \right ) x}+\frac {2 b^{2} \arctanh \left (b x +a \right ) a \ln \left (b x \right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}}-\frac {b^{2} \ln \left (b x +a -1\right )^{2}}{8 \left (a -1\right )^{2}}+\frac {b^{2} \dilog \left (\frac {1}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{4 \left (a -1\right )^{2}}+\frac {b^{2} \ln \left (b x +a -1\right ) \ln \left (\frac {1}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{4 \left (a -1\right )^{2}}-\frac {b^{2} \ln \left (b x +a +1\right )^{2}}{8 \left (1+a \right )^{2}}+\frac {b^{2} \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) \ln \left (b x +a +1\right )}{4 \left (1+a \right )^{2}}-\frac {b^{2} \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{4 \left (1+a \right )^{2}}-\frac {b^{2} \dilog \left (\frac {1}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{4 \left (1+a \right )^{2}}-\frac {b^{2} \ln \left (b x +a -1\right )}{\left (a -1\right ) \left (1+a \right ) \left (2 a -2\right )}+\frac {b^{2} \ln \left (b x +a +1\right )}{\left (a -1\right ) \left (1+a \right ) \left (2+2 a \right )}+\frac {b^{2} \ln \left (b x \right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}}+\frac {b^{2} a \dilog \left (\frac {b x +a -1}{a -1}\right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}}+\frac {b^{2} a \ln \left (b x \right ) \ln \left (\frac {b x +a -1}{a -1}\right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}}-\frac {b^{2} a \dilog \left (\frac {b x +a +1}{1+a}\right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}}-\frac {b^{2} a \ln \left (b x \right ) \ln \left (\frac {b x +a +1}{1+a}\right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 360, normalized size = 0.97 \[ \frac {1}{8} \, {\left (\frac {8 \, {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )} a}{a^{4} - 2 \, a^{2} + 1} - \frac {8 \, {\left (\log \left (\frac {b x}{a + 1} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {b x}{a + 1}\right )\right )} a}{a^{4} - 2 \, a^{2} + 1} + \frac {8 \, {\left (\log \left (\frac {b x}{a - 1} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {b x}{a - 1}\right )\right )} a}{a^{4} - 2 \, a^{2} + 1} - \frac {{\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 2 \, {\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + {\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a - 1\right )^{2}}{a^{4} - 2 \, a^{2} + 1} + \frac {4 \, \log \left (b x + a + 1\right )}{a^{3} + a^{2} - a - 1} - \frac {4 \, \log \left (b x + a - 1\right )}{a^{3} - a^{2} - a + 1} + \frac {8 \, \log \relax (x)}{a^{4} - 2 \, a^{2} + 1}\right )} b^{2} + \frac {1}{2} \, {\left (\frac {4 \, a b \log \relax (x)}{a^{4} - 2 \, a^{2} + 1} + \frac {b \log \left (b x + a + 1\right )}{a^{2} + 2 \, a + 1} - \frac {b \log \left (b x + a - 1\right )}{a^{2} - 2 \, a + 1} + \frac {2}{{\left (a^{2} - 1\right )} x}\right )} b \operatorname {artanh}\left (b x + a\right ) - \frac {\operatorname {artanh}\left (b x + a\right )^{2}}{2 \, x^{2}} \]
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 \[ \int \frac {{\mathrm {atanh}\left (a+b\,x\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {atanh}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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