Optimal. Leaf size=150 \[ \frac {1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+3 a c^2 d^3 x+3 a c d^3 \log (x)+b c d^3 \log \left (1-c^2 x^2\right )+\frac {1}{2} b c^2 d^3 x+3 b c^2 d^3 x \tanh ^{-1}(c x)-\frac {3}{2} b c d^3 \text {Li}_2(-c x)+\frac {3}{2} b c d^3 \text {Li}_2(c x)+b c d^3 \log (x)-\frac {1}{2} b c d^3 \tanh ^{-1}(c x) \]
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Rubi [A] time = 0.16, antiderivative size = 150, 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 = {5940, 5910, 260, 5916, 266, 36, 29, 31, 5912, 321, 206} \[ -\frac {3}{2} b c d^3 \text {PolyLog}(2,-c x)+\frac {3}{2} b c d^3 \text {PolyLog}(2,c x)+\frac {1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+3 a c^2 d^3 x+3 a c d^3 \log (x)+b c d^3 \log \left (1-c^2 x^2\right )+\frac {1}{2} b c^2 d^3 x+3 b c^2 d^3 x \tanh ^{-1}(c x)+b c d^3 \log (x)-\frac {1}{2} b c d^3 \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 206
Rule 260
Rule 266
Rule 321
Rule 5910
Rule 5912
Rule 5916
Rule 5940
Rubi steps
\begin {align*} \int \frac {(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^2} \, dx &=\int \left (3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+c^3 d^3 x \left (a+b \tanh ^{-1}(c x)\right )\right ) \, dx\\ &=d^3 \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (3 c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx+\left (3 c^2 d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (c^3 d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=3 a c^2 d^3 x-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+3 a c d^3 \log (x)-\frac {3}{2} b c d^3 \text {Li}_2(-c x)+\frac {3}{2} b c d^3 \text {Li}_2(c x)+\left (b c d^3\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx+\left (3 b c^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx-\frac {1}{2} \left (b c^4 d^3\right ) \int \frac {x^2}{1-c^2 x^2} \, dx\\ &=3 a c^2 d^3 x+\frac {1}{2} b c^2 d^3 x+3 b c^2 d^3 x \tanh ^{-1}(c x)-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+3 a c d^3 \log (x)-\frac {3}{2} b c d^3 \text {Li}_2(-c x)+\frac {3}{2} b c d^3 \text {Li}_2(c x)+\frac {1}{2} \left (b c d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{2} \left (b c^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx-\left (3 b c^3 d^3\right ) \int \frac {x}{1-c^2 x^2} \, dx\\ &=3 a c^2 d^3 x+\frac {1}{2} b c^2 d^3 x-\frac {1}{2} b c d^3 \tanh ^{-1}(c x)+3 b c^2 d^3 x \tanh ^{-1}(c x)-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+3 a c d^3 \log (x)+\frac {3}{2} b c d^3 \log \left (1-c^2 x^2\right )-\frac {3}{2} b c d^3 \text {Li}_2(-c x)+\frac {3}{2} b c d^3 \text {Li}_2(c x)+\frac {1}{2} \left (b c d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} \left (b c^3 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=3 a c^2 d^3 x+\frac {1}{2} b c^2 d^3 x-\frac {1}{2} b c d^3 \tanh ^{-1}(c x)+3 b c^2 d^3 x \tanh ^{-1}(c x)-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+3 a c d^3 \log (x)+b c d^3 \log (x)+b c d^3 \log \left (1-c^2 x^2\right )-\frac {3}{2} b c d^3 \text {Li}_2(-c x)+\frac {3}{2} b c d^3 \text {Li}_2(c x)\\ \end {align*}
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Mathematica [A] time = 0.14, size = 149, normalized size = 0.99 \[ \frac {d^3 \left (2 a c^3 x^3+12 a c^2 x^2+12 a c x \log (x)-4 a+2 b c^3 x^3 \tanh ^{-1}(c x)+2 b c^2 x^2+4 b c x \log \left (1-c^2 x^2\right )+12 b c^2 x^2 \tanh ^{-1}(c x)-6 b c x \text {Li}_2(-c x)+6 b c x \text {Li}_2(c x)+4 b c x \log (c x)+b c x \log (1-c x)-b c x \log (c x+1)-4 b \tanh ^{-1}(c x)\right )}{4 x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{3} x^{2} + 3 \, a c d^{3} x + a d^{3} + {\left (b c^{3} d^{3} x^{3} + 3 \, b c^{2} d^{3} x^{2} + 3 \, b c d^{3} x + b d^{3}\right )} \operatorname {artanh}\left (c x\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 189, normalized size = 1.26 \[ \frac {d^{3} a \,c^{3} x^{2}}{2}+3 d^{3} a \,c^{2} x +3 c \,d^{3} a \ln \left (c x \right )-\frac {d^{3} a}{x}+\frac {d^{3} b \arctanh \left (c x \right ) c^{3} x^{2}}{2}+3 b \,c^{2} d^{3} x \arctanh \left (c x \right )+3 c \,d^{3} b \arctanh \left (c x \right ) \ln \left (c x \right )-\frac {d^{3} b \arctanh \left (c x \right )}{x}-\frac {3 c \,d^{3} b \dilog \left (c x \right )}{2}-\frac {3 c \,d^{3} b \dilog \left (c x +1\right )}{2}-\frac {3 c \,d^{3} b \ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {b \,c^{2} d^{3} x}{2}+c \,d^{3} b \ln \left (c x \right )+\frac {5 c \,d^{3} b \ln \left (c x -1\right )}{4}+\frac {3 c \,d^{3} b \ln \left (c x +1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 229, normalized size = 1.53 \[ \frac {1}{4} \, b c^{3} d^{3} x^{2} \log \left (c x + 1\right ) - \frac {1}{4} \, b c^{3} d^{3} x^{2} \log \left (-c x + 1\right ) + \frac {1}{2} \, a c^{3} d^{3} x^{2} + 3 \, a c^{2} d^{3} x + \frac {1}{2} \, b c^{2} d^{3} x + \frac {3}{2} \, {\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b c d^{3} - \frac {3}{2} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b c d^{3} + \frac {3}{2} \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b c d^{3} - \frac {1}{4} \, b c d^{3} \log \left (c x + 1\right ) + \frac {1}{4} \, b c d^{3} \log \left (c x - 1\right ) + 3 \, a c d^{3} \log \relax (x) - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} b d^{3} - \frac {a d^{3}}{x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^3}{x^2} \,d x \]
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 \[ d^{3} \left (\int 3 a c^{2}\, dx + \int \frac {a}{x^{2}}\, dx + \int \frac {3 a c}{x}\, dx + \int a c^{3} x\, dx + \int 3 b c^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {3 b c \operatorname {atanh}{\left (c x \right )}}{x}\, dx + \int b c^{3} x \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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