Optimal. Leaf size=227 \[ \frac {4 \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3 (c x+1)}-\frac {a+b \tanh ^{-1}(c x)}{2 c^5 d^3 (c x+1)^2}-\frac {6 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^3 d^3}-\frac {3 a x}{c^4 d^3}+\frac {3 b \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{c^5 d^3}+\frac {15 b}{8 c^5 d^3 (c x+1)}-\frac {b}{8 c^5 d^3 (c x+1)^2}-\frac {19 b \tanh ^{-1}(c x)}{8 c^5 d^3}+\frac {b x}{2 c^4 d^3}-\frac {3 b x \tanh ^{-1}(c x)}{c^4 d^3}-\frac {3 b \log \left (1-c^2 x^2\right )}{2 c^5 d^3} \]
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Rubi [A] time = 0.29, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {5940, 5910, 260, 5916, 321, 206, 5926, 627, 44, 207, 5918, 2402, 2315} \[ \frac {3 b \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{c^5 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^3 d^3}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3 (c x+1)}-\frac {a+b \tanh ^{-1}(c x)}{2 c^5 d^3 (c x+1)^2}-\frac {6 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3}-\frac {3 a x}{c^4 d^3}-\frac {3 b \log \left (1-c^2 x^2\right )}{2 c^5 d^3}+\frac {b x}{2 c^4 d^3}+\frac {15 b}{8 c^5 d^3 (c x+1)}-\frac {b}{8 c^5 d^3 (c x+1)^2}-\frac {3 b x \tanh ^{-1}(c x)}{c^4 d^3}-\frac {19 b \tanh ^{-1}(c x)}{8 c^5 d^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 207
Rule 260
Rule 321
Rule 627
Rule 2315
Rule 2402
Rule 5910
Rule 5916
Rule 5918
Rule 5926
Rule 5940
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \tanh ^{-1}(c x)\right )}{(d+c d x)^3} \, dx &=\int \left (-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^3}+\frac {x \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3}+\frac {a+b \tanh ^{-1}(c x)}{c^4 d^3 (1+c x)^3}-\frac {4 \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^3 (1+c x)^2}+\frac {6 \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^3 (1+c x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx}{c^4 d^3}-\frac {3 \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^4 d^3}-\frac {4 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c^4 d^3}+\frac {6 \int \frac {a+b \tanh ^{-1}(c x)}{1+c x} \, dx}{c^4 d^3}+\frac {\int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^3 d^3}\\ &=-\frac {3 a x}{c^4 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^3 d^3}-\frac {a+b \tanh ^{-1}(c x)}{2 c^5 d^3 (1+c x)^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3 (1+c x)}-\frac {6 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^5 d^3}+\frac {b \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx}{2 c^4 d^3}-\frac {(3 b) \int \tanh ^{-1}(c x) \, dx}{c^4 d^3}-\frac {(4 b) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c^4 d^3}+\frac {(6 b) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^4 d^3}-\frac {b \int \frac {x^2}{1-c^2 x^2} \, dx}{2 c^2 d^3}\\ &=-\frac {3 a x}{c^4 d^3}+\frac {b x}{2 c^4 d^3}-\frac {3 b x \tanh ^{-1}(c x)}{c^4 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^3 d^3}-\frac {a+b \tanh ^{-1}(c x)}{2 c^5 d^3 (1+c x)^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3 (1+c x)}-\frac {6 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^5 d^3}+\frac {(6 b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{c^5 d^3}+\frac {b \int \frac {1}{(1-c x) (1+c x)^3} \, dx}{2 c^4 d^3}-\frac {b \int \frac {1}{1-c^2 x^2} \, dx}{2 c^4 d^3}-\frac {(4 b) \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{c^4 d^3}+\frac {(3 b) \int \frac {x}{1-c^2 x^2} \, dx}{c^3 d^3}\\ &=-\frac {3 a x}{c^4 d^3}+\frac {b x}{2 c^4 d^3}-\frac {b \tanh ^{-1}(c x)}{2 c^5 d^3}-\frac {3 b x \tanh ^{-1}(c x)}{c^4 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^3 d^3}-\frac {a+b \tanh ^{-1}(c x)}{2 c^5 d^3 (1+c x)^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3 (1+c x)}-\frac {6 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^5 d^3}-\frac {3 b \log \left (1-c^2 x^2\right )}{2 c^5 d^3}+\frac {3 b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 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^4 d^3}-\frac {(4 b) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^4 d^3}\\ &=-\frac {3 a x}{c^4 d^3}+\frac {b x}{2 c^4 d^3}-\frac {b}{8 c^5 d^3 (1+c x)^2}+\frac {15 b}{8 c^5 d^3 (1+c x)}-\frac {b \tanh ^{-1}(c x)}{2 c^5 d^3}-\frac {3 b x \tanh ^{-1}(c x)}{c^4 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^3 d^3}-\frac {a+b \tanh ^{-1}(c x)}{2 c^5 d^3 (1+c x)^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3 (1+c x)}-\frac {6 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^5 d^3}-\frac {3 b \log \left (1-c^2 x^2\right )}{2 c^5 d^3}+\frac {3 b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^3}-\frac {b \int \frac {1}{-1+c^2 x^2} \, dx}{8 c^4 d^3}+\frac {(2 b) \int \frac {1}{-1+c^2 x^2} \, dx}{c^4 d^3}\\ &=-\frac {3 a x}{c^4 d^3}+\frac {b x}{2 c^4 d^3}-\frac {b}{8 c^5 d^3 (1+c x)^2}+\frac {15 b}{8 c^5 d^3 (1+c x)}-\frac {19 b \tanh ^{-1}(c x)}{8 c^5 d^3}-\frac {3 b x \tanh ^{-1}(c x)}{c^4 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^3 d^3}-\frac {a+b \tanh ^{-1}(c x)}{2 c^5 d^3 (1+c x)^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3 (1+c x)}-\frac {6 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^5 d^3}-\frac {3 b \log \left (1-c^2 x^2\right )}{2 c^5 d^3}+\frac {3 b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^3}\\ \end {align*}
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Mathematica [A] time = 0.82, size = 189, normalized size = 0.83 \[ \frac {16 a c^2 x^2-96 a c x+\frac {128 a}{c x+1}-\frac {16 a}{(c x+1)^2}+192 a \log (c x+1)+b \left (-48 \log \left (1-c^2 x^2\right )+4 \tanh ^{-1}(c x) \left (4 c^2 x^2-24 c x-48 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-14 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )+14 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )-4\right )+96 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+16 c x-28 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )+28 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )\right )}{32 c^5 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{4} \operatorname {artanh}\left (c x\right ) + a x^{4}}{c^{3} d^{3} x^{3} + 3 \, c^{2} d^{3} x^{2} + 3 \, c d^{3} x + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{4}}{{\left (c d x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 319, normalized size = 1.41 \[ \frac {a \,x^{2}}{2 c^{3} d^{3}}-\frac {3 a x}{c^{4} d^{3}}-\frac {a}{2 c^{5} d^{3} \left (c x +1\right )^{2}}+\frac {4 a}{c^{5} d^{3} \left (c x +1\right )}+\frac {6 a \ln \left (c x +1\right )}{c^{5} d^{3}}+\frac {b \arctanh \left (c x \right ) x^{2}}{2 c^{3} d^{3}}-\frac {3 b x \arctanh \left (c x \right )}{c^{4} d^{3}}-\frac {b \arctanh \left (c x \right )}{2 c^{5} d^{3} \left (c x +1\right )^{2}}+\frac {4 b \arctanh \left (c x \right )}{c^{5} d^{3} \left (c x +1\right )}+\frac {6 b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{c^{5} d^{3}}-\frac {3 b \ln \left (c x +1\right )^{2}}{2 c^{5} d^{3}}+\frac {3 b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{c^{5} d^{3}}-\frac {3 b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{c^{5} d^{3}}-\frac {3 b \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{c^{5} d^{3}}+\frac {b x}{2 c^{4} d^{3}}+\frac {b}{2 c^{5} d^{3}}-\frac {b}{8 c^{5} d^{3} \left (c x +1\right )^{2}}+\frac {15 b}{8 c^{5} d^{3} \left (c x +1\right )}-\frac {43 b \ln \left (c x +1\right )}{16 c^{5} d^{3}}-\frac {5 b \ln \left (c x -1\right )}{16 c^{5} d^{3}} \]
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 (c^{5} {\left (\frac {2 \, {\left (9 \, c x + 8\right )}}{c^{12} d^{3} x^{2} + 2 \, c^{11} d^{3} x + c^{10} d^{3}} + \frac {4 \, {\left (c x^{2} - 4 \, x\right )}}{c^{9} d^{3}} + \frac {31 \, \log \left (c x + 1\right )}{c^{10} d^{3}} + \frac {\log \left (c x - 1\right )}{c^{10} d^{3}}\right )} + 32 \, c^{5} \int \frac {x^{5} \log \left (c x + 1\right )}{2 \, {\left (c^{8} d^{3} x^{4} + 2 \, c^{7} d^{3} x^{3} - 2 \, c^{5} d^{3} x - c^{4} d^{3}\right )}}\,{d x} + 3 \, c^{4} {\left (\frac {2 \, {\left (7 \, c x + 6\right )}}{c^{11} d^{3} x^{2} + 2 \, c^{10} d^{3} x + c^{9} d^{3}} - \frac {8 \, x}{c^{8} d^{3}} + \frac {17 \, \log \left (c x + 1\right )}{c^{9} d^{3}} - \frac {\log \left (c x - 1\right )}{c^{9} d^{3}}\right )} - 32 \, c^{4} \int \frac {x^{4} \log \left (c x + 1\right )}{2 \, {\left (c^{8} d^{3} x^{4} + 2 \, c^{7} d^{3} x^{3} - 2 \, c^{5} d^{3} x - c^{4} d^{3}\right )}}\,{d x} - 15 \, c^{3} {\left (\frac {2 \, {\left (5 \, c x + 4\right )}}{c^{10} d^{3} x^{2} + 2 \, c^{9} d^{3} x + c^{8} d^{3}} + \frac {7 \, \log \left (c x + 1\right )}{c^{8} d^{3}} + \frac {\log \left (c x - 1\right )}{c^{8} d^{3}}\right )} + 192 \, c^{3} \int \frac {x^{3} \log \left (c x + 1\right )}{2 \, {\left (c^{8} d^{3} x^{4} + 2 \, c^{7} d^{3} x^{3} - 2 \, c^{5} d^{3} x - c^{4} d^{3}\right )}}\,{d x} + 9 \, c^{2} {\left (\frac {2 \, {\left (3 \, c x + 2\right )}}{c^{9} d^{3} x^{2} + 2 \, c^{8} d^{3} x + c^{7} d^{3}} + \frac {\log \left (c x + 1\right )}{c^{7} d^{3}} - \frac {\log \left (c x - 1\right )}{c^{7} d^{3}}\right )} + 576 \, c^{2} \int \frac {x^{2} \log \left (c x + 1\right )}{2 \, {\left (c^{8} d^{3} x^{4} + 2 \, c^{7} d^{3} x^{3} - 2 \, c^{5} d^{3} x - c^{4} d^{3}\right )}}\,{d x} + 9 \, c {\left (\frac {2 \, x}{c^{7} d^{3} x^{2} + 2 \, c^{6} d^{3} x + c^{5} d^{3}} - \frac {\log \left (c x + 1\right )}{c^{6} d^{3}} + \frac {\log \left (c x - 1\right )}{c^{6} d^{3}}\right )} + 576 \, c \int \frac {x \log \left (c x + 1\right )}{2 \, {\left (c^{8} d^{3} x^{4} + 2 \, c^{7} d^{3} x^{3} - 2 \, c^{5} d^{3} x - c^{4} d^{3}\right )}}\,{d x} - \frac {8 \, {\left (c^{4} x^{4} - 4 \, c^{3} x^{3} - 11 \, c^{2} x^{2} + 2 \, c x + 12 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x + 1\right ) + 7\right )} \log \left (-c x + 1\right )}{c^{7} d^{3} x^{2} + 2 \, c^{6} d^{3} x + c^{5} d^{3}} + \frac {14 \, {\left (c x + 2\right )}}{c^{7} d^{3} x^{2} + 2 \, c^{6} d^{3} x + c^{5} d^{3}} - \frac {7 \, \log \left (c x + 1\right )}{c^{5} d^{3}} + \frac {7 \, \log \left (c x - 1\right )}{c^{5} d^{3}} + 192 \, \int \frac {\log \left (c x + 1\right )}{2 \, {\left (c^{8} d^{3} x^{4} + 2 \, c^{7} d^{3} x^{3} - 2 \, c^{5} d^{3} x - c^{4} d^{3}\right )}}\,{d x}\right )} b + \frac {1}{2} \, a {\left (\frac {8 \, c x + 7}{c^{7} d^{3} x^{2} + 2 \, c^{6} d^{3} x + c^{5} d^{3}} + \frac {c x^{2} - 6 \, x}{c^{4} d^{3}} + \frac {12 \, \log \left (c x + 1\right )}{c^{5} 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 = -0.00 \[ \int \frac {x^4\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a x^{4}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {b x^{4} \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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