Optimal. Leaf size=315 \[ -\frac {e (4 a+3 b) \log (1-c x)}{20 c^5}+\frac {e (4 a-3 b) \log (c x+1)}{20 c^5}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {2 a e x}{5 c^4}-\frac {2 a e x^3}{15 c^2}-\frac {2}{25} a e x^5+\frac {b e \tanh ^{-1}(c x)^2}{5 c^5}-\frac {2 b e x \tanh ^{-1}(c x)}{5 c^4}-\frac {77 b e x^2}{300 c^3}+\frac {b x^4 \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 c}-\frac {2 b e x^3 \tanh ^{-1}(c x)}{15 c^2}+\frac {b \log \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^5}-\frac {b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}-\frac {23 b e \log \left (1-c^2 x^2\right )}{75 c^5}+\frac {b x^2 \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^3}-\frac {2}{25} b e x^5 \tanh ^{-1}(c x)-\frac {9 b e x^4}{200 c} \]
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Rubi [A] time = 0.78, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5916, 266, 43, 6085, 6725, 1802, 633, 31, 5980, 5910, 260, 5948, 2475, 2390, 2301} \[ \frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {e (4 a+3 b) \log (1-c x)}{20 c^5}+\frac {e (4 a-3 b) \log (c x+1)}{20 c^5}-\frac {2 a e x^3}{15 c^2}-\frac {2 a e x}{5 c^4}-\frac {2}{25} a e x^5+\frac {b x^4 \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 c}+\frac {b x^2 \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^3}+\frac {b \log \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^5}-\frac {77 b e x^2}{300 c^3}-\frac {b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}-\frac {23 b e \log \left (1-c^2 x^2\right )}{75 c^5}-\frac {2 b e x^3 \tanh ^{-1}(c x)}{15 c^2}-\frac {2 b e x \tanh ^{-1}(c x)}{5 c^4}+\frac {b e \tanh ^{-1}(c x)^2}{5 c^5}-\frac {9 b e x^4}{200 c}-\frac {2}{25} b e x^5 \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 260
Rule 266
Rule 633
Rule 1802
Rule 2301
Rule 2390
Rule 2475
Rule 5910
Rule 5916
Rule 5948
Rule 5980
Rule 6085
Rule 6725
Rubi steps
\begin {align*} \int x^4 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx &=\frac {b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac {b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}+\left (2 c^2 e\right ) \int \left (\frac {-2 b x^3-b c^2 x^5-4 a c^3 x^6-4 b c^3 x^6 \tanh ^{-1}(c x)}{20 c^3 \left (-1+c^2 x^2\right )}-\frac {b x \log \left (1-c^2 x^2\right )}{10 c^5 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=\frac {b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac {b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac {(b e) \int \frac {x \log \left (1-c^2 x^2\right )}{-1+c^2 x^2} \, dx}{5 c^3}+\frac {e \int \frac {-2 b x^3-b c^2 x^5-4 a c^3 x^6-4 b c^3 x^6 \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{10 c}\\ &=\frac {b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac {b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\log \left (1-c^2 x\right )}{-1+c^2 x} \, dx,x,x^2\right )}{10 c^3}+\frac {e \int \left (\frac {x^3 \left (2 b+b c^2 x^2+4 a c^3 x^3\right )}{1-c^2 x^2}-\frac {4 b c^3 x^6 \tanh ^{-1}(c x)}{-1+c^2 x^2}\right ) \, dx}{10 c}\\ &=\frac {b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac {b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-c^2 x^2\right )}{10 c^5}+\frac {e \int \frac {x^3 \left (2 b+b c^2 x^2+4 a c^3 x^3\right )}{1-c^2 x^2} \, dx}{10 c}-\frac {1}{5} \left (2 b c^2 e\right ) \int \frac {x^6 \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx\\ &=-\frac {b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac {b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac {1}{5} (2 b e) \int x^4 \tanh ^{-1}(c x) \, dx-\frac {1}{5} (2 b e) \int \frac {x^4 \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac {e \int \left (-\frac {4 a}{c^3}-\frac {3 b x}{c^2}-\frac {4 a x^2}{c}-b x^3-4 a c x^4+\frac {4 a+3 b c x}{c^3 \left (1-c^2 x^2\right )}\right ) \, dx}{10 c}\\ &=-\frac {2 a e x}{5 c^4}-\frac {3 b e x^2}{20 c^3}-\frac {2 a e x^3}{15 c^2}-\frac {b e x^4}{40 c}-\frac {2}{25} a e x^5-\frac {2}{25} b e x^5 \tanh ^{-1}(c x)-\frac {b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac {b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}+\frac {e \int \frac {4 a+3 b c x}{1-c^2 x^2} \, dx}{10 c^4}-\frac {(2 b e) \int x^2 \tanh ^{-1}(c x) \, dx}{5 c^2}-\frac {(2 b e) \int \frac {x^2 \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{5 c^2}+\frac {1}{25} (2 b c e) \int \frac {x^5}{1-c^2 x^2} \, dx\\ &=-\frac {2 a e x}{5 c^4}-\frac {3 b e x^2}{20 c^3}-\frac {2 a e x^3}{15 c^2}-\frac {b e x^4}{40 c}-\frac {2}{25} a e x^5-\frac {2 b e x^3 \tanh ^{-1}(c x)}{15 c^2}-\frac {2}{25} b e x^5 \tanh ^{-1}(c x)-\frac {b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac {b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac {(2 b e) \int \tanh ^{-1}(c x) \, dx}{5 c^4}-\frac {(2 b e) \int \frac {\tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{5 c^4}-\frac {((4 a-3 b) e) \int \frac {1}{-c-c^2 x} \, dx}{20 c^3}+\frac {((4 a+3 b) e) \int \frac {1}{c-c^2 x} \, dx}{20 c^3}+\frac {(2 b e) \int \frac {x^3}{1-c^2 x^2} \, dx}{15 c}+\frac {1}{25} (b c e) \operatorname {Subst}\left (\int \frac {x^2}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {2 a e x}{5 c^4}-\frac {3 b e x^2}{20 c^3}-\frac {2 a e x^3}{15 c^2}-\frac {b e x^4}{40 c}-\frac {2}{25} a e x^5-\frac {2 b e x \tanh ^{-1}(c x)}{5 c^4}-\frac {2 b e x^3 \tanh ^{-1}(c x)}{15 c^2}-\frac {2}{25} b e x^5 \tanh ^{-1}(c x)+\frac {b e \tanh ^{-1}(c x)^2}{5 c^5}-\frac {(4 a+3 b) e \log (1-c x)}{20 c^5}+\frac {(4 a-3 b) e \log (1+c x)}{20 c^5}-\frac {b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac {b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}+\frac {(2 b e) \int \frac {x}{1-c^2 x^2} \, dx}{5 c^3}+\frac {(b e) \operatorname {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )}{15 c}+\frac {1}{25} (b c e) \operatorname {Subst}\left (\int \left (-\frac {1}{c^4}-\frac {x}{c^2}-\frac {1}{c^4 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {2 a e x}{5 c^4}-\frac {19 b e x^2}{100 c^3}-\frac {2 a e x^3}{15 c^2}-\frac {9 b e x^4}{200 c}-\frac {2}{25} a e x^5-\frac {2 b e x \tanh ^{-1}(c x)}{5 c^4}-\frac {2 b e x^3 \tanh ^{-1}(c x)}{15 c^2}-\frac {2}{25} b e x^5 \tanh ^{-1}(c x)+\frac {b e \tanh ^{-1}(c x)^2}{5 c^5}-\frac {(4 a+3 b) e \log (1-c x)}{20 c^5}+\frac {(4 a-3 b) e \log (1+c x)}{20 c^5}-\frac {6 b e \log \left (1-c^2 x^2\right )}{25 c^5}-\frac {b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac {b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}+\frac {(b e) \operatorname {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{15 c}\\ &=-\frac {2 a e x}{5 c^4}-\frac {77 b e x^2}{300 c^3}-\frac {2 a e x^3}{15 c^2}-\frac {9 b e x^4}{200 c}-\frac {2}{25} a e x^5-\frac {2 b e x \tanh ^{-1}(c x)}{5 c^4}-\frac {2 b e x^3 \tanh ^{-1}(c x)}{15 c^2}-\frac {2}{25} b e x^5 \tanh ^{-1}(c x)+\frac {b e \tanh ^{-1}(c x)^2}{5 c^5}-\frac {(4 a+3 b) e \log (1-c x)}{20 c^5}+\frac {(4 a-3 b) e \log (1+c x)}{20 c^5}-\frac {23 b e \log \left (1-c^2 x^2\right )}{75 c^5}-\frac {b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac {b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 236, normalized size = 0.75 \[ \frac {30 c^2 e x^2 \log \left (1-c^2 x^2\right ) \left (4 a c^3 x^3+4 b c^3 x^3 \tanh ^{-1}(c x)+b \left (c^2 x^2+2\right )\right )+2 \log (1-c x) (-60 a e+30 b d-137 b e)+2 \log (c x+1) (60 a e+30 b d-137 b e)+24 a c^5 x^5 (5 d-2 e)-80 a c^3 e x^3-240 a c e x+3 b c^4 x^4 (10 d-9 e)+2 b c^2 x^2 (30 d-77 e)+30 b e \log ^2\left (1-c^2 x^2\right )-8 b c x \tanh ^{-1}(c x) \left (2 e \left (3 c^4 x^4+5 c^2 x^2+15\right )-15 c^4 d x^4\right )+120 b e \tanh ^{-1}(c x)^2}{600 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 251, normalized size = 0.80 \[ -\frac {80 \, a c^{3} e x^{3} - 24 \, {\left (5 \, a c^{5} d - 2 \, a c^{5} e\right )} x^{5} - 3 \, {\left (10 \, b c^{4} d - 9 \, b c^{4} e\right )} x^{4} + 240 \, a c e x - 30 \, b e \log \left (-c^{2} x^{2} + 1\right )^{2} - 30 \, b e \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} - 2 \, {\left (30 \, b c^{2} d - 77 \, b c^{2} e\right )} x^{2} - 2 \, {\left (60 \, a c^{5} e x^{5} + 15 \, b c^{4} e x^{4} + 30 \, b c^{2} e x^{2} + 30 \, b d - 137 \, b e\right )} \log \left (-c^{2} x^{2} + 1\right ) - 4 \, {\left (15 \, b c^{5} e x^{5} \log \left (-c^{2} x^{2} + 1\right ) - 10 \, b c^{3} e x^{3} + 3 \, {\left (5 \, b c^{5} d - 2 \, b c^{5} e\right )} x^{5} - 30 \, b c e x + 30 \, a e\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{600 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 455, normalized size = 1.44 \[ \frac {60 \, b c^{5} x^{5} e \log \left (c x + 1\right )^{2} - 60 \, b c^{5} x^{5} e \log \left (-c x + 1\right )^{2} + 120 \, a c^{5} x^{5} e \log \left (c x + 1\right ) - 24 \, b c^{5} x^{5} e \log \left (c x + 1\right ) + 120 \, a c^{5} x^{5} e \log \left (-c x + 1\right ) + 24 \, b c^{5} x^{5} e \log \left (-c x + 1\right ) + 60 \, b c^{5} d x^{5} \log \left (-\frac {c x + 1}{c x - 1}\right ) + 120 \, a c^{5} d x^{5} - 48 \, a c^{5} x^{5} e + 30 \, b c^{4} x^{4} e \log \left (c x + 1\right ) + 30 \, b c^{4} x^{4} e \log \left (-c x + 1\right ) + 30 \, b c^{4} d x^{4} - 27 \, b c^{4} x^{4} e - 40 \, b c^{3} x^{3} e \log \left (c x + 1\right ) + 40 \, b c^{3} x^{3} e \log \left (-c x + 1\right ) - 80 \, a c^{3} x^{3} e + 60 \, b c^{2} x^{2} e \log \left (c x + 1\right ) + 60 \, b c^{2} x^{2} e \log \left (-c x + 1\right ) + 60 \, b c^{2} d x^{2} - 154 \, b c^{2} x^{2} e - 120 \, b c x e \log \left (c x + 1\right ) + 120 \, b c x e \log \left (-c x + 1\right ) - 240 \, a c x e + 60 \, b e \log \left (c x + 1\right )^{2} - 60 \, b e \log \left (c x - 1\right )^{2} + 120 \, b e \log \left (c x - 1\right ) \log \left (-c x + 1\right ) + 60 \, b d \log \left (c^{2} x^{2} - 1\right ) + 120 \, a e \log \left (c x + 1\right ) - 274 \, b e \log \left (c x + 1\right ) - 120 \, a e \log \left (c x - 1\right ) - 274 \, b e \log \left (c x - 1\right )}{600 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.09, size = 4757, normalized size = 15.10 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.34, size = 314, normalized size = 1.00 \[ \frac {1}{5} \, a d x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b e \operatorname {artanh}\left (c x\right ) + \frac {1}{20} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b d + \frac {1}{75} \, {\left (15 \, x^{5} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} a e + \frac {{\left ({\left (30 i \, \pi c^{4} - 27 \, c^{4}\right )} x^{4} + {\left (60 i \, \pi c^{2} - 154 \, c^{2}\right )} x^{2} + {\left (60 i \, \pi + 30 \, c^{4} x^{4} + 60 \, c^{2} x^{2} + 120 \, \log \left (c x - 1\right ) - 274\right )} \log \left (c x + 1\right ) - 2 \, {\left (-30 i \, \pi - 15 \, c^{4} x^{4} - 30 \, c^{2} x^{2} + 137\right )} \log \left (c x - 1\right )\right )} b e}{600 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.59, size = 599, normalized size = 1.90 \[ \frac {a\,d\,x^5}{5}-\frac {2\,a\,e\,x^5}{25}+\frac {b\,d\,x^5\,\ln \left (c\,x+1\right )}{10}-\frac {b\,d\,x^5\,\ln \left (1-c\,x\right )}{10}-\frac {b\,e\,x^5\,\ln \left (c\,x+1\right )}{25}+\frac {b\,e\,x^5\,\ln \left (1-c\,x\right )}{25}+\frac {b\,e\,{\ln \left (c\,x+1\right )}^2}{10\,c^5}+\frac {b\,e\,{\ln \left (1-c\,x\right )}^2}{10\,c^5}-\frac {2\,a\,e\,x}{5\,c^4}-\frac {2\,a\,e\,x^3}{15\,c^2}+\frac {b\,d\,x^4}{20\,c}+\frac {b\,d\,x^2}{10\,c^3}-\frac {9\,b\,e\,x^4}{200\,c}-\frac {77\,b\,e\,x^2}{300\,c^3}+\frac {a\,e\,x^5\,\ln \left (1-c^2\,x^2\right )}{5}-\frac {a\,e\,\ln \left (c\,x-1\right )}{5\,c^5}+\frac {a\,e\,\ln \left (c\,x+1\right )}{5\,c^5}+\frac {b\,d\,\ln \left (c\,x-1\right )}{10\,c^5}+\frac {b\,d\,\ln \left (c\,x+1\right )}{10\,c^5}-\frac {137\,b\,e\,\ln \left (c\,x-1\right )}{300\,c^5}-\frac {137\,b\,e\,\ln \left (c\,x+1\right )}{300\,c^5}-\frac {b\,e\,\ln \left (c\,x+1\right )\,\ln \left (-\frac {2\,a\,e-2\,a\,c\,e\,x}{5\,c^4}\right )}{10\,c^5}-\frac {b\,e\,\ln \left (c\,x+1\right )\,\ln \left (-\frac {2\,a\,e+2\,a\,c\,e\,x}{5\,c^4}\right )}{10\,c^5}-\frac {b\,e\,\ln \left (1-c\,x\right )\,\ln \left (-\frac {2\,a\,e-2\,a\,c\,e\,x}{5\,c^4}\right )}{10\,c^5}-\frac {b\,e\,\ln \left (1-c\,x\right )\,\ln \left (-\frac {2\,a\,e+2\,a\,c\,e\,x}{5\,c^4}\right )}{10\,c^5}-\frac {b\,e\,x\,\ln \left (c\,x+1\right )}{5\,c^4}+\frac {b\,e\,x\,\ln \left (1-c\,x\right )}{5\,c^4}+\frac {b\,e\,x^4\,\ln \left (1-c^2\,x^2\right )}{20\,c}+\frac {b\,e\,x^2\,\ln \left (1-c^2\,x^2\right )}{10\,c^3}+\frac {b\,e\,\ln \left (-\frac {2\,a\,e-2\,a\,c\,e\,x}{5\,c^4}\right )\,\ln \left (1-c^2\,x^2\right )}{10\,c^5}+\frac {b\,e\,\ln \left (-\frac {2\,a\,e+2\,a\,c\,e\,x}{5\,c^4}\right )\,\ln \left (1-c^2\,x^2\right )}{10\,c^5}-\frac {b\,e\,x^3\,\ln \left (c\,x+1\right )}{15\,c^2}+\frac {b\,e\,x^3\,\ln \left (1-c\,x\right )}{15\,c^2}+\frac {b\,e\,x^5\,\ln \left (c\,x+1\right )\,\ln \left (1-c^2\,x^2\right )}{10}-\frac {b\,e\,x^5\,\ln \left (1-c\,x\right )\,\ln \left (1-c^2\,x^2\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.18, size = 338, normalized size = 1.07 \[ \begin {cases} \frac {a d x^{5}}{5} + \frac {a e x^{5} \log {\left (- c^{2} x^{2} + 1 \right )}}{5} - \frac {2 a e x^{5}}{25} - \frac {2 a e x^{3}}{15 c^{2}} - \frac {2 a e x}{5 c^{4}} + \frac {2 a e \operatorname {atanh}{\left (c x \right )}}{5 c^{5}} + \frac {b d x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {b e x^{5} \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {atanh}{\left (c x \right )}}{5} - \frac {2 b e x^{5} \operatorname {atanh}{\left (c x \right )}}{25} + \frac {b d x^{4}}{20 c} + \frac {b e x^{4} \log {\left (- c^{2} x^{2} + 1 \right )}}{20 c} - \frac {9 b e x^{4}}{200 c} - \frac {2 b e x^{3} \operatorname {atanh}{\left (c x \right )}}{15 c^{2}} + \frac {b d x^{2}}{10 c^{3}} + \frac {b e x^{2} \log {\left (- c^{2} x^{2} + 1 \right )}}{10 c^{3}} - \frac {77 b e x^{2}}{300 c^{3}} - \frac {2 b e x \operatorname {atanh}{\left (c x \right )}}{5 c^{4}} + \frac {b d \log {\left (- c^{2} x^{2} + 1 \right )}}{10 c^{5}} + \frac {b e \log {\left (- c^{2} x^{2} + 1 \right )}^{2}}{20 c^{5}} - \frac {137 b e \log {\left (- c^{2} x^{2} + 1 \right )}}{300 c^{5}} + \frac {b e \operatorname {atanh}^{2}{\left (c x \right )}}{5 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d x^{5}}{5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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