Optimal. Leaf size=225 \[ \frac {b (2 d-3 e) x}{8 c^3}-\frac {2 b e x}{3 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {b e x^3}{18 c}-\frac {b (2 d-3 e) \tanh ^{-1}(c x)}{8 c^4}+\frac {2 b e \tanh ^{-1}(c x)}{3 c^4}-\frac {e x^2 \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2504, 2442,
45, 6230, 470, 327, 212, 2521, 2498, 2505, 308} \begin {gather*} \frac {1}{4} x^4 \left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {e x^2 \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2}-\frac {e \log \left (1-c^2 x^2\right ) \left (a+b \tanh ^{-1}(c x)\right )}{4 c^4}-\frac {1}{8} e x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {b (2 d-3 e) \tanh ^{-1}(c x)}{8 c^4}+\frac {2 b e \tanh ^{-1}(c x)}{3 c^4}+\frac {b x (2 d-3 e)}{8 c^3}-\frac {2 b e x}{3 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b x^3 (2 d-e)}{24 c}-\frac {b e x^3}{18 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 212
Rule 308
Rule 327
Rule 470
Rule 2442
Rule 2498
Rule 2504
Rule 2505
Rule 2521
Rule 6230
Rubi steps
\begin {align*} \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx &=-\frac {e x^2 \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-(b c) \int \left (\frac {x^2 \left (-2 e+c^2 (2 d-e) x^2\right )}{8 c^2 \left (1-c^2 x^2\right )}-\frac {e \left (1+c^2 x^2\right ) \log \left (1-c^2 x^2\right )}{4 c^4}\right ) \, dx\\ &=-\frac {e x^2 \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {b \int \frac {x^2 \left (-2 e+c^2 (2 d-e) x^2\right )}{1-c^2 x^2} \, dx}{8 c}+\frac {(b e) \int \left (1+c^2 x^2\right ) \log \left (1-c^2 x^2\right ) \, dx}{4 c^3}\\ &=\frac {b (2 d-e) x^3}{24 c}-\frac {e x^2 \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {(b (2 d-3 e)) \int \frac {x^2}{1-c^2 x^2} \, dx}{8 c}+\frac {(b e) \int \left (\log \left (1-c^2 x^2\right )+c^2 x^2 \log \left (1-c^2 x^2\right )\right ) \, dx}{4 c^3}\\ &=\frac {b (2 d-3 e) x}{8 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {e x^2 \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {(b (2 d-3 e)) \int \frac {1}{1-c^2 x^2} \, dx}{8 c^3}+\frac {(b e) \int \log \left (1-c^2 x^2\right ) \, dx}{4 c^3}+\frac {(b e) \int x^2 \log \left (1-c^2 x^2\right ) \, dx}{4 c}\\ &=\frac {b (2 d-3 e) x}{8 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {b (2 d-3 e) \tanh ^{-1}(c x)}{8 c^4}-\frac {e x^2 \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {(b e) \int \frac {x^2}{1-c^2 x^2} \, dx}{2 c}+\frac {1}{6} (b c e) \int \frac {x^4}{1-c^2 x^2} \, dx\\ &=\frac {b (2 d-3 e) x}{8 c^3}-\frac {b e x}{2 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {b (2 d-3 e) \tanh ^{-1}(c x)}{8 c^4}-\frac {e x^2 \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {(b e) \int \frac {1}{1-c^2 x^2} \, dx}{2 c^3}+\frac {1}{6} (b c e) \int \left (-\frac {1}{c^4}-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac {b (2 d-3 e) x}{8 c^3}-\frac {2 b e x}{3 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {b e x^3}{18 c}-\frac {b (2 d-3 e) \tanh ^{-1}(c x)}{8 c^4}+\frac {b e \tanh ^{-1}(c x)}{2 c^4}-\frac {e x^2 \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {(b e) \int \frac {1}{1-c^2 x^2} \, dx}{6 c^3}\\ &=\frac {b (2 d-3 e) x}{8 c^3}-\frac {2 b e x}{3 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {b e x^3}{18 c}-\frac {b (2 d-3 e) \tanh ^{-1}(c x)}{8 c^4}+\frac {2 b e \tanh ^{-1}(c x)}{3 c^4}-\frac {e x^2 \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 192, normalized size = 0.85 \begin {gather*} \frac {6 b c (6 d-25 e) x-36 a c^2 e x^2+2 b c^3 (6 d-7 e) x^3+18 a c^4 (2 d-e) x^4-18 b c^2 x^2 \left (-2 c^2 d x^2+e \left (2+c^2 x^2\right )\right ) \tanh ^{-1}(c x)+3 (6 b d-12 a e-25 b e) \log (1-c x)-3 (6 b d+12 a e-25 b e) \log (1+c x)+12 e \left (3 a c^4 x^4+b c x \left (3+c^2 x^2\right )+3 b \left (-1+c^4 x^4\right ) \tanh ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{144 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 16.66, size = 3739, normalized size = 16.62
method | result | size |
default | \(\text {Expression too large to display}\) | \(3739\) |
risch | \(\text {Expression too large to display}\) | \(8859\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.27, size = 274, normalized size = 1.22 \begin {gather*} \frac {1}{4} \, a d x^{4} + \frac {1}{8} \, {\left (2 \, x^{4} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b \operatorname {artanh}\left (c x\right ) e + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b d + \frac {1}{8} \, {\left (2 \, x^{4} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} a e - \frac {{\left (2 \, {\left (-6 i \, \pi c^{3} + 7 \, c^{3}\right )} x^{3} + 6 \, {\left (-6 i \, \pi c + 25 \, c\right )} x + 3 \, {\left (6 i \, \pi - 4 \, c^{3} x^{3} - 12 \, c x - 25\right )} \log \left (c x + 1\right ) + 3 \, {\left (-6 i \, \pi - 4 \, c^{3} x^{3} - 12 \, c x + 25\right )} \log \left (c x - 1\right )\right )} b e}{144 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 306, normalized size = 1.36 \begin {gather*} \frac {36 \, a c^{4} d x^{4} + 12 \, b c^{3} d x^{3} + 36 \, b c d x - 2 \, {\left (9 \, a c^{4} x^{4} + 7 \, b c^{3} x^{3} + 18 \, a c^{2} x^{2} + 75 \, b c x\right )} \cosh \left (1\right ) + 12 \, {\left ({\left (3 \, a c^{4} x^{4} + b c^{3} x^{3} + 3 \, b c x - 3 \, a\right )} \cosh \left (1\right ) + {\left (3 \, a c^{4} x^{4} + b c^{3} x^{3} + 3 \, b c x - 3 \, a\right )} \sinh \left (1\right )\right )} \log \left (-c^{2} x^{2} + 1\right ) + 3 \, {\left (6 \, b c^{4} d x^{4} - 6 \, b d - {\left (3 \, b c^{4} x^{4} + 6 \, b c^{2} x^{2} - 25 \, b\right )} \cosh \left (1\right ) + 6 \, {\left ({\left (b c^{4} x^{4} - b\right )} \cosh \left (1\right ) + {\left (b c^{4} x^{4} - b\right )} \sinh \left (1\right )\right )} \log \left (-c^{2} x^{2} + 1\right ) - {\left (3 \, b c^{4} x^{4} + 6 \, b c^{2} x^{2} - 25 \, b\right )} \sinh \left (1\right )\right )} \log \left (-\frac {c x + 1}{c x - 1}\right ) - 2 \, {\left (9 \, a c^{4} x^{4} + 7 \, b c^{3} x^{3} + 18 \, a c^{2} x^{2} + 75 \, b c x\right )} \sinh \left (1\right )}{144 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.48, size = 279, normalized size = 1.24 \begin {gather*} \begin {cases} \frac {a d x^{4}}{4} + \frac {a e x^{4} \log {\left (- c^{2} x^{2} + 1 \right )}}{4} - \frac {a e x^{4}}{8} - \frac {a e x^{2}}{4 c^{2}} - \frac {a e \log {\left (- c^{2} x^{2} + 1 \right )}}{4 c^{4}} + \frac {b d x^{4} \operatorname {atanh}{\left (c x \right )}}{4} + \frac {b e x^{4} \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {atanh}{\left (c x \right )}}{4} - \frac {b e x^{4} \operatorname {atanh}{\left (c x \right )}}{8} + \frac {b d x^{3}}{12 c} + \frac {b e x^{3} \log {\left (- c^{2} x^{2} + 1 \right )}}{12 c} - \frac {7 b e x^{3}}{72 c} - \frac {b e x^{2} \operatorname {atanh}{\left (c x \right )}}{4 c^{2}} + \frac {b d x}{4 c^{3}} + \frac {b e x \log {\left (- c^{2} x^{2} + 1 \right )}}{4 c^{3}} - \frac {25 b e x}{24 c^{3}} - \frac {b d \operatorname {atanh}{\left (c x \right )}}{4 c^{4}} - \frac {b e \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {atanh}{\left (c x \right )}}{4 c^{4}} + \frac {25 b e \operatorname {atanh}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d x^{4}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.70, size = 851, normalized size = 3.78 \begin {gather*} {\ln \left (1-c\,x\right )}^2\,\left (\frac {b\,e}{8\,c^4}-\frac {b\,e\,x^4}{8}\right )-{\ln \left (c\,x+1\right )}^2\,\left (\frac {b\,e}{8\,c^4}-\frac {b\,e\,x^4}{8}\right )+\ln \left (1-c\,x\right )\,\left (\frac {x^4\,\left (a\,e-\frac {b\,d}{2}+\frac {b\,e}{4}+\frac {b\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )}{2}\right )}{4}-\frac {x^2\,\left (\frac {16\,a\,e-8\,b\,d+8\,b\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )}{c}-\frac {16\,a\,e-8\,b\,d+4\,b\,e+8\,b\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )}{c}\right )}{32\,c}+\frac {b\,e\,x}{4\,c^3}+\frac {b\,e\,x^3}{12\,c}\right )-x^2\,\left (\frac {a\,\left (e-2\,d+2\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )\right )}{4\,c^2}+\frac {a\,\left (d-e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )\right )}{2\,c^2}\right )-x\,\left (\frac {b\,\left (7\,e-6\,d+6\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )\right )}{24\,c^3}+\frac {3\,b\,e}{4\,c^3}\right )-\frac {a\,x^4\,\left (e-2\,d+2\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )\right )}{8}-\frac {\ln \left (\frac {x\,\left (12\,a\,e-6\,b\,d+25\,b\,e+6\,b\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )\right )}{24\,c^2}-\frac {25\,b\,e-6\,b\,d+6\,b\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )}{24\,c^3}-\frac {a\,e\,x}{2\,c^2}\right )\,\left (12\,a\,e-6\,b\,d+25\,b\,e+6\,b\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )\right )}{48\,c^4}-\frac {\ln \left (\frac {x\,\left (12\,a\,e+6\,b\,d-25\,b\,e-6\,b\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )\right )}{24\,c^2}-\frac {25\,b\,e-6\,b\,d+6\,b\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )}{24\,c^3}-\frac {a\,e\,x}{2\,c^2}\right )\,\left (12\,a\,e+6\,b\,d-25\,b\,e-6\,b\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )\right )}{48\,c^4}+c\,\ln \left (c\,x+1\right )\,\left (\frac {x^4\,\left (4\,a\,e+2\,b\,d-b\,e-2\,b\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )\right )}{16\,c}+\frac {b\,e\,x}{4\,c^4}+\frac {b\,e\,x^3}{12\,c^2}-\frac {b\,e\,x^2}{8\,c^3}\right )-\frac {b\,x^3\,\left (7\,e-6\,d+6\,e\,\left (\ln \left (c\,x+1\right )+\ln \left (1-c\,x\right )-\ln \left (1-c^2\,x^2\right )\right )\right )}{72\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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