Optimal. Leaf size=189 \[ -\frac {b c d^4}{6 x^2}-\frac {2 b c^2 d^4}{x}+a c^4 d^4 x+2 b c^3 d^4 \tanh ^{-1}(c x)+b c^4 d^4 x \tanh ^{-1}(c x)-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+4 a c^3 d^4 \log (x)+\frac {19}{3} b c^3 d^4 \log (x)-\frac {8}{3} b c^3 d^4 \log \left (1-c^2 x^2\right )-2 b c^3 d^4 \text {PolyLog}(2,-c x)+2 b c^3 d^4 \text {PolyLog}(2,c x) \]
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Rubi [A]
time = 0.16, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6087, 6021,
266, 6037, 272, 46, 331, 212, 36, 29, 31, 6031} \begin {gather*} -\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+a c^4 d^4 x+4 a c^3 d^4 \log (x)+b c^4 d^4 x \tanh ^{-1}(c x)-2 b c^3 d^4 \text {Li}_2(-c x)+2 b c^3 d^4 \text {Li}_2(c x)+\frac {19}{3} b c^3 d^4 \log (x)+2 b c^3 d^4 \tanh ^{-1}(c x)-\frac {2 b c^2 d^4}{x}-\frac {8}{3} b c^3 d^4 \log \left (1-c^2 x^2\right )-\frac {b c d^4}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 212
Rule 266
Rule 272
Rule 331
Rule 6021
Rule 6031
Rule 6037
Rule 6087
Rubi steps
\begin {align*} \int \frac {(d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4} \, dx &=\int \left (c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}+\frac {4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^3}+\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+\frac {4 c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^4 \int \frac {a+b \tanh ^{-1}(c x)}{x^4} \, dx+\left (4 c d^4\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx+\left (6 c^2 d^4\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (4 c^3 d^4\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx+\left (c^4 d^4\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=a c^4 d^4 x-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+4 a c^3 d^4 \log (x)-2 b c^3 d^4 \text {Li}_2(-c x)+2 b c^3 d^4 \text {Li}_2(c x)+\frac {1}{3} \left (b c d^4\right ) \int \frac {1}{x^3 \left (1-c^2 x^2\right )} \, dx+\left (2 b c^2 d^4\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (6 b c^3 d^4\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx+\left (b c^4 d^4\right ) \int \tanh ^{-1}(c x) \, dx\\ &=-\frac {2 b c^2 d^4}{x}+a c^4 d^4 x+b c^4 d^4 x \tanh ^{-1}(c x)-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+4 a c^3 d^4 \log (x)-2 b c^3 d^4 \text {Li}_2(-c x)+2 b c^3 d^4 \text {Li}_2(c x)+\frac {1}{6} \left (b c d^4\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )+\left (3 b c^3 d^4\right ) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\left (2 b c^4 d^4\right ) \int \frac {1}{1-c^2 x^2} \, dx-\left (b c^5 d^4\right ) \int \frac {x}{1-c^2 x^2} \, dx\\ &=-\frac {2 b c^2 d^4}{x}+a c^4 d^4 x+2 b c^3 d^4 \tanh ^{-1}(c x)+b c^4 d^4 x \tanh ^{-1}(c x)-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+4 a c^3 d^4 \log (x)+\frac {1}{2} b c^3 d^4 \log \left (1-c^2 x^2\right )-2 b c^3 d^4 \text {Li}_2(-c x)+2 b c^3 d^4 \text {Li}_2(c x)+\frac {1}{6} \left (b c d^4\right ) \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 )+\left (3 b c^3 d^4\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\left (3 b c^5 d^4\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b c d^4}{6 x^2}-\frac {2 b c^2 d^4}{x}+a c^4 d^4 x+2 b c^3 d^4 \tanh ^{-1}(c x)+b c^4 d^4 x \tanh ^{-1}(c x)-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+4 a c^3 d^4 \log (x)+\frac {19}{3} b c^3 d^4 \log (x)-\frac {8}{3} b c^3 d^4 \log \left (1-c^2 x^2\right )-2 b c^3 d^4 \text {Li}_2(-c x)+2 b c^3 d^4 \text {Li}_2(c x)\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 197, normalized size = 1.04 \begin {gather*} \frac {d^4 \left (-2 a-12 a c x-b c x-36 a c^2 x^2-12 b c^2 x^2+6 a c^4 x^4-2 b \tanh ^{-1}(c x)-12 b c x \tanh ^{-1}(c x)-36 b c^2 x^2 \tanh ^{-1}(c x)+6 b c^4 x^4 \tanh ^{-1}(c x)+24 a c^3 x^3 \log (x)+38 b c^3 x^3 \log (c x)-6 b c^3 x^3 \log (1-c x)+6 b c^3 x^3 \log (1+c x)-16 b c^3 x^3 \log \left (1-c^2 x^2\right )-12 b c^3 x^3 \text {PolyLog}(2,-c x)+12 b c^3 x^3 \text {PolyLog}(2,c x)\right )}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 228, normalized size = 1.21
method | result | size |
derivativedivides | \(c^{3} \left (a c \,d^{4} x -\frac {6 d^{4} a}{c x}+4 d^{4} a \ln \left (c x \right )-\frac {d^{4} a}{3 c^{3} x^{3}}-\frac {2 d^{4} a}{c^{2} x^{2}}+b c \,d^{4} x \arctanh \left (c x \right )-\frac {6 d^{4} b \arctanh \left (c x \right )}{c x}+4 d^{4} b \arctanh \left (c x \right ) \ln \left (c x \right )-\frac {d^{4} b \arctanh \left (c x \right )}{3 c^{3} x^{3}}-\frac {2 d^{4} b \arctanh \left (c x \right )}{c^{2} x^{2}}-2 d^{4} b \dilog \left (c x \right )-2 d^{4} b \dilog \left (c x +1\right )-2 d^{4} b \ln \left (c x \right ) \ln \left (c x +1\right )-\frac {d^{4} b}{6 c^{2} x^{2}}-\frac {2 d^{4} b}{c x}+\frac {19 d^{4} b \ln \left (c x \right )}{3}-\frac {11 d^{4} b \ln \left (c x -1\right )}{3}-\frac {5 d^{4} b \ln \left (c x +1\right )}{3}\right )\) | \(228\) |
default | \(c^{3} \left (a c \,d^{4} x -\frac {6 d^{4} a}{c x}+4 d^{4} a \ln \left (c x \right )-\frac {d^{4} a}{3 c^{3} x^{3}}-\frac {2 d^{4} a}{c^{2} x^{2}}+b c \,d^{4} x \arctanh \left (c x \right )-\frac {6 d^{4} b \arctanh \left (c x \right )}{c x}+4 d^{4} b \arctanh \left (c x \right ) \ln \left (c x \right )-\frac {d^{4} b \arctanh \left (c x \right )}{3 c^{3} x^{3}}-\frac {2 d^{4} b \arctanh \left (c x \right )}{c^{2} x^{2}}-2 d^{4} b \dilog \left (c x \right )-2 d^{4} b \dilog \left (c x +1\right )-2 d^{4} b \ln \left (c x \right ) \ln \left (c x +1\right )-\frac {d^{4} b}{6 c^{2} x^{2}}-\frac {2 d^{4} b}{c x}+\frac {19 d^{4} b \ln \left (c x \right )}{3}-\frac {11 d^{4} b \ln \left (c x -1\right )}{3}-\frac {5 d^{4} b \ln \left (c x +1\right )}{3}\right )\) | \(228\) |
risch | \(\frac {b \,c^{4} d^{4} \ln \left (c x +1\right ) x}{2}-\frac {3 b \,c^{2} d^{4} \ln \left (c x +1\right )}{x}-\frac {b c \,d^{4} \ln \left (c x +1\right )}{x^{2}}-\frac {b c \,d^{4}}{6 x^{2}}-\frac {2 b \,c^{2} d^{4}}{x}+\frac {c \,d^{4} b \ln \left (-c x +1\right )}{x^{2}}-\frac {c^{4} d^{4} b \ln \left (-c x +1\right ) x}{2}+\frac {3 c^{2} d^{4} b \ln \left (-c x +1\right )}{x}-\frac {d^{4} a}{3 x^{3}}-c^{3} d^{4} a +2 c^{3} d^{4} \dilog \left (-c x +1\right ) b -\frac {2 c \,d^{4} a}{x^{2}}-b \,c^{3} d^{4}-\frac {5 b \,c^{3} d^{4} \ln \left (c x +1\right )}{3}+\frac {13 b \,c^{3} d^{4} \ln \left (c x \right )}{6}-\frac {b \,d^{4} \ln \left (c x +1\right )}{6 x^{3}}-2 b \,c^{3} d^{4} \dilog \left (c x +1\right )+\frac {d^{4} b \ln \left (-c x +1\right )}{6 x^{3}}+\frac {25 c^{3} d^{4} b \ln \left (-c x \right )}{6}+4 c^{3} d^{4} a \ln \left (-c x \right )-\frac {6 c^{2} d^{4} a}{x}-\frac {11 c^{3} d^{4} b \ln \left (-c x +1\right )}{3}+a \,c^{4} d^{4} x\) | \(318\) |
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*} d^{4} \left (\int a c^{4}\, dx + \int \frac {a}{x^{4}}\, dx + \int \frac {4 a c}{x^{3}}\, dx + \int \frac {6 a c^{2}}{x^{2}}\, dx + \int \frac {4 a c^{3}}{x}\, dx + \int b c^{4} \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {4 b c \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {6 b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {4 b c^{3} \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \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 {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^4}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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