Optimal. Leaf size=150 \[ -\frac {b}{8 c^3 d^3 (1+c x)^2}+\frac {7 b}{8 c^3 d^3 (1+c x)}-\frac {7 b \tanh ^{-1}(c x)}{8 c^3 d^3}-\frac {a+b \tanh ^{-1}(c x)}{2 c^3 d^3 (1+c x)^2}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d^3}+\frac {b \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 c^3 d^3} \]
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Rubi [A]
time = 0.17, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6087, 6063,
641, 46, 213, 6055, 2449, 2352} \begin {gather*} \frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3 (c x+1)}-\frac {a+b \tanh ^{-1}(c x)}{2 c^3 d^3 (c x+1)^2}-\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3}+\frac {b \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 c^3 d^3}+\frac {7 b}{8 c^3 d^3 (c x+1)}-\frac {b}{8 c^3 d^3 (c x+1)^2}-\frac {7 b \tanh ^{-1}(c x)}{8 c^3 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 213
Rule 641
Rule 2352
Rule 2449
Rule 6055
Rule 6063
Rule 6087
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{(d+c d x)^3} \, dx &=\int \left (\frac {a+b \tanh ^{-1}(c x)}{c^2 d^3 (1+c x)^3}-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^3 (1+c x)^2}+\frac {a+b \tanh ^{-1}(c x)}{c^2 d^3 (1+c x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx}{c^2 d^3}+\frac {\int \frac {a+b \tanh ^{-1}(c x)}{1+c x} \, dx}{c^2 d^3}-\frac {2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c^2 d^3}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{2 c^3 d^3 (1+c x)^2}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d^3}+\frac {b \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx}{2 c^2 d^3}+\frac {b \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^2 d^3}-\frac {(2 b) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c^2 d^3}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{2 c^3 d^3 (1+c x)^2}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d^3}+\frac {b \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{c^3 d^3}+\frac {b \int \frac {1}{(1-c x) (1+c x)^3} \, dx}{2 c^2 d^3}-\frac {(2 b) \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{c^2 d^3}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{2 c^3 d^3 (1+c x)^2}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d^3}+\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 c^3 d^3}+\frac {b \int \left (\frac {1}{2 (1+c x)^3}+\frac {1}{4 (1+c x)^2}-\frac {1}{4 \left (-1+c^2 x^2\right )}\right ) \, dx}{2 c^2 d^3}-\frac {(2 b) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^2 d^3}\\ &=-\frac {b}{8 c^3 d^3 (1+c x)^2}+\frac {7 b}{8 c^3 d^3 (1+c x)}-\frac {a+b \tanh ^{-1}(c x)}{2 c^3 d^3 (1+c x)^2}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d^3}+\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 c^3 d^3}-\frac {b \int \frac {1}{-1+c^2 x^2} \, dx}{8 c^2 d^3}+\frac {b \int \frac {1}{-1+c^2 x^2} \, dx}{c^2 d^3}\\ &=-\frac {b}{8 c^3 d^3 (1+c x)^2}+\frac {7 b}{8 c^3 d^3 (1+c x)}-\frac {7 b \tanh ^{-1}(c x)}{8 c^3 d^3}-\frac {a+b \tanh ^{-1}(c x)}{2 c^3 d^3 (1+c x)^2}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d^3}+\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 c^3 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 145, normalized size = 0.97 \begin {gather*} \frac {-\frac {16 a}{(1+c x)^2}+\frac {64 a}{1+c x}+32 a \log (1+c x)+b \left (12 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )+16 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-12 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )+4 \tanh ^{-1}(c x) \left (6 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )-8 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-6 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )\right )\right )}{32 c^3 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 208, normalized size = 1.39
method | result | size |
derivativedivides | \(\frac {\frac {a \ln \left (c x +1\right )}{d^{3}}+\frac {2 a}{d^{3} \left (c x +1\right )}-\frac {a}{2 d^{3} \left (c x +1\right )^{2}}+\frac {b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d^{3}}+\frac {2 b \arctanh \left (c x \right )}{d^{3} \left (c x +1\right )}-\frac {b \arctanh \left (c x \right )}{2 d^{3} \left (c x +1\right )^{2}}+\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d^{3}}-\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3}}-\frac {b \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3}}-\frac {b \ln \left (c x +1\right )^{2}}{4 d^{3}}+\frac {7 b \ln \left (c x -1\right )}{16 d^{3}}-\frac {b}{8 d^{3} \left (c x +1\right )^{2}}+\frac {7 b}{8 d^{3} \left (c x +1\right )}-\frac {7 b \ln \left (c x +1\right )}{16 d^{3}}}{c^{3}}\) | \(208\) |
default | \(\frac {\frac {a \ln \left (c x +1\right )}{d^{3}}+\frac {2 a}{d^{3} \left (c x +1\right )}-\frac {a}{2 d^{3} \left (c x +1\right )^{2}}+\frac {b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d^{3}}+\frac {2 b \arctanh \left (c x \right )}{d^{3} \left (c x +1\right )}-\frac {b \arctanh \left (c x \right )}{2 d^{3} \left (c x +1\right )^{2}}+\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d^{3}}-\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3}}-\frac {b \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3}}-\frac {b \ln \left (c x +1\right )^{2}}{4 d^{3}}+\frac {7 b \ln \left (c x -1\right )}{16 d^{3}}-\frac {b}{8 d^{3} \left (c x +1\right )^{2}}+\frac {7 b}{8 d^{3} \left (c x +1\right )}-\frac {7 b \ln \left (c x +1\right )}{16 d^{3}}}{c^{3}}\) | \(208\) |
risch | \(\frac {b \ln \left (c x +1\right )^{2}}{4 c^{3} d^{3}}+\frac {\left (\frac {b x}{c^{2}}+\frac {3 b}{4 c^{3}}\right ) \ln \left (c x +1\right )}{d^{3} \left (c x +1\right )^{2}}-\frac {7 b \ln \left (-c x -1\right )}{16 d^{3} c^{3}}-\frac {b \ln \left (-c x +1\right ) x}{2 d^{3} c^{2} \left (-c x -1\right )}+\frac {b \ln \left (-c x +1\right )}{2 d^{3} c^{3} \left (-c x -1\right )}+\frac {b}{8 d^{3} c^{3} \left (-c x -1\right )}-\frac {b \ln \left (-c x +1\right ) x^{2}}{16 d^{3} c \left (-c x -1\right )^{2}}-\frac {b \ln \left (-c x +1\right ) x}{8 d^{3} c^{2} \left (-c x -1\right )^{2}}+\frac {3 b \ln \left (-c x +1\right )}{16 d^{3} c^{3} \left (-c x -1\right )^{2}}-\frac {b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{2 d^{3} c^{3}}+\frac {b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3} c^{3}}+\frac {b \dilog \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3} c^{3}}-\frac {2 a}{d^{3} c^{3} \left (-c x -1\right )}-\frac {a}{2 d^{3} c^{3} \left (-c x -1\right )^{2}}+\frac {a \ln \left (-c x -1\right )}{d^{3} c^{3}}-\frac {b}{8 c^{3} d^{3} \left (c x +1\right )^{2}}+\frac {b}{c^{3} d^{3} \left (c x +1\right )}\) | \(349\) |
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*} \frac {\int \frac {a x^{2}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {b x^{2} \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx}{d^{3}} \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 {x^2\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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