Optimal. Leaf size=185 \[ 4 a c d^4 x+\frac {13}{4} b c d^4 x+\frac {2}{3} b c^2 d^4 x^2+\frac {1}{12} b c^3 d^4 x^3-\frac {13}{4} b d^4 \tanh ^{-1}(c x)+4 b c d^4 x \tanh ^{-1}(c x)+3 c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{3} c^3 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} c^4 d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+a d^4 \log (x)+\frac {8}{3} b d^4 \log \left (1-c^2 x^2\right )-\frac {1}{2} b d^4 \text {PolyLog}(2,-c x)+\frac {1}{2} b d^4 \text {PolyLog}(2,c x) \]
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Rubi [A]
time = 0.15, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6087, 6021,
266, 6031, 6037, 327, 212, 272, 45, 308} \begin {gather*} \frac {1}{4} c^4 d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{3} c^3 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+3 c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+4 a c d^4 x+a d^4 \log (x)+\frac {1}{12} b c^3 d^4 x^3+\frac {2}{3} b c^2 d^4 x^2+\frac {8}{3} b d^4 \log \left (1-c^2 x^2\right )-\frac {1}{2} b d^4 \text {Li}_2(-c x)+\frac {1}{2} b d^4 \text {Li}_2(c x)+\frac {13}{4} b c d^4 x-\frac {13}{4} b d^4 \tanh ^{-1}(c x)+4 b c d^4 x \tanh ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 212
Rule 266
Rule 272
Rule 308
Rule 327
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} \, dx &=\int \left (4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+6 c^2 d^4 x \left (a+b \tanh ^{-1}(c x)\right )+4 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )\right ) \, dx\\ &=d^4 \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx+\left (4 c d^4\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (6 c^2 d^4\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (4 c^3 d^4\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (c^4 d^4\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=4 a c d^4 x+3 c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{3} c^3 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} c^4 d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+a d^4 \log (x)-\frac {1}{2} b d^4 \text {Li}_2(-c x)+\frac {1}{2} b d^4 \text {Li}_2(c x)+\left (4 b c d^4\right ) \int \tanh ^{-1}(c x) \, dx-\left (3 b c^3 d^4\right ) \int \frac {x^2}{1-c^2 x^2} \, dx-\frac {1}{3} \left (4 b c^4 d^4\right ) \int \frac {x^3}{1-c^2 x^2} \, dx-\frac {1}{4} \left (b c^5 d^4\right ) \int \frac {x^4}{1-c^2 x^2} \, dx\\ &=4 a c d^4 x+3 b c d^4 x+4 b c d^4 x \tanh ^{-1}(c x)+3 c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{3} c^3 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} c^4 d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+a d^4 \log (x)-\frac {1}{2} b d^4 \text {Li}_2(-c x)+\frac {1}{2} b d^4 \text {Li}_2(c x)-\left (3 b c d^4\right ) \int \frac {1}{1-c^2 x^2} \, dx-\left (4 b c^2 d^4\right ) \int \frac {x}{1-c^2 x^2} \, dx-\frac {1}{3} \left (2 b c^4 d^4\right ) \text {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {1}{4} \left (b c^5 d^4\right ) \int \left (-\frac {1}{c^4}-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=4 a c d^4 x+\frac {13}{4} b c d^4 x+\frac {1}{12} b c^3 d^4 x^3-3 b d^4 \tanh ^{-1}(c x)+4 b c d^4 x \tanh ^{-1}(c x)+3 c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{3} c^3 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} c^4 d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+a d^4 \log (x)+2 b d^4 \log \left (1-c^2 x^2\right )-\frac {1}{2} b d^4 \text {Li}_2(-c x)+\frac {1}{2} b d^4 \text {Li}_2(c x)-\frac {1}{4} \left (b c d^4\right ) \int \frac {1}{1-c^2 x^2} \, dx-\frac {1}{3} \left (2 b c^4 d^4\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=4 a c d^4 x+\frac {13}{4} b c d^4 x+\frac {2}{3} b c^2 d^4 x^2+\frac {1}{12} b c^3 d^4 x^3-\frac {13}{4} b d^4 \tanh ^{-1}(c x)+4 b c d^4 x \tanh ^{-1}(c x)+3 c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{3} c^3 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} c^4 d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+a d^4 \log (x)+\frac {8}{3} b d^4 \log \left (1-c^2 x^2\right )-\frac {1}{2} b d^4 \text {Li}_2(-c x)+\frac {1}{2} b d^4 \text {Li}_2(c x)\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 179, normalized size = 0.97 \begin {gather*} \frac {1}{24} d^4 \left (96 a c x+78 b c x+72 a c^2 x^2+16 b c^2 x^2+32 a c^3 x^3+2 b c^3 x^3+6 a c^4 x^4+96 b c x \tanh ^{-1}(c x)+72 b c^2 x^2 \tanh ^{-1}(c x)+32 b c^3 x^3 \tanh ^{-1}(c x)+6 b c^4 x^4 \tanh ^{-1}(c x)+24 a \log (x)+39 b \log (1-c x)-39 b \log (1+c x)+48 b \log \left (1-c^2 x^2\right )+16 b \log \left (-1+c^2 x^2\right )-12 b \text {PolyLog}(2,-c x)+12 b \text {PolyLog}(2,c x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 222, normalized size = 1.20
method | result | size |
derivativedivides | \(\frac {d^{4} a \,c^{4} x^{4}}{4}+\frac {4 d^{4} a \,c^{3} x^{3}}{3}+3 d^{4} a \,c^{2} x^{2}+4 a c \,d^{4} x +d^{4} a \ln \left (c x \right )+\frac {d^{4} b \arctanh \left (c x \right ) c^{4} x^{4}}{4}+\frac {4 d^{4} b \arctanh \left (c x \right ) c^{3} x^{3}}{3}+3 d^{4} b \arctanh \left (c x \right ) c^{2} x^{2}+4 b c \,d^{4} x \arctanh \left (c x \right )+d^{4} b \arctanh \left (c x \right ) \ln \left (c x \right )-\frac {d^{4} b \dilog \left (c x \right )}{2}-\frac {d^{4} b \dilog \left (c x +1\right )}{2}-\frac {d^{4} b \ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {b \,c^{3} d^{4} x^{3}}{12}+\frac {2 b \,c^{2} d^{4} x^{2}}{3}+\frac {13 b c \,d^{4} x}{4}+\frac {103 d^{4} b \ln \left (c x -1\right )}{24}+\frac {25 d^{4} b \ln \left (c x +1\right )}{24}\) | \(222\) |
default | \(\frac {d^{4} a \,c^{4} x^{4}}{4}+\frac {4 d^{4} a \,c^{3} x^{3}}{3}+3 d^{4} a \,c^{2} x^{2}+4 a c \,d^{4} x +d^{4} a \ln \left (c x \right )+\frac {d^{4} b \arctanh \left (c x \right ) c^{4} x^{4}}{4}+\frac {4 d^{4} b \arctanh \left (c x \right ) c^{3} x^{3}}{3}+3 d^{4} b \arctanh \left (c x \right ) c^{2} x^{2}+4 b c \,d^{4} x \arctanh \left (c x \right )+d^{4} b \arctanh \left (c x \right ) \ln \left (c x \right )-\frac {d^{4} b \dilog \left (c x \right )}{2}-\frac {d^{4} b \dilog \left (c x +1\right )}{2}-\frac {d^{4} b \ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {b \,c^{3} d^{4} x^{3}}{12}+\frac {2 b \,c^{2} d^{4} x^{2}}{3}+\frac {13 b c \,d^{4} x}{4}+\frac {103 d^{4} b \ln \left (c x -1\right )}{24}+\frac {25 d^{4} b \ln \left (c x +1\right )}{24}\) | \(222\) |
risch | \(4 a c \,d^{4} x +\frac {13 b c \,d^{4} x}{4}+\frac {2 b \,c^{2} d^{4} x^{2}}{3}+\frac {b \,c^{3} d^{4} x^{3}}{12}+\frac {25 d^{4} b \ln \left (c x +1\right )}{24}-\frac {58 d^{4} b}{9}+\frac {d^{4} a \,c^{4} x^{4}}{4}+\frac {4 d^{4} a \,c^{3} x^{3}}{3}+3 d^{4} a \,c^{2} x^{2}-\frac {103 d^{4} a}{12}-\frac {d^{4} b \dilog \left (c x +1\right )}{2}+\frac {d^{4} \dilog \left (-c x +1\right ) b}{2}+d^{4} a \ln \left (-c x \right )+\frac {103 d^{4} b \ln \left (-c x +1\right )}{24}-\frac {3 d^{4} \ln \left (-c x +1\right ) x^{2} b \,c^{2}}{2}-\frac {d^{4} \ln \left (-c x +1\right ) x^{4} b \,c^{4}}{8}-\frac {2 d^{4} \ln \left (-c x +1\right ) x^{3} b \,c^{3}}{3}+\frac {2 d^{4} b \ln \left (c x +1\right ) x^{3} c^{3}}{3}+\frac {3 d^{4} b \ln \left (c x +1\right ) x^{2} c^{2}}{2}+2 d^{4} b \ln \left (c x +1\right ) c x +\frac {d^{4} b \ln \left (c x +1\right ) x^{4} c^{4}}{8}-2 d^{4} b \ln \left (-c x +1\right ) c x\) | \(290\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 276, normalized size = 1.49 \begin {gather*} \frac {1}{4} \, a c^{4} d^{4} x^{4} + \frac {4}{3} \, a c^{3} d^{4} x^{3} + \frac {1}{12} \, b c^{3} d^{4} x^{3} + 3 \, a c^{2} d^{4} x^{2} + \frac {2}{3} \, b c^{2} d^{4} x^{2} + 4 \, a c d^{4} x + \frac {13}{4} \, b c d^{4} x + 2 \, {\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{4} - \frac {1}{2} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b d^{4} + \frac {1}{2} \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b d^{4} - \frac {23}{24} \, b d^{4} \log \left (c x + 1\right ) + \frac {55}{24} \, b d^{4} \log \left (c x - 1\right ) + a d^{4} \log \left (x\right ) + \frac {1}{24} \, {\left (3 \, b c^{4} d^{4} x^{4} + 16 \, b c^{3} d^{4} x^{3} + 36 \, b c^{2} d^{4} x^{2}\right )} \log \left (c x + 1\right ) - \frac {1}{24} \, {\left (3 \, b c^{4} d^{4} x^{4} + 16 \, b c^{3} d^{4} x^{3} + 36 \, b c^{2} d^{4} x^{2}\right )} \log \left (-c x + 1\right ) \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 4 a c\, dx + \int \frac {a}{x}\, dx + \int 6 a c^{2} x\, dx + \int 4 a c^{3} x^{2}\, dx + \int a c^{4} x^{3}\, dx + \int 4 b c \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{x}\, dx + \int 6 b c^{2} x \operatorname {atanh}{\left (c x \right )}\, dx + \int 4 b c^{3} x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int b c^{4} x^{3} \operatorname {atanh}{\left (c x \right )}\, 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} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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