Optimal. Leaf size=185 \[ -\frac {b c}{6 d x^2}+\frac {b c^2}{2 d x}-\frac {b c^3 \tanh ^{-1}(c x)}{2 d}-\frac {a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d x}+\frac {4 b c^3 \log (x)}{3 d}-\frac {2 b c^3 \log \left (1-c^2 x^2\right )}{3 d}-\frac {c^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b c^3 \text {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{2 d} \]
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Rubi [A]
time = 0.25, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {6081, 6037,
272, 46, 331, 212, 36, 29, 31, 6079, 2497} \begin {gather*} -\frac {c^3 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac {a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}+\frac {b c^3 \text {Li}_2\left (\frac {2}{c x+1}-1\right )}{2 d}+\frac {4 b c^3 \log (x)}{3 d}-\frac {b c^3 \tanh ^{-1}(c x)}{2 d}+\frac {b c^2}{2 d x}-\frac {2 b c^3 \log \left (1-c^2 x^2\right )}{3 d}-\frac {b c}{6 d x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 212
Rule 272
Rule 331
Rule 2497
Rule 6037
Rule 6079
Rule 6081
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{x^4 (d+c d x)} \, dx &=-\left (c \int \frac {a+b \tanh ^{-1}(c x)}{x^3 (d+c d x)} \, dx\right )+\frac {\int \frac {a+b \tanh ^{-1}(c x)}{x^4} \, dx}{d}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{3 d x^3}+c^2 \int \frac {a+b \tanh ^{-1}(c x)}{x^2 (d+c d x)} \, dx-\frac {c \int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx}{d}+\frac {(b c) \int \frac {1}{x^3 \left (1-c^2 x^2\right )} \, dx}{3 d}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-c^3 \int \frac {a+b \tanh ^{-1}(c x)}{x (d+c d x)} \, dx+\frac {(b c) \text {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )}{6 d}+\frac {c^2 \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx}{d}-\frac {\left (b c^2\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx}{2 d}\\ &=\frac {b c^2}{2 d x}-\frac {a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac {c^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {(b c) \text {Subst}\left (\int \left (\frac {1}{x^2}+\frac {c^2}{x}-\frac {c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d}+\frac {\left (b c^3\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx}{d}-\frac {\left (b c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{2 d}+\frac {\left (b c^4\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac {b c}{6 d x^2}+\frac {b c^2}{2 d x}-\frac {b c^3 \tanh ^{-1}(c x)}{2 d}-\frac {a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d x}+\frac {b c^3 \log (x)}{3 d}-\frac {b c^3 \log \left (1-c^2 x^2\right )}{6 d}-\frac {c^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b c^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{2 d}+\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {b c}{6 d x^2}+\frac {b c^2}{2 d x}-\frac {b c^3 \tanh ^{-1}(c x)}{2 d}-\frac {a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d x}+\frac {b c^3 \log (x)}{3 d}-\frac {b c^3 \log \left (1-c^2 x^2\right )}{6 d}-\frac {c^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b c^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{2 d}+\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}+\frac {\left (b c^5\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {b c}{6 d x^2}+\frac {b c^2}{2 d x}-\frac {b c^3 \tanh ^{-1}(c x)}{2 d}-\frac {a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 d x^2}-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d x}+\frac {4 b c^3 \log (x)}{3 d}-\frac {2 b c^3 \log \left (1-c^2 x^2\right )}{3 d}-\frac {c^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b c^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 172, normalized size = 0.93 \begin {gather*} \frac {-2 a+3 a c x-b c x-6 a c^2 x^2+3 b c^2 x^2+b c^3 x^3-b \tanh ^{-1}(c x) \left (2-3 c x+6 c^2 x^2+3 c^3 x^3+6 c^3 x^3 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )-6 a c^3 x^3 \log (x)+6 a c^3 x^3 \log (1+c x)+8 b c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+3 b c^3 x^3 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )}{6 d x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 302, normalized size = 1.63
method | result | size |
derivativedivides | \(c^{3} \left (\frac {a \ln \left (c x +1\right )}{d}-\frac {a}{3 d \,c^{3} x^{3}}-\frac {a}{d c x}+\frac {a}{2 d \,c^{2} x^{2}}-\frac {a \ln \left (c x \right )}{d}+\frac {b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d}-\frac {b \arctanh \left (c x \right )}{3 d \,c^{3} x^{3}}-\frac {b \arctanh \left (c x \right )}{d c x}+\frac {b \arctanh \left (c x \right )}{2 d \,c^{2} x^{2}}-\frac {b \arctanh \left (c x \right ) \ln \left (c x \right )}{d}-\frac {5 b \ln \left (c x -1\right )}{12 d}-\frac {11 b \ln \left (c x +1\right )}{12 d}-\frac {b}{6 d \,c^{2} x^{2}}+\frac {b}{2 d c x}+\frac {4 b \ln \left (c x \right )}{3 d}+\frac {b \dilog \left (c x \right )}{2 d}+\frac {b \dilog \left (c x +1\right )}{2 d}+\frac {b \ln \left (c x \right ) \ln \left (c x +1\right )}{2 d}+\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d}-\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d}-\frac {b \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d}-\frac {b \ln \left (c x +1\right )^{2}}{4 d}\right )\) | \(302\) |
default | \(c^{3} \left (\frac {a \ln \left (c x +1\right )}{d}-\frac {a}{3 d \,c^{3} x^{3}}-\frac {a}{d c x}+\frac {a}{2 d \,c^{2} x^{2}}-\frac {a \ln \left (c x \right )}{d}+\frac {b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d}-\frac {b \arctanh \left (c x \right )}{3 d \,c^{3} x^{3}}-\frac {b \arctanh \left (c x \right )}{d c x}+\frac {b \arctanh \left (c x \right )}{2 d \,c^{2} x^{2}}-\frac {b \arctanh \left (c x \right ) \ln \left (c x \right )}{d}-\frac {5 b \ln \left (c x -1\right )}{12 d}-\frac {11 b \ln \left (c x +1\right )}{12 d}-\frac {b}{6 d \,c^{2} x^{2}}+\frac {b}{2 d c x}+\frac {4 b \ln \left (c x \right )}{3 d}+\frac {b \dilog \left (c x \right )}{2 d}+\frac {b \dilog \left (c x +1\right )}{2 d}+\frac {b \ln \left (c x \right ) \ln \left (c x +1\right )}{2 d}+\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d}-\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d}-\frac {b \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d}-\frac {b \ln \left (c x +1\right )^{2}}{4 d}\right )\) | \(302\) |
risch | \(-\frac {c b \ln \left (-c x +1\right )}{4 d \,x^{2}}+\frac {c^{3} b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d}-\frac {c^{3} b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{2 d}-\frac {b c}{6 d \,x^{2}}+\frac {b \,c^{2}}{2 d x}+\frac {c^{2} b \ln \left (-c x +1\right )}{2 d x}-\frac {a}{3 d \,x^{3}}+\frac {11 b \,c^{3} \ln \left (c x \right )}{12 d}-\frac {11 b \,c^{3} \ln \left (c x +1\right )}{12 d}+\frac {b \,c^{3} \ln \left (c x +1\right )^{2}}{4 d}+\frac {b \,c^{3} \dilog \left (c x +1\right )}{2 d}-\frac {b \ln \left (c x +1\right )}{6 d \,x^{3}}+\frac {b c \ln \left (c x +1\right )}{4 d \,x^{2}}-\frac {b \,c^{2} \ln \left (c x +1\right )}{2 d x}+\frac {b \ln \left (-c x +1\right )}{6 d \,x^{3}}+\frac {5 c^{3} b \ln \left (-c x \right )}{12 d}-\frac {5 c^{3} b \ln \left (-c x +1\right )}{12 d}-\frac {c^{3} \dilog \left (-c x +1\right ) b}{2 d}+\frac {c^{3} b \dilog \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d}-\frac {c^{3} a \ln \left (-c x \right )}{d}+\frac {c^{3} a \ln \left (-c x -1\right )}{d}-\frac {c^{2} a}{d x}+\frac {c a}{2 d \,x^{2}}\) | \(353\) |
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 x^{5} + x^{4}}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c x^{5} + x^{4}}\, dx}{d} \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 = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x^4\,\left (d+c\,d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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