Optimal. Leaf size=275 \[ \frac {a d^2 x}{e^3}-\frac {b d x}{2 c e^2}+\frac {b x^2}{6 c e}+\frac {b d \tanh ^{-1}(c x)}{2 c^2 e^2}+\frac {b d^2 x \tanh ^{-1}(c x)}{e^3}-\frac {d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{e^4}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^4}+\frac {b d^2 \log \left (1-c^2 x^2\right )}{2 c e^3}+\frac {b \log \left (1-c^2 x^2\right )}{6 c^3 e}-\frac {b d^3 \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 e^4}+\frac {b d^3 \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^4} \]
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Rubi [A]
time = 0.20, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 12, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {6087, 6021,
266, 6037, 327, 212, 272, 45, 6057, 2449, 2352, 2497} \begin {gather*} \frac {d^3 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{e^4}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^4}-\frac {d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac {a d^2 x}{e^3}+\frac {b d^2 \log \left (1-c^2 x^2\right )}{2 c e^3}+\frac {b d \tanh ^{-1}(c x)}{2 c^2 e^2}+\frac {b \log \left (1-c^2 x^2\right )}{6 c^3 e}-\frac {b d^3 \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 e^4}+\frac {b d^3 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e^4}+\frac {b d^2 x \tanh ^{-1}(c x)}{e^3}-\frac {b d x}{2 c e^2}+\frac {b x^2}{6 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 212
Rule 266
Rule 272
Rule 327
Rule 2352
Rule 2449
Rule 2497
Rule 6021
Rule 6037
Rule 6057
Rule 6087
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{d+e x} \, dx &=\int \left (\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{e^3}-\frac {d x \left (a+b \tanh ^{-1}(c x)\right )}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{e}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {d^2 \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{e^3}-\frac {d^3 \int \frac {a+b \tanh ^{-1}(c x)}{d+e x} \, dx}{e^3}-\frac {d \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{e^2}+\frac {\int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{e}\\ &=\frac {a d^2 x}{e^3}-\frac {d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{e^4}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^4}-\frac {\left (b c d^3\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{e^4}+\frac {\left (b c d^3\right ) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{1-c^2 x^2} \, dx}{e^4}+\frac {\left (b d^2\right ) \int \tanh ^{-1}(c x) \, dx}{e^3}+\frac {(b c d) \int \frac {x^2}{1-c^2 x^2} \, dx}{2 e^2}-\frac {(b c) \int \frac {x^3}{1-c^2 x^2} \, dx}{3 e}\\ &=\frac {a d^2 x}{e^3}-\frac {b d x}{2 c e^2}+\frac {b d^2 x \tanh ^{-1}(c x)}{e^3}-\frac {d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{e^4}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^4}+\frac {b d^3 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^4}-\frac {\left (b d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{e^4}-\frac {\left (b c d^2\right ) \int \frac {x}{1-c^2 x^2} \, dx}{e^3}+\frac {(b d) \int \frac {1}{1-c^2 x^2} \, dx}{2 c e^2}-\frac {(b c) \text {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )}{6 e}\\ &=\frac {a d^2 x}{e^3}-\frac {b d x}{2 c e^2}+\frac {b d \tanh ^{-1}(c x)}{2 c^2 e^2}+\frac {b d^2 x \tanh ^{-1}(c x)}{e^3}-\frac {d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{e^4}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^4}+\frac {b d^2 \log \left (1-c^2 x^2\right )}{2 c e^3}-\frac {b d^3 \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 e^4}+\frac {b d^3 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^4}-\frac {(b c) \text {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 e}\\ &=\frac {a d^2 x}{e^3}-\frac {b d x}{2 c e^2}+\frac {b x^2}{6 c e}+\frac {b d \tanh ^{-1}(c x)}{2 c^2 e^2}+\frac {b d^2 x \tanh ^{-1}(c x)}{e^3}-\frac {d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{e^4}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^4}+\frac {b d^2 \log \left (1-c^2 x^2\right )}{2 c e^3}+\frac {b \log \left (1-c^2 x^2\right )}{6 c^3 e}-\frac {b d^3 \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 e^4}+\frac {b d^3 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^4}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.53, size = 474, normalized size = 1.72 \begin {gather*} \frac {-\frac {b e^3}{c^3}+6 a d^2 e x-\frac {3 b d e^2 x}{c}-3 a d e^2 x^2+\frac {b e^3 x^2}{c}+2 a e^3 x^3+\frac {3 b d e^2 \tanh ^{-1}(c x)}{c^2}-3 i b d^3 \pi \tanh ^{-1}(c x)+6 b d^2 e x \tanh ^{-1}(c x)-3 b d e^2 x^2 \tanh ^{-1}(c x)+2 b e^3 x^3 \tanh ^{-1}(c x)-6 b d^3 \tanh ^{-1}\left (\frac {c d}{e}\right ) \tanh ^{-1}(c x)+3 b d^3 \tanh ^{-1}(c x)^2-\frac {3 b d^2 e \tanh ^{-1}(c x)^2}{c}+\frac {3 b d^2 \sqrt {1-\frac {c^2 d^2}{e^2}} e e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )} \tanh ^{-1}(c x)^2}{c}+6 b d^3 \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+3 i b d^3 \pi \log \left (1+e^{2 \tanh ^{-1}(c x)}\right )-6 b d^3 \tanh ^{-1}\left (\frac {c d}{e}\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-6 b d^3 \tanh ^{-1}(c x) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-6 a d^3 \log (d+e x)+\frac {3 b d^2 e \log \left (1-c^2 x^2\right )}{c}+\frac {b e^3 \log \left (1-c^2 x^2\right )}{c^3}+\frac {3}{2} i b d^3 \pi \log \left (1-c^2 x^2\right )+6 b d^3 \tanh ^{-1}\left (\frac {c d}{e}\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )-3 b d^3 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+3 b d^3 \text {PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )}{6 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.74, size = 423, normalized size = 1.54
method | result | size |
derivativedivides | \(\frac {\frac {a \,c^{4} d^{2} x}{e^{3}}-\frac {a \,c^{4} d \,x^{2}}{2 e^{2}}+\frac {a \,c^{4} x^{3}}{3 e}-\frac {a \,c^{4} d^{3} \ln \left (c e x +d c \right )}{e^{4}}+\frac {b \,c^{4} \arctanh \left (c x \right ) d^{2} x}{e^{3}}-\frac {b \,c^{4} \arctanh \left (c x \right ) d \,x^{2}}{2 e^{2}}+\frac {b \,c^{4} \arctanh \left (c x \right ) x^{3}}{3 e}-\frac {b \,c^{4} \arctanh \left (c x \right ) d^{3} \ln \left (c e x +d c \right )}{e^{4}}-\frac {b \,c^{4} d^{3} \ln \left (c e x +d c \right ) \ln \left (\frac {c e x -e}{-d c -e}\right )}{2 e^{4}}-\frac {b \,c^{4} d^{3} \dilog \left (\frac {c e x -e}{-d c -e}\right )}{2 e^{4}}+\frac {b \,c^{4} d^{3} \ln \left (c e x +d c \right ) \ln \left (\frac {c e x +e}{-d c +e}\right )}{2 e^{4}}+\frac {b \,c^{4} d^{3} \dilog \left (\frac {c e x +e}{-d c +e}\right )}{2 e^{4}}-\frac {2 b \,c^{3} d^{2}}{3 e^{3}}-\frac {b \,c^{3} d x}{2 e^{2}}+\frac {b \,c^{3} x^{2}}{6 e}+\frac {b \,c^{3} \ln \left (-c e x +e \right ) d^{2}}{2 e^{3}}-\frac {b \,c^{2} \ln \left (-c e x +e \right ) d}{4 e^{2}}+\frac {b c \ln \left (-c e x +e \right )}{6 e}+\frac {b \,c^{3} \ln \left (-c e x -e \right ) d^{2}}{2 e^{3}}+\frac {b \,c^{2} \ln \left (-c e x -e \right ) d}{4 e^{2}}+\frac {b c \ln \left (-c e x -e \right )}{6 e}}{c^{4}}\) | \(423\) |
default | \(\frac {\frac {a \,c^{4} d^{2} x}{e^{3}}-\frac {a \,c^{4} d \,x^{2}}{2 e^{2}}+\frac {a \,c^{4} x^{3}}{3 e}-\frac {a \,c^{4} d^{3} \ln \left (c e x +d c \right )}{e^{4}}+\frac {b \,c^{4} \arctanh \left (c x \right ) d^{2} x}{e^{3}}-\frac {b \,c^{4} \arctanh \left (c x \right ) d \,x^{2}}{2 e^{2}}+\frac {b \,c^{4} \arctanh \left (c x \right ) x^{3}}{3 e}-\frac {b \,c^{4} \arctanh \left (c x \right ) d^{3} \ln \left (c e x +d c \right )}{e^{4}}-\frac {b \,c^{4} d^{3} \ln \left (c e x +d c \right ) \ln \left (\frac {c e x -e}{-d c -e}\right )}{2 e^{4}}-\frac {b \,c^{4} d^{3} \dilog \left (\frac {c e x -e}{-d c -e}\right )}{2 e^{4}}+\frac {b \,c^{4} d^{3} \ln \left (c e x +d c \right ) \ln \left (\frac {c e x +e}{-d c +e}\right )}{2 e^{4}}+\frac {b \,c^{4} d^{3} \dilog \left (\frac {c e x +e}{-d c +e}\right )}{2 e^{4}}-\frac {2 b \,c^{3} d^{2}}{3 e^{3}}-\frac {b \,c^{3} d x}{2 e^{2}}+\frac {b \,c^{3} x^{2}}{6 e}+\frac {b \,c^{3} \ln \left (-c e x +e \right ) d^{2}}{2 e^{3}}-\frac {b \,c^{2} \ln \left (-c e x +e \right ) d}{4 e^{2}}+\frac {b c \ln \left (-c e x +e \right )}{6 e}+\frac {b \,c^{3} \ln \left (-c e x -e \right ) d^{2}}{2 e^{3}}+\frac {b \,c^{2} \ln \left (-c e x -e \right ) d}{4 e^{2}}+\frac {b c \ln \left (-c e x -e \right )}{6 e}}{c^{4}}\) | \(423\) |
risch | \(\frac {b \,d^{3} \dilog \left (\frac {\left (-c x +1\right ) e -d c -e}{-d c -e}\right )}{2 e^{4}}-\frac {b \,d^{3} \dilog \left (\frac {\left (c x +1\right ) e +d c -e}{d c -e}\right )}{2 e^{4}}-\frac {b d \ln \left (c x +1\right ) x^{2}}{4 e^{2}}+\frac {b d \ln \left (c x +1\right )}{4 c^{2} e^{2}}+\frac {b \ln \left (c x +1\right ) x \,d^{2}}{2 e^{3}}+\frac {b \ln \left (c x +1\right ) d^{2}}{2 c \,e^{3}}-\frac {b \,d^{3} \ln \left (c x +1\right ) \ln \left (\frac {\left (c x +1\right ) e +d c -e}{d c -e}\right )}{2 e^{4}}-\frac {a d \,x^{2}}{2 e^{2}}-\frac {a \,d^{3} \ln \left (\left (-c x +1\right ) e -d c -e \right )}{e^{4}}+\frac {b \ln \left (-c x +1\right )}{6 c^{3} e}-\frac {b \ln \left (-c x +1\right ) x^{3}}{6 e}-\frac {a \,d^{2}}{c \,e^{3}}+\frac {a d}{2 c^{2} e^{2}}-\frac {b \,d^{2}}{c \,e^{3}}+\frac {b \ln \left (c x +1\right )}{6 c^{3} e}+\frac {b \ln \left (c x +1\right ) x^{3}}{6 e}+\frac {b \,d^{3} \ln \left (-c x +1\right ) \ln \left (\frac {\left (-c x +1\right ) e -d c -e}{-d c -e}\right )}{2 e^{4}}-\frac {b d \ln \left (-c x +1\right )}{4 c^{2} e^{2}}+\frac {a \,d^{2} x}{e^{3}}+\frac {b d \ln \left (-c x +1\right ) x^{2}}{4 e^{2}}+\frac {b \ln \left (-c x +1\right ) d^{2}}{2 c \,e^{3}}-\frac {b \ln \left (-c x +1\right ) d^{2} x}{2 e^{3}}-\frac {11 b}{18 c^{3} e}-\frac {a}{3 c^{3} e}+\frac {a \,x^{3}}{3 e}-\frac {b d x}{2 c \,e^{2}}+\frac {b \,x^{2}}{6 c e}\) | \(484\) |
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*} \int \frac {x^{3} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )}{d + e x}\, dx \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 = -0.00 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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