Optimal. Leaf size=218 \[ -\frac {b c}{8 d^3 (1+c x)^2}-\frac {9 b c}{8 d^3 (1+c x)}+\frac {9 b c \tanh ^{-1}(c x)}{8 d^3}-\frac {a+b \tanh ^{-1}(c x)}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )}{d^3 (1+c x)}-\frac {3 a c \log (x)}{d^3}+\frac {b c \log (x)}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^3}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d^3}+\frac {3 b c \text {PolyLog}(2,-c x)}{2 d^3}-\frac {3 b c \text {PolyLog}(2,c x)}{2 d^3}+\frac {3 b c \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 d^3} \]
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Rubi [A]
time = 0.20, 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 = {6087, 6037,
272, 36, 29, 31, 6031, 6063, 641, 46, 213, 6055, 2449, 2352} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 213
Rule 272
Rule 641
Rule 2352
Rule 2449
Rule 6031
Rule 6037
Rule 6055
Rule 6063
Rule 6087
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) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d^3}+\frac {(3 b c) \text {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) \text {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 ) \text {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 = 0.81, size = 186, normalized size = 0.85 \begin {gather*} \frac {-\frac {32 a}{x}-\frac {16 a c}{(1+c x)^2}-\frac {64 a c}{1+c x}-96 a c \log (x)+96 a c \log (1+c x)+b c \left (-20 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )+32 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+48 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+20 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )+4 \tanh ^{-1}(c x) \left (-\frac {8}{c x}-10 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )-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 )\right )\right )}{32 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 307, normalized size = 1.41
method | result | size |
derivativedivides | \(c \left (-\frac {a}{d^{3} c x}-\frac {3 a \ln \left (c x \right )}{d^{3}}-\frac {a}{2 d^{3} \left (c x +1\right )^{2}}-\frac {2 a}{d^{3} \left (c x +1\right )}+\frac {3 a \ln \left (c x +1\right )}{d^{3}}-\frac {b \arctanh \left (c x \right )}{d^{3} c x}-\frac {3 b \arctanh \left (c x \right ) \ln \left (c x \right )}{d^{3}}-\frac {b \arctanh \left (c x \right )}{2 d^{3} \left (c x +1\right )^{2}}-\frac {2 b \arctanh \left (c x \right )}{d^{3} \left (c x +1\right )}+\frac {3 b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d^{3}}-\frac {17 b \ln \left (c x -1\right )}{16 d^{3}}+\frac {b \ln \left (c x \right )}{d^{3}}-\frac {b}{8 d^{3} \left (c x +1\right )^{2}}-\frac {9 b}{8 d^{3} \left (c x +1\right )}+\frac {b \ln \left (c x +1\right )}{16 d^{3}}+\frac {3 b \dilog \left (c x \right )}{2 d^{3}}+\frac {3 b \dilog \left (c x +1\right )}{2 d^{3}}+\frac {3 b \ln \left (c x \right ) \ln \left (c x +1\right )}{2 d^{3}}-\frac {3 b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3}}+\frac {3 b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d^{3}}-\frac {3 b \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3}}-\frac {3 b \ln \left (c x +1\right )^{2}}{4 d^{3}}\right )\) | \(307\) |
default | \(c \left (-\frac {a}{d^{3} c x}-\frac {3 a \ln \left (c x \right )}{d^{3}}-\frac {a}{2 d^{3} \left (c x +1\right )^{2}}-\frac {2 a}{d^{3} \left (c x +1\right )}+\frac {3 a \ln \left (c x +1\right )}{d^{3}}-\frac {b \arctanh \left (c x \right )}{d^{3} c x}-\frac {3 b \arctanh \left (c x \right ) \ln \left (c x \right )}{d^{3}}-\frac {b \arctanh \left (c x \right )}{2 d^{3} \left (c x +1\right )^{2}}-\frac {2 b \arctanh \left (c x \right )}{d^{3} \left (c x +1\right )}+\frac {3 b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d^{3}}-\frac {17 b \ln \left (c x -1\right )}{16 d^{3}}+\frac {b \ln \left (c x \right )}{d^{3}}-\frac {b}{8 d^{3} \left (c x +1\right )^{2}}-\frac {9 b}{8 d^{3} \left (c x +1\right )}+\frac {b \ln \left (c x +1\right )}{16 d^{3}}+\frac {3 b \dilog \left (c x \right )}{2 d^{3}}+\frac {3 b \dilog \left (c x +1\right )}{2 d^{3}}+\frac {3 b \ln \left (c x \right ) \ln \left (c x +1\right )}{2 d^{3}}-\frac {3 b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3}}+\frac {3 b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d^{3}}-\frac {3 b \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3}}-\frac {3 b \ln \left (c x +1\right )^{2}}{4 d^{3}}\right )\) | \(307\) |
risch | \(-\frac {b c \ln \left (c x +1\right )}{4 d^{3} \left (c x +1\right )^{2}}-\frac {b c \ln \left (c x +1\right )}{d^{3} \left (c x +1\right )}-\frac {c b \ln \left (-c x +1\right )}{2 d^{3} \left (-c x -1\right )}+\frac {3 c b \ln \left (-c x +1\right )}{16 d^{3} \left (-c x -1\right )^{2}}-\frac {3 c b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{2 d^{3}}+\frac {3 c b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3}}-\frac {b c}{8 d^{3} \left (c x +1\right )^{2}}-\frac {b c}{d^{3} \left (c x +1\right )}+\frac {c^{2} b \ln \left (-c x +1\right ) x}{2 d^{3} \left (-c x -1\right )}-\frac {c^{3} b \ln \left (-c x +1\right ) x^{2}}{16 d^{3} \left (-c x -1\right )^{2}}-\frac {c^{2} b \ln \left (-c x +1\right ) x}{8 d^{3} \left (-c x -1\right )^{2}}-\frac {a}{d^{3} x}+\frac {3 b c \dilog \left (c x +1\right )}{2 d^{3}}-\frac {3 c \dilog \left (-c x +1\right ) b}{2 d^{3}}+\frac {3 c b \dilog \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3}}+\frac {b c \ln \left (c x \right )}{2 d^{3}}-\frac {b c \ln \left (c x +1\right )}{2 d^{3}}-\frac {b \ln \left (c x +1\right )}{2 d^{3} x}+\frac {3 b c \ln \left (c x +1\right )^{2}}{4 d^{3}}-\frac {c b \ln \left (-c x +1\right )}{2 d^{3}}+\frac {b \ln \left (-c x +1\right )}{2 d^{3} x}+\frac {c b}{8 d^{3} \left (-c x -1\right )}-\frac {3 c a \ln \left (-c x \right )}{d^{3}}-\frac {c a}{2 d^{3} \left (-c x -1\right )^{2}}+\frac {2 c a}{d^{3} \left (-c x -1\right )}+\frac {3 c a \ln \left (-c x -1\right )}{d^{3}}+\frac {c b \ln \left (-c x \right )}{2 d^{3}}+\frac {9 b c \ln \left (-c x -1\right )}{16 d^{3}}\) | \(464\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x^2\,{\left (d+c\,d\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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