Optimal. Leaf size=206 \[ \frac {5}{6} c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {2}{3} b c^3 d \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {b c^2 d \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {b c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {1}{3} b^2 c^3 d \text {Li}_2\left (\frac {2}{c x+1}-1\right )+b^2 c^3 d \log (x)+\frac {1}{3} b^2 c^3 d \tanh ^{-1}(c x)-\frac {b^2 c^2 d}{3 x}-\frac {1}{2} b^2 c^3 d \log \left (1-c^2 x^2\right ) \]
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Rubi [A] time = 0.45, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {5940, 5916, 5982, 325, 206, 5988, 5932, 2447, 266, 36, 29, 31, 5948} \[ -\frac {1}{3} b^2 c^3 d \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+\frac {5}{6} c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c^2 d \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {2}{3} b c^3 d \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {b c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {1}{2} b^2 c^3 d \log \left (1-c^2 x^2\right )-\frac {b^2 c^2 d}{3 x}+b^2 c^3 d \log (x)+\frac {1}{3} b^2 c^3 d \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 206
Rule 266
Rule 325
Rule 2447
Rule 5916
Rule 5932
Rule 5940
Rule 5948
Rule 5982
Rule 5988
Rubi steps
\begin {align*} \int \frac {(d+c d x) \left (a+b \tanh ^{-1}(c x)\right )^2}{x^4} \, dx &=\int \left (\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{x^4}+\frac {c d \left (a+b \tanh ^{-1}(c x)\right )^2}{x^3}\right ) \, dx\\ &=d \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^4} \, dx+(c d) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx\\ &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} (2 b c d) \int \frac {a+b \tanh ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )} \, dx+\left (b c^2 d\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} (2 b c d) \int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx+\left (b c^2 d\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\frac {1}{3} \left (2 b c^3 d\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx+\left (b c^4 d\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx\\ &=-\frac {b c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {b c^2 d \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {5}{6} c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} \left (b^2 c^2 d\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{3} \left (2 b c^3 d\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx+\left (b^2 c^3 d\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {b^2 c^2 d}{3 x}-\frac {b c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {b c^2 d \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {5}{6} c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {2}{3} b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )+\frac {1}{2} \left (b^2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{3} \left (b^2 c^4 d\right ) \int \frac {1}{1-c^2 x^2} \, dx-\frac {1}{3} \left (2 b^2 c^4 d\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^2 d}{3 x}+\frac {1}{3} b^2 c^3 d \tanh ^{-1}(c x)-\frac {b c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {b c^2 d \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {5}{6} c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {2}{3} b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-\frac {1}{3} b^2 c^3 d \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+\frac {1}{2} \left (b^2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} \left (b^2 c^5 d\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2 d}{3 x}+\frac {1}{3} b^2 c^3 d \tanh ^{-1}(c x)-\frac {b c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {b c^2 d \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {5}{6} c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+b^2 c^3 d \log (x)-\frac {1}{2} b^2 c^3 d \log \left (1-c^2 x^2\right )+\frac {2}{3} b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-\frac {1}{3} b^2 c^3 d \text {Li}_2\left (-1+\frac {2}{1+c x}\right )\\ \end {align*}
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Mathematica [A] time = 0.48, size = 246, normalized size = 1.19 \[ -\frac {d \left (3 a^2 c x+2 a^2-4 a b c^3 x^3 \log (c x)+3 a b c^3 x^3 \log (1-c x)-3 a b c^3 x^3 \log (c x+1)+6 a b c^2 x^2+2 a b c^3 x^3 \log \left (1-c^2 x^2\right )+2 b \tanh ^{-1}(c x) \left (a (3 c x+2)-2 b c^3 x^3 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+b c x \left (-c^2 x^2+3 c x+1\right )\right )+2 a b c x+2 b^2 c^3 x^3 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )+b^2 \left (-5 c^3 x^3+3 c x+2\right ) \tanh ^{-1}(c x)^2+2 b^2 c^2 x^2-6 b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )\right )}{6 x^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} c d x + a^{2} d + {\left (b^{2} c d x + b^{2} d\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c d x + a b d\right )} \operatorname {artanh}\left (c x\right )}{x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 440, normalized size = 2.14 \[ -\frac {c d a b \arctanh \left (c x \right )}{x^{2}}-\frac {c^{3} d \,b^{2} \dilog \left (c x +1\right )}{3}-\frac {c^{3} d \,b^{2} \dilog \left (c x \right )}{3}-\frac {d \,b^{2} \arctanh \left (c x \right )^{2}}{3 x^{3}}+\frac {c^{3} d \,b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{3}-\frac {5 c^{3} d \,b^{2} \ln \left (c x -1\right )^{2}}{24}+c^{3} d \,b^{2} \ln \left (c x \right )-\frac {c^{3} d \,b^{2} \ln \left (c x +1\right )^{2}}{24}-\frac {2 c^{3} d \,b^{2} \ln \left (c x -1\right )}{3}-\frac {c^{3} d \,b^{2} \ln \left (c x +1\right )}{3}-\frac {c \,a^{2} d}{2 x^{2}}-\frac {b^{2} c^{2} d}{3 x}-\frac {c^{3} d \,b^{2} \ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {a^{2} d}{3 x^{3}}-\frac {c^{3} d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{12}+\frac {2 c^{3} d a b \ln \left (c x \right )}{3}+\frac {c^{3} d a b \ln \left (c x +1\right )}{6}-\frac {5 c^{3} d a b \ln \left (c x -1\right )}{6}-\frac {c^{2} d a b}{x}-\frac {c d a b}{3 x^{2}}+\frac {5 c^{3} d \,b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{12}+\frac {c^{3} d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{12}-\frac {c d \,b^{2} \arctanh \left (c x \right )^{2}}{2 x^{2}}-\frac {c^{2} d \,b^{2} \arctanh \left (c x \right )}{x}-\frac {c d \,b^{2} \arctanh \left (c x \right )}{3 x^{2}}-\frac {2 d a b \arctanh \left (c x \right )}{3 x^{3}}-\frac {5 c^{3} d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{6}+\frac {c^{3} d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{6}+\frac {2 c^{3} d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x \right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.92, size = 417, normalized size = 2.02 \[ -\frac {1}{3} \, {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} c^{3} d - \frac {1}{3} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b^{2} c^{3} d + \frac {1}{3} \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b^{2} c^{3} d - \frac {1}{3} \, b^{2} c^{3} d \log \left (c x + 1\right ) - \frac {2}{3} \, b^{2} c^{3} d \log \left (c x - 1\right ) + b^{2} c^{3} d \log \relax (x) + \frac {1}{2} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} a b c d - \frac {1}{3} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} a b d - \frac {a^{2} c d}{2 \, x^{2}} - \frac {a^{2} d}{3 \, x^{3}} - \frac {8 \, b^{2} c^{2} d x^{2} - {\left (b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x - 2 \, b^{2} d\right )} \log \left (c x + 1\right )^{2} - {\left (5 \, b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x - 2 \, b^{2} d\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (3 \, b^{2} c^{2} d x^{2} + b^{2} c d x\right )} \log \left (c x + 1\right ) - 2 \, {\left (6 \, b^{2} c^{2} d x^{2} + 2 \, b^{2} c d x - {\left (b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x - 2 \, b^{2} d\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{24 \, x^{3}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right )}{x^4} \,d x \]
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 \[ d \left (\int \frac {a^{2}}{x^{4}}\, dx + \int \frac {a^{2} c}{x^{3}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b c \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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