Optimal. Leaf size=137 \[ -\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^2 d^2 \log (x)-\frac {1}{2} b c^2 d^2 \text {Li}_2(-c x)+\frac {1}{2} b c^2 d^2 \text {Li}_2(c x)-b c^2 d^2 \log \left (1-c^2 x^2\right )+2 b c^2 d^2 \log (x)+\frac {1}{2} b c^2 d^2 \tanh ^{-1}(c x)-\frac {b c d^2}{2 x} \]
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Rubi [A] time = 0.14, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5940, 5916, 325, 206, 266, 36, 29, 31, 5912} \[ -\frac {1}{2} b c^2 d^2 \text {PolyLog}(2,-c x)+\frac {1}{2} b c^2 d^2 \text {PolyLog}(2,c x)-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^2 d^2 \log (x)-b c^2 d^2 \log \left (1-c^2 x^2\right )+2 b c^2 d^2 \log (x)+\frac {1}{2} b c^2 d^2 \tanh ^{-1}(c x)-\frac {b c d^2}{2 x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 206
Rule 266
Rule 325
Rule 5912
Rule 5916
Rule 5940
Rubi steps
\begin {align*} \int \frac {(d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{x^3} \, dx &=\int \left (\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x^3}+\frac {2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+\frac {c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^2 \int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx+\left (2 c d^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (c^2 d^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx\\ &=-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^2 d^2 \log (x)-\frac {1}{2} b c^2 d^2 \text {Li}_2(-c x)+\frac {1}{2} b c^2 d^2 \text {Li}_2(c x)+\frac {1}{2} \left (b c d^2\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (2 b c^2 d^2\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {b c d^2}{2 x}-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^2 d^2 \log (x)-\frac {1}{2} b c^2 d^2 \text {Li}_2(-c x)+\frac {1}{2} b c^2 d^2 \text {Li}_2(c x)+\left (b c^2 d^2\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^3 d^2\right ) \int \frac {1}{1-c^2 x^2} \, dx\\ &=-\frac {b c d^2}{2 x}+\frac {1}{2} b c^2 d^2 \tanh ^{-1}(c x)-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^2 d^2 \log (x)-\frac {1}{2} b c^2 d^2 \text {Li}_2(-c x)+\frac {1}{2} b c^2 d^2 \text {Li}_2(c x)+\left (b c^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\left (b c^4 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b c d^2}{2 x}+\frac {1}{2} b c^2 d^2 \tanh ^{-1}(c x)-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^2 d^2 \log (x)+2 b c^2 d^2 \log (x)-b c^2 d^2 \log \left (1-c^2 x^2\right )-\frac {1}{2} b c^2 d^2 \text {Li}_2(-c x)+\frac {1}{2} b c^2 d^2 \text {Li}_2(c x)\\ \end {align*}
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Mathematica [A] time = 0.10, size = 143, normalized size = 1.04 \[ \frac {d^2 \left (4 a c^2 x^2 \log (x)-8 a c x-2 a-2 b c^2 x^2 \text {Li}_2(-c x)+2 b c^2 x^2 \text {Li}_2(c x)+8 b c^2 x^2 \log (c x)-b c^2 x^2 \log (1-c x)+b c^2 x^2 \log (c x+1)-4 b c^2 x^2 \log \left (1-c^2 x^2\right )-2 b c x-8 b c x \tanh ^{-1}(c x)-2 b \tanh ^{-1}(c x)\right )}{4 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a c^{2} d^{2} x^{2} + 2 \, a c d^{2} x + a d^{2} + {\left (b c^{2} d^{2} x^{2} + 2 \, b c d^{2} x + b d^{2}\right )} \operatorname {artanh}\left (c x\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 176, normalized size = 1.28 \[ c^{2} d^{2} a \ln \left (c x \right )-\frac {2 c \,d^{2} a}{x}-\frac {a \,d^{2}}{2 x^{2}}+c^{2} d^{2} b \arctanh \left (c x \right ) \ln \left (c x \right )-\frac {2 c \,d^{2} b \arctanh \left (c x \right )}{x}-\frac {d^{2} b \arctanh \left (c x \right )}{2 x^{2}}-\frac {b c \,d^{2}}{2 x}+2 c^{2} d^{2} b \ln \left (c x \right )-\frac {5 c^{2} d^{2} b \ln \left (c x -1\right )}{4}-\frac {3 c^{2} d^{2} b \ln \left (c x +1\right )}{4}-\frac {c^{2} d^{2} b \dilog \left (c x \right )}{2}-\frac {c^{2} d^{2} b \dilog \left (c x +1\right )}{2}-\frac {c^{2} d^{2} b \ln \left (c x \right ) \ln \left (c x +1\right )}{2} \]
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 \[ \frac {1}{2} \, b c^{2} d^{2} \int \frac {\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{x}\,{d x} + a c^{2} d^{2} \log \relax (x) - {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} b c d^{2} + \frac {1}{4} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} b d^{2} - \frac {2 \, a c d^{2}}{x} - \frac {a d^{2}}{2 \, x^{2}} \]
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 \[ \int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^2}{x^3} \,d x \]
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 \[ d^{2} \left (\int \frac {a}{x^{3}}\, dx + \int \frac {2 a c}{x^{2}}\, dx + \int \frac {a c^{2}}{x}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 b c \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {b c^{2} \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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