Optimal. Leaf size=146 \[ \frac {c^2 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {a+b \tanh ^{-1}(c x)}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac {b c^2 \text {Li}_2\left (\frac {2}{c x+1}-1\right )}{2 d}+\frac {b c^2 \log \left (1-c^2 x^2\right )}{2 d}-\frac {b c^2 \log (x)}{d}+\frac {b c^2 \tanh ^{-1}(c x)}{2 d}-\frac {b c}{2 d x} \]
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Rubi [A] time = 0.23, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5934, 5916, 325, 206, 266, 36, 29, 31, 5932, 2447} \[ -\frac {b c^2 \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{2 d}+\frac {c^2 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {a+b \tanh ^{-1}(c x)}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{d x}+\frac {b c^2 \log \left (1-c^2 x^2\right )}{2 d}-\frac {b c^2 \log (x)}{d}+\frac {b c^2 \tanh ^{-1}(c x)}{2 d}-\frac {b c}{2 d x} \]
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 5934
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{x^3 (d+c d x)} \, dx &=-\left (c \int \frac {a+b \tanh ^{-1}(c x)}{x^2 (d+c d x)} \, dx\right )+\frac {\int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx}{d}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{2 d x^2}+c^2 \int \frac {a+b \tanh ^{-1}(c x)}{x (d+c d x)} \, dx-\frac {c \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx}{d}+\frac {(b c) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx}{2 d}\\ &=-\frac {b c}{2 d x}-\frac {a+b \tanh ^{-1}(c x)}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{d x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {\left (b c^2\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx}{d}+\frac {\left (b c^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{2 d}-\frac {\left (b c^3\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac {b c}{2 d x}+\frac {b c^2 \tanh ^{-1}(c x)}{2 d}-\frac {a+b \tanh ^{-1}(c x)}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{d x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {b c^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{2 d}-\frac {\left (b c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {b c}{2 d x}+\frac {b c^2 \tanh ^{-1}(c x)}{2 d}-\frac {a+b \tanh ^{-1}(c x)}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{d x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {b c^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{2 d}-\frac {\left (b c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}-\frac {\left (b c^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {b c}{2 d x}+\frac {b c^2 \tanh ^{-1}(c x)}{2 d}-\frac {a+b \tanh ^{-1}(c x)}{2 d x^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac {b c^2 \log (x)}{d}+\frac {b c^2 \log \left (1-c^2 x^2\right )}{2 d}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {b c^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 133, normalized size = 0.91 \[ -\frac {-2 a c^2 x^2 \log (x)+2 a c^2 x^2 \log (c x+1)-2 a c x+a+b c^2 x^2 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )+2 b c^2 x^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-b \tanh ^{-1}(c x) \left (c^2 x^2+2 c^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+2 c x-1\right )+b c x}{2 d x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c x\right ) + a}{c d x^{4} + d x^{3}}, x\right ) \]
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 \[ \int \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 286, normalized size = 1.96 \[ -\frac {a}{2 d \,x^{2}}+\frac {c^{2} a \ln \left (c x \right )}{d}+\frac {c a}{d x}-\frac {c^{2} a \ln \left (c x +1\right )}{d}-\frac {b \arctanh \left (c x \right )}{2 d \,x^{2}}+\frac {c^{2} b \arctanh \left (c x \right ) \ln \left (c x \right )}{d}+\frac {c b \arctanh \left (c x \right )}{d x}-\frac {c^{2} b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d}-\frac {c^{2} b \dilog \left (c x \right )}{2 d}-\frac {c^{2} b \dilog \left (c x +1\right )}{2 d}-\frac {c^{2} b \ln \left (c x \right ) \ln \left (c x +1\right )}{2 d}+\frac {c^{2} b \ln \left (c x +1\right )^{2}}{4 d}-\frac {c^{2} b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d}+\frac {c^{2} b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 d}+\frac {c^{2} b \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 d}-\frac {b c}{2 d x}-\frac {c^{2} b \ln \left (c x \right )}{d}+\frac {c^{2} b \ln \left (c x -1\right )}{4 d}+\frac {3 c^{2} b \ln \left (c x +1\right )}{4 d} \]
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 \[ -\frac {1}{2} \, {\left (\frac {2 \, c^{2} \log \left (c x + 1\right )}{d} - \frac {2 \, c^{2} \log \relax (x)}{d} - \frac {2 \, c x - 1}{d x^{2}}\right )} a + \frac {1}{2} \, b \int \frac {\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{c d x^{4} + d x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x^3\,\left (d+c\,d\,x\right )} \,d x \]
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 \[ \frac {\int \frac {a}{c x^{4} + x^{3}}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c x^{4} + x^{3}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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