Optimal. Leaf size=160 \[ -\frac {b c d^3}{2 x}+a c^3 d^3 x+\frac {1}{2} b c^2 d^3 \tanh ^{-1}(c x)+b c^3 d^3 x \tanh ^{-1}(c x)-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+3 a c^2 d^3 \log (x)+3 b c^2 d^3 \log (x)-b c^2 d^3 \log \left (1-c^2 x^2\right )-\frac {3}{2} b c^2 d^3 \text {PolyLog}(2,-c x)+\frac {3}{2} b c^2 d^3 \text {PolyLog}(2,c x) \]
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Rubi [A]
time = 0.12, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {6087, 6021,
266, 6037, 331, 212, 272, 36, 29, 31, 6031} \begin {gather*} -\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^3 d^3 x+3 a c^2 d^3 \log (x)+b c^3 d^3 x \tanh ^{-1}(c x)-\frac {3}{2} b c^2 d^3 \text {Li}_2(-c x)+\frac {3}{2} b c^2 d^3 \text {Li}_2(c x)-b c^2 d^3 \log \left (1-c^2 x^2\right )+3 b c^2 d^3 \log (x)+\frac {1}{2} b c^2 d^3 \tanh ^{-1}(c x)-\frac {b c d^3}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
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)^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^3} \, dx &=\int \left (c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^3}+\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^3 \int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx+\left (3 c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (3 c^2 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx+\left (c^3 d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=a c^3 d^3 x-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+3 a c^2 d^3 \log (x)-\frac {3}{2} b c^2 d^3 \text {Li}_2(-c x)+\frac {3}{2} b c^2 d^3 \text {Li}_2(c x)+\frac {1}{2} \left (b c d^3\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (3 b c^2 d^3\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx+\left (b c^3 d^3\right ) \int \tanh ^{-1}(c x) \, dx\\ &=-\frac {b c d^3}{2 x}+a c^3 d^3 x+b c^3 d^3 x \tanh ^{-1}(c x)-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+3 a c^2 d^3 \log (x)-\frac {3}{2} b c^2 d^3 \text {Li}_2(-c x)+\frac {3}{2} b c^2 d^3 \text {Li}_2(c x)+\frac {1}{2} \left (3 b c^2 d^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{2} \left (b c^3 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx-\left (b c^4 d^3\right ) \int \frac {x}{1-c^2 x^2} \, dx\\ &=-\frac {b c d^3}{2 x}+a c^3 d^3 x+\frac {1}{2} b c^2 d^3 \tanh ^{-1}(c x)+b c^3 d^3 x \tanh ^{-1}(c x)-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+3 a c^2 d^3 \log (x)+\frac {1}{2} b c^2 d^3 \log \left (1-c^2 x^2\right )-\frac {3}{2} b c^2 d^3 \text {Li}_2(-c x)+\frac {3}{2} b c^2 d^3 \text {Li}_2(c x)+\frac {1}{2} \left (3 b c^2 d^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} \left (3 b c^4 d^3\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b c d^3}{2 x}+a c^3 d^3 x+\frac {1}{2} b c^2 d^3 \tanh ^{-1}(c x)+b c^3 d^3 x \tanh ^{-1}(c x)-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+3 a c^2 d^3 \log (x)+3 b c^2 d^3 \log (x)-b c^2 d^3 \log \left (1-c^2 x^2\right )-\frac {3}{2} b c^2 d^3 \text {Li}_2(-c x)+\frac {3}{2} b c^2 d^3 \text {Li}_2(c x)\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 165, normalized size = 1.03 \begin {gather*} \frac {d^3 \left (-2 a-12 a c x-2 b c x+4 a c^3 x^3-2 b \tanh ^{-1}(c x)-12 b c x \tanh ^{-1}(c x)+4 b c^3 x^3 \tanh ^{-1}(c x)+12 a c^2 x^2 \log (x)+12 b c^2 x^2 \log (c x)-b c^2 x^2 \log (1-c x)+b c^2 x^2 \log (1+c x)-4 b c^2 x^2 \log \left (1-c^2 x^2\right )-6 b c^2 x^2 \text {PolyLog}(2,-c x)+6 b c^2 x^2 \text {PolyLog}(2,c x)\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 188, normalized size = 1.18
method | result | size |
derivativedivides | \(c^{2} \left (d^{3} a c x -\frac {3 d^{3} a}{c x}-\frac {d^{3} a}{2 c^{2} x^{2}}+3 d^{3} a \ln \left (c x \right )+b c \,d^{3} x \arctanh \left (c x \right )-\frac {3 d^{3} b \arctanh \left (c x \right )}{c x}-\frac {d^{3} b \arctanh \left (c x \right )}{2 c^{2} x^{2}}+3 d^{3} b \arctanh \left (c x \right ) \ln \left (c x \right )-\frac {3 d^{3} b \dilog \left (c x \right )}{2}-\frac {3 d^{3} b \dilog \left (c x +1\right )}{2}-\frac {3 d^{3} b \ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {5 d^{3} b \ln \left (c x -1\right )}{4}-\frac {3 d^{3} b \ln \left (c x +1\right )}{4}-\frac {d^{3} b}{2 c x}+3 d^{3} b \ln \left (c x \right )\right )\) | \(188\) |
default | \(c^{2} \left (d^{3} a c x -\frac {3 d^{3} a}{c x}-\frac {d^{3} a}{2 c^{2} x^{2}}+3 d^{3} a \ln \left (c x \right )+b c \,d^{3} x \arctanh \left (c x \right )-\frac {3 d^{3} b \arctanh \left (c x \right )}{c x}-\frac {d^{3} b \arctanh \left (c x \right )}{2 c^{2} x^{2}}+3 d^{3} b \arctanh \left (c x \right ) \ln \left (c x \right )-\frac {3 d^{3} b \dilog \left (c x \right )}{2}-\frac {3 d^{3} b \dilog \left (c x +1\right )}{2}-\frac {3 d^{3} b \ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {5 d^{3} b \ln \left (c x -1\right )}{4}-\frac {3 d^{3} b \ln \left (c x +1\right )}{4}-\frac {d^{3} b}{2 c x}+3 d^{3} b \ln \left (c x \right )\right )\) | \(188\) |
risch | \(-\frac {c^{3} d^{3} b \ln \left (-c x +1\right ) x}{2}-\frac {5 c^{2} d^{3} b \ln \left (-c x +1\right )}{4}-b \,c^{2} d^{3}-\frac {b c \,d^{3}}{2 x}+\frac {7 c^{2} d^{3} b \ln \left (-c x \right )}{4}+\frac {3 c \,d^{3} b \ln \left (-c x +1\right )}{2 x}+\frac {d^{3} b \ln \left (-c x +1\right )}{4 x^{2}}+\frac {3 c^{2} d^{3} \dilog \left (-c x +1\right ) b}{2}+a \,c^{3} d^{3} x -c^{2} d^{3} a -\frac {d^{3} a}{2 x^{2}}-\frac {3 c \,d^{3} a}{x}+3 c^{2} d^{3} a \ln \left (-c x \right )+\frac {b \,c^{3} d^{3} \ln \left (c x +1\right ) x}{2}-\frac {3 b \,c^{2} d^{3} \ln \left (c x +1\right )}{4}+\frac {5 b \,c^{2} d^{3} \ln \left (c x \right )}{4}-\frac {b \,d^{3} \ln \left (c x +1\right )}{4 x^{2}}-\frac {3 b c \,d^{3} \ln \left (c x +1\right )}{2 x}-\frac {3 b \,c^{2} d^{3} \dilog \left (c x +1\right )}{2}\) | \(258\) |
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^{3} \left (\int a c^{3}\, dx + \int \frac {a}{x^{3}}\, dx + \int \frac {3 a c}{x^{2}}\, dx + \int \frac {3 a c^{2}}{x}\, dx + \int b c^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {3 b c \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {3 b c^{2} \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 )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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