Optimal. Leaf size=218 \[ -\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )}{d^3 (c x+1)}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d^3 (c x+1)^2}-\frac {a+b \tanh ^{-1}(c x)}{d^3 x}-\frac {3 c \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^3}-\frac {3 a c \log (x)}{d^3}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d^3}+\frac {3 b c \text {Li}_2(-c x)}{2 d^3}-\frac {3 b c \text {Li}_2(c x)}{2 d^3}+\frac {3 b c \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 d^3}-\frac {9 b c}{8 d^3 (c x+1)}-\frac {b c}{8 d^3 (c x+1)^2}+\frac {b c \log (x)}{d^3}+\frac {9 b c \tanh ^{-1}(c x)}{8 d^3} \]
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Rubi [A] time = 0.27, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5940, 5916, 266, 36, 29, 31, 5912, 5926, 627, 44, 207, 5918, 2402, 2315} \[ \frac {3 b c \text {PolyLog}(2,-c x)}{2 d^3}-\frac {3 b c \text {PolyLog}(2,c x)}{2 d^3}+\frac {3 b c \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 d^3}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )}{d^3 (c x+1)}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d^3 (c x+1)^2}-\frac {a+b \tanh ^{-1}(c x)}{d^3 x}-\frac {3 c \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^3}-\frac {3 a c \log (x)}{d^3}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d^3}-\frac {9 b c}{8 d^3 (c x+1)}-\frac {b c}{8 d^3 (c x+1)^2}+\frac {b c \log (x)}{d^3}+\frac {9 b c \tanh ^{-1}(c x)}{8 d^3} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 207
Rule 266
Rule 627
Rule 2315
Rule 2402
Rule 5912
Rule 5916
Rule 5918
Rule 5926
Rule 5940
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{x^2 (d+c d x)^3} \, dx &=\int \left (\frac {a+b \tanh ^{-1}(c x)}{d^3 x^2}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right )}{d^3 x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^3 (1+c x)^3}+\frac {2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^3 (1+c x)^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^3 (1+c x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx}{d^3}-\frac {(3 c) \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx}{d^3}+\frac {c^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx}{d^3}+\frac {\left (2 c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{d^3}+\frac {\left (3 c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1+c x} \, dx}{d^3}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )}{d^3 (1+c x)}-\frac {3 a c \log (x)}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {3 b c \text {Li}_2(-c x)}{2 d^3}-\frac {3 b c \text {Li}_2(c x)}{2 d^3}+\frac {(b c) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx}{d^3}+\frac {\left (b c^2\right ) \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx}{2 d^3}+\frac {\left (2 b c^2\right ) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{d^3}+\frac {\left (3 b c^2\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d^3}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )}{d^3 (1+c x)}-\frac {3 a c \log (x)}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {3 b c \text {Li}_2(-c x)}{2 d^3}-\frac {3 b c \text {Li}_2(c x)}{2 d^3}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d^3}+\frac {(3 b c) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{d^3}+\frac {\left (b c^2\right ) \int \frac {1}{(1-c x) (1+c x)^3} \, dx}{2 d^3}+\frac {\left (2 b c^2\right ) \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{d^3}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )}{d^3 (1+c x)}-\frac {3 a c \log (x)}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {3 b c \text {Li}_2(-c x)}{2 d^3}-\frac {3 b c \text {Li}_2(c x)}{2 d^3}+\frac {3 b c \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 d^3}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d^3}+\frac {\left (b c^2\right ) \int \left (\frac {1}{2 (1+c x)^3}+\frac {1}{4 (1+c x)^2}-\frac {1}{4 \left (-1+c^2 x^2\right )}\right ) \, dx}{2 d^3}+\frac {\left (2 b c^2\right ) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{d^3}+\frac {\left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )}{2 d^3}\\ &=-\frac {b c}{8 d^3 (1+c x)^2}-\frac {9 b c}{8 d^3 (1+c x)}-\frac {a+b \tanh ^{-1}(c x)}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )}{d^3 (1+c x)}-\frac {3 a c \log (x)}{d^3}+\frac {b c \log (x)}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^3}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d^3}+\frac {3 b c \text {Li}_2(-c x)}{2 d^3}-\frac {3 b c \text {Li}_2(c x)}{2 d^3}+\frac {3 b c \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 d^3}-\frac {\left (b c^2\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{8 d^3}-\frac {\left (b c^2\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{d^3}\\ &=-\frac {b c}{8 d^3 (1+c x)^2}-\frac {9 b c}{8 d^3 (1+c x)}+\frac {9 b c \tanh ^{-1}(c x)}{8 d^3}-\frac {a+b \tanh ^{-1}(c x)}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )}{d^3 (1+c x)}-\frac {3 a c \log (x)}{d^3}+\frac {b c \log (x)}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^3}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d^3}+\frac {3 b c \text {Li}_2(-c x)}{2 d^3}-\frac {3 b c \text {Li}_2(c x)}{2 d^3}+\frac {3 b c \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 d^3}\\ \end {align*}
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Mathematica [A] time = 1.24, size = 186, normalized size = 0.85 \[ \frac {-\frac {64 a c}{c x+1}-\frac {16 a c}{(c x+1)^2}-96 a c \log (x)+96 a c \log (c x+1)-\frac {32 a}{x}+b c \left (32 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+48 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )+20 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )-20 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )+4 \tanh ^{-1}(c x) \left (-\frac {8}{c x}-24 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+10 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )-10 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )\right )\right )}{32 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c x\right ) + a}{c^{3} d^{3} x^{5} + 3 \, c^{2} d^{3} x^{4} + 3 \, c d^{3} x^{3} + d^{3} x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 319, normalized size = 1.46 \[ -\frac {a}{d^{3} x}-\frac {3 c a \ln \left (c x \right )}{d^{3}}-\frac {c a}{2 d^{3} \left (c x +1\right )^{2}}-\frac {2 c a}{d^{3} \left (c x +1\right )}+\frac {3 c a \ln \left (c x +1\right )}{d^{3}}-\frac {b \arctanh \left (c x \right )}{d^{3} x}-\frac {3 c b \arctanh \left (c x \right ) \ln \left (c x \right )}{d^{3}}-\frac {c b \arctanh \left (c x \right )}{2 d^{3} \left (c x +1\right )^{2}}-\frac {2 c b \arctanh \left (c x \right )}{d^{3} \left (c x +1\right )}+\frac {3 c b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d^{3}}+\frac {c b \ln \left (c x \right )}{d^{3}}-\frac {17 c b \ln \left (c x -1\right )}{16 d^{3}}-\frac {b c}{8 d^{3} \left (c x +1\right )^{2}}-\frac {9 b c}{8 d^{3} \left (c x +1\right )}+\frac {c b \ln \left (c x +1\right )}{16 d^{3}}+\frac {3 c b \dilog \left (c x \right )}{2 d^{3}}+\frac {3 c b \dilog \left (c x +1\right )}{2 d^{3}}+\frac {3 c b \ln \left (c x \right ) \ln \left (c x +1\right )}{2 d^{3}}-\frac {3 c b \ln \left (c x +1\right )^{2}}{4 d^{3}}+\frac {3 c b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d^{3}}-\frac {3 c b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 d^{3}}-\frac {3 c b \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 d^{3}} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {6 \, c^{2} x^{2} + 9 \, c x + 2}{c^{2} d^{3} x^{3} + 2 \, c d^{3} x^{2} + d^{3} x} - \frac {6 \, c \log \left (c x + 1\right )}{d^{3}} + \frac {6 \, c \log \relax (x)}{d^{3}}\right )} + \frac {1}{2} \, b \int \frac {\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{c^{3} d^{3} x^{5} + 3 \, c^{2} d^{3} x^{4} + 3 \, c d^{3} x^{3} + d^{3} x^{2}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x^2\,{\left (d+c\,d\,x\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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