Optimal. Leaf size=250 \[ -\frac {b c^2 \text {Li}_2\left (\frac {2}{c x+1}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d}+\frac {c^2 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d}-\frac {2 b c^2 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d x}+\frac {b^2 c^2 \text {Li}_2\left (\frac {2}{c x+1}-1\right )}{d}-\frac {b^2 c^2 \text {Li}_3\left (\frac {2}{c x+1}-1\right )}{2 d}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d}+\frac {b^2 c^2 \log (x)}{d} \]
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Rubi [A] time = 0.63, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5934, 5916, 5982, 266, 36, 29, 31, 5948, 5988, 5932, 2447, 6056, 6610} \[ -\frac {b c^2 \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}+\frac {b^2 c^2 \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{d}-\frac {b^2 c^2 \text {PolyLog}\left (3,\frac {2}{c x+1}-1\right )}{2 d}-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d}+\frac {c^2 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d}-\frac {2 b c^2 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d}+\frac {b^2 c^2 \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 2447
Rule 5916
Rule 5932
Rule 5934
Rule 5948
Rule 5982
Rule 5988
Rule 6056
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3 (d+c d x)} \, dx &=-\left (c \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2 (d+c d x)} \, dx\right )+\frac {\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx}{d}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+c^2 \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x (d+c d x)} \, dx-\frac {c \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx}{d}+\frac {(b c) \int \frac {a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx}{d}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {(b c) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx}{d}-\frac {\left (2 b c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{d}+\frac {\left (b c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{d}-\frac {\left (2 b c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d}-\frac {\left (2 b c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx}{d}+\frac {\left (b^2 c^2\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx}{d}+\frac {\left (b^2 c^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}-\frac {2 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d}-\frac {b^2 c^2 \text {Li}_3\left (-1+\frac {2}{1+c x}\right )}{2 d}+\frac {\left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d}+\frac {\left (2 b^2 c^3\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}-\frac {2 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b^2 c^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d}-\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d}-\frac {b^2 c^2 \text {Li}_3\left (-1+\frac {2}{1+c x}\right )}{2 d}+\frac {\left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}+\frac {\left (b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}+\frac {b^2 c^2 \log (x)}{d}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d}-\frac {2 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b^2 c^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d}-\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d}-\frac {b^2 c^2 \text {Li}_3\left (-1+\frac {2}{1+c x}\right )}{2 d}\\ \end {align*}
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Mathematica [C] time = 1.04, size = 317, normalized size = 1.27 \[ \frac {2 a^2 c^2 \log (x)-2 a^2 c^2 \log (c x+1)+\frac {2 a^2 c}{x}-\frac {a^2}{x^2}+\frac {2 a b \left (-c^2 x^2 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )-c x \left (2 c x \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+1\right )+\tanh ^{-1}(c x) \left (c^2 x^2+2 c^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+2 c x-1\right )\right )}{x^2}+2 b^2 c^2 \left (\log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-\frac {\tanh ^{-1}(c x)^2}{2 c^2 x^2}+\tanh ^{-1}(c x) \text {Li}_2\left (e^{2 \tanh ^{-1}(c x)}\right )+\text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )-\frac {1}{2} \text {Li}_3\left (e^{2 \tanh ^{-1}(c x)}\right )-\frac {2}{3} \tanh ^{-1}(c x)^3+\frac {\tanh ^{-1}(c x)^2}{c x}-\frac {1}{2} \tanh ^{-1}(c x)^2-\frac {\tanh ^{-1}(c x)}{c x}+\tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )-2 \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+\frac {i \pi ^3}{24}\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (c x\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x\right ) + a^{2}}{c d x^{4} + d x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.10, size = 1841, normalized size = 7.36 \[ \text {result too large to display} \]
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 \[ -\frac {1}{2} \, {\left (\frac {2 \, c^{2} \log \left (c x + 1\right )}{d} - \frac {2 \, c^{2} \log \relax (x)}{d} - \frac {2 \, c x - 1}{d x^{2}}\right )} a^{2} - \frac {{\left (2 \, b^{2} c^{2} x^{2} \log \left (c x + 1\right ) - 2 \, b^{2} c x + b^{2}\right )} \log \left (-c x + 1\right )^{2}}{8 \, d x^{2}} + \int \frac {{\left (b^{2} c x - b^{2}\right )} \log \left (c x + 1\right )^{2} + 4 \, {\left (a b c x - a b\right )} \log \left (c x + 1\right ) - {\left (2 \, b^{2} c^{3} x^{3} + b^{2} c^{2} x^{2} - 4 \, a b + {\left (4 \, a b c - b^{2} c\right )} x - 2 \, {\left (b^{2} c^{4} x^{4} + b^{2} c^{3} x^{3} - b^{2} c x + b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \, {\left (c^{2} d x^{5} - d x^{3}\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x^3\,\left (d+c\,d\,x\right )} \,d x \]
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 \[ \frac {\int \frac {a^{2}}{c x^{4} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c x^{4} + x^{3}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c x^{4} + x^{3}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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