Optimal. Leaf size=194 \[ -\frac {3 \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^3 (c x+1)}+\frac {a+b \tanh ^{-1}(c x)}{2 c^4 d^3 (c x+1)^2}+\frac {3 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^3}+\frac {a x}{c^3 d^3}-\frac {3 b \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 c^4 d^3}-\frac {11 b}{8 c^4 d^3 (c x+1)}+\frac {b}{8 c^4 d^3 (c x+1)^2}+\frac {11 b \tanh ^{-1}(c x)}{8 c^4 d^3}+\frac {b x \tanh ^{-1}(c x)}{c^3 d^3}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^4 d^3} \]
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Rubi [A] time = 0.25, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5940, 5910, 260, 5926, 627, 44, 207, 5918, 2402, 2315} \[ -\frac {3 b \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 c^4 d^3}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^3 (c x+1)}+\frac {a+b \tanh ^{-1}(c x)}{2 c^4 d^3 (c x+1)^2}+\frac {3 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^3}+\frac {a x}{c^3 d^3}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^4 d^3}-\frac {11 b}{8 c^4 d^3 (c x+1)}+\frac {b}{8 c^4 d^3 (c x+1)^2}+\frac {b x \tanh ^{-1}(c x)}{c^3 d^3}+\frac {11 b \tanh ^{-1}(c x)}{8 c^4 d^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 207
Rule 260
Rule 627
Rule 2315
Rule 2402
Rule 5910
Rule 5918
Rule 5926
Rule 5940
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{(d+c d x)^3} \, dx &=\int \left (\frac {a+b \tanh ^{-1}(c x)}{c^3 d^3}-\frac {a+b \tanh ^{-1}(c x)}{c^3 d^3 (1+c x)^3}+\frac {3 \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3 (1+c x)^2}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3 (1+c x)}\right ) \, dx\\ &=\frac {\int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^3 d^3}-\frac {\int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx}{c^3 d^3}+\frac {3 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c^3 d^3}-\frac {3 \int \frac {a+b \tanh ^{-1}(c x)}{1+c x} \, dx}{c^3 d^3}\\ &=\frac {a x}{c^3 d^3}+\frac {a+b \tanh ^{-1}(c x)}{2 c^4 d^3 (1+c x)^2}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^3 (1+c x)}+\frac {3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^4 d^3}-\frac {b \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx}{2 c^3 d^3}+\frac {b \int \tanh ^{-1}(c x) \, dx}{c^3 d^3}+\frac {(3 b) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c^3 d^3}-\frac {(3 b) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^3 d^3}\\ &=\frac {a x}{c^3 d^3}+\frac {b x \tanh ^{-1}(c x)}{c^3 d^3}+\frac {a+b \tanh ^{-1}(c x)}{2 c^4 d^3 (1+c x)^2}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^3 (1+c x)}+\frac {3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^4 d^3}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{c^4 d^3}-\frac {b \int \frac {1}{(1-c x) (1+c x)^3} \, dx}{2 c^3 d^3}+\frac {(3 b) \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{c^3 d^3}-\frac {b \int \frac {x}{1-c^2 x^2} \, dx}{c^2 d^3}\\ &=\frac {a x}{c^3 d^3}+\frac {b x \tanh ^{-1}(c x)}{c^3 d^3}+\frac {a+b \tanh ^{-1}(c x)}{2 c^4 d^3 (1+c x)^2}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^3 (1+c x)}+\frac {3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^4 d^3}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^4 d^3}-\frac {3 b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 c^4 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^3 d^3}+\frac {(3 b) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^3 d^3}\\ &=\frac {a x}{c^3 d^3}+\frac {b}{8 c^4 d^3 (1+c x)^2}-\frac {11 b}{8 c^4 d^3 (1+c x)}+\frac {b x \tanh ^{-1}(c x)}{c^3 d^3}+\frac {a+b \tanh ^{-1}(c x)}{2 c^4 d^3 (1+c x)^2}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^3 (1+c x)}+\frac {3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^4 d^3}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^4 d^3}-\frac {3 b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 c^4 d^3}+\frac {b \int \frac {1}{-1+c^2 x^2} \, dx}{8 c^3 d^3}-\frac {(3 b) \int \frac {1}{-1+c^2 x^2} \, dx}{2 c^3 d^3}\\ &=\frac {a x}{c^3 d^3}+\frac {b}{8 c^4 d^3 (1+c x)^2}-\frac {11 b}{8 c^4 d^3 (1+c x)}+\frac {11 b \tanh ^{-1}(c x)}{8 c^4 d^3}+\frac {b x \tanh ^{-1}(c x)}{c^3 d^3}+\frac {a+b \tanh ^{-1}(c x)}{2 c^4 d^3 (1+c x)^2}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^3 (1+c x)}+\frac {3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^4 d^3}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^4 d^3}-\frac {3 b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 c^4 d^3}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 167, normalized size = 0.86 \[ \frac {32 a c x-\frac {96 a}{c x+1}+\frac {16 a}{(c x+1)^2}-96 a \log (c x+1)+b \left (16 \log \left (1-c^2 x^2\right )-48 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+20 \sinh \left (2 \tanh ^{-1}(c x)\right )-\sinh \left (4 \tanh ^{-1}(c x)\right )-20 \cosh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (4 \tanh ^{-1}(c x)\right )+4 \tanh ^{-1}(c x) \left (8 c x+24 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+10 \sinh \left (2 \tanh ^{-1}(c x)\right )-\sinh \left (4 \tanh ^{-1}(c x)\right )-10 \cosh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (4 \tanh ^{-1}(c x)\right )\right )\right )}{32 c^4 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{3} \operatorname {artanh}\left (c x\right ) + a x^{3}}{c^{3} d^{3} x^{3} + 3 \, c^{2} d^{3} x^{2} + 3 \, c d^{3} x + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{3}}{{\left (c d x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 270, normalized size = 1.39 \[ \frac {a x}{c^{3} d^{3}}+\frac {a}{2 c^{4} d^{3} \left (c x +1\right )^{2}}-\frac {3 a}{c^{4} d^{3} \left (c x +1\right )}-\frac {3 a \ln \left (c x +1\right )}{c^{4} d^{3}}+\frac {b x \arctanh \left (c x \right )}{c^{3} d^{3}}+\frac {b \arctanh \left (c x \right )}{2 c^{4} d^{3} \left (c x +1\right )^{2}}-\frac {3 b \arctanh \left (c x \right )}{c^{4} d^{3} \left (c x +1\right )}-\frac {3 b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{c^{4} d^{3}}+\frac {3 b \ln \left (c x +1\right )^{2}}{4 c^{4} d^{3}}-\frac {3 b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 c^{4} d^{3}}+\frac {3 b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 c^{4} d^{3}}+\frac {3 b \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 c^{4} d^{3}}+\frac {b}{8 c^{4} d^{3} \left (c x +1\right )^{2}}-\frac {11 b}{8 c^{4} d^{3} \left (c x +1\right )}+\frac {19 b \ln \left (c x +1\right )}{16 c^{4} d^{3}}-\frac {3 b \ln \left (c x -1\right )}{16 c^{4} d^{3}} \]
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}{32} \, {\left (2 \, c^{4} {\left (\frac {2 \, {\left (7 \, c x + 6\right )}}{c^{10} d^{3} x^{2} + 2 \, c^{9} d^{3} x + c^{8} d^{3}} - \frac {8 \, x}{c^{7} d^{3}} + \frac {17 \, \log \left (c x + 1\right )}{c^{8} d^{3}} - \frac {\log \left (c x - 1\right )}{c^{8} d^{3}}\right )} - 32 \, c^{4} \int \frac {x^{4} \log \left (c x + 1\right )}{2 \, {\left (c^{7} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{3} - 2 \, c^{4} d^{3} x - c^{3} d^{3}\right )}}\,{d x} - 6 \, c^{3} {\left (\frac {2 \, {\left (5 \, c x + 4\right )}}{c^{9} d^{3} x^{2} + 2 \, c^{8} d^{3} x + c^{7} d^{3}} + \frac {7 \, \log \left (c x + 1\right )}{c^{7} d^{3}} + \frac {\log \left (c x - 1\right )}{c^{7} d^{3}}\right )} + 128 \, c^{3} \int \frac {x^{3} \log \left (c x + 1\right )}{2 \, {\left (c^{7} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{3} - 2 \, c^{4} d^{3} x - c^{3} d^{3}\right )}}\,{d x} + 288 \, c^{2} \int \frac {x^{2} \log \left (c x + 1\right )}{2 \, {\left (c^{7} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{3} - 2 \, c^{4} d^{3} x - c^{3} d^{3}\right )}}\,{d x} + 9 \, c {\left (\frac {2 \, x}{c^{6} d^{3} x^{2} + 2 \, c^{5} d^{3} x + c^{4} d^{3}} - \frac {\log \left (c x + 1\right )}{c^{5} d^{3}} + \frac {\log \left (c x - 1\right )}{c^{5} d^{3}}\right )} + 288 \, c \int \frac {x \log \left (c x + 1\right )}{2 \, {\left (c^{7} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{3} - 2 \, c^{4} d^{3} x - c^{3} d^{3}\right )}}\,{d x} + \frac {8 \, {\left (2 \, c^{3} x^{3} + 4 \, c^{2} x^{2} - 4 \, c x - 6 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x + 1\right ) - 5\right )} \log \left (-c x + 1\right )}{c^{6} d^{3} x^{2} + 2 \, c^{5} d^{3} x + c^{4} d^{3}} + \frac {10 \, {\left (c x + 2\right )}}{c^{6} d^{3} x^{2} + 2 \, c^{5} d^{3} x + c^{4} d^{3}} - \frac {5 \, \log \left (c x + 1\right )}{c^{4} d^{3}} + \frac {5 \, \log \left (c x - 1\right )}{c^{4} d^{3}} + 96 \, \int \frac {\log \left (c x + 1\right )}{2 \, {\left (c^{7} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{3} - 2 \, c^{4} d^{3} x - c^{3} d^{3}\right )}}\,{d x}\right )} b - \frac {1}{2} \, a {\left (\frac {6 \, c x + 5}{c^{6} d^{3} x^{2} + 2 \, c^{5} d^{3} x + c^{4} d^{3}} - \frac {2 \, x}{c^{3} d^{3}} + \frac {6 \, \log \left (c x + 1\right )}{c^{4} d^{3}}\right )} \]
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 \[ \int \frac {x^3\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a x^{3}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {b x^{3} \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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