Optimal. Leaf size=315 \[ -\frac {2 a e x}{5 c^4}-\frac {77 b e x^2}{300 c^3}-\frac {2 a e x^3}{15 c^2}-\frac {9 b e x^4}{200 c}-\frac {2}{25} a e x^5-\frac {2 b e x \tanh ^{-1}(c x)}{5 c^4}-\frac {2 b e x^3 \tanh ^{-1}(c x)}{15 c^2}-\frac {2}{25} b e x^5 \tanh ^{-1}(c x)+\frac {b e \tanh ^{-1}(c x)^2}{5 c^5}-\frac {(4 a+3 b) e \log (1-c x)}{20 c^5}+\frac {(4 a-3 b) e \log (1+c x)}{20 c^5}-\frac {23 b e \log \left (1-c^2 x^2\right )}{75 c^5}-\frac {b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac {b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5} \]
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Rubi [A]
time = 0.53, 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 =
{6037, 272, 45, 6232, 6857, 1816, 647, 31, 6127, 6021, 266, 6095, 2525, 2437, 2338}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 45
Rule 266
Rule 272
Rule 647
Rule 1816
Rule 2338
Rule 2437
Rule 2525
Rule 6021
Rule 6037
Rule 6095
Rule 6127
Rule 6232
Rule 6857
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) \text {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) \text {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) \text {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) \text {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )}{15 c}+\frac {1}{25} (b c e) \text {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) \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 )}{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 = 0.09, size = 236, normalized size = 0.75 \begin {gather*} \frac {-240 a c e x+2 b c^2 (30 d-77 e) x^2-80 a c^3 e x^3+3 b c^4 (10 d-9 e) x^4+24 a c^5 (5 d-2 e) x^5-8 b c x \left (-15 c^4 d x^4+2 e \left (15+5 c^2 x^2+3 c^4 x^4\right )\right ) \tanh ^{-1}(c x)+120 b e \tanh ^{-1}(c x)^2+2 (30 b d-60 a e-137 b e) \log (1-c x)+2 (30 b d+60 a e-137 b e) \log (1+c x)+30 c^2 e x^2 \left (4 a c^3 x^3+b \left (2+c^2 x^2\right )+4 b c^3 x^3 \tanh ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )+30 b e \log ^2\left (1-c^2 x^2\right )}{600 c^5} \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 = 17.53, size = 4757, normalized size = 15.10
method | result | size |
risch | \(\text {Expression too large to display}\) | \(4341\) |
default | \(\text {Expression too large to display}\) | \(4757\) |
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 = 320, normalized size = 1.02 \begin {gather*} \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 \operatorname {artanh}\left (c x\right ) e + \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 (3 \, {\left (-10 i \, \pi c^{4} + 9 \, c^{4}\right )} x^{4} + 2 \, {\left (-30 i \, \pi c^{2} + 77 \, c^{2}\right )} x^{2} + 2 \, {\left (-30 i \, \pi - 15 \, c^{4} x^{4} - 30 \, c^{2} x^{2} - 60 \, \log \left (c x - 1\right ) + 137\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 390, normalized size = 1.24 \begin {gather*} \frac {120 \, a c^{5} d x^{5} + 30 \, b c^{4} d x^{4} + 60 \, b c^{2} d x^{2} + 30 \, {\left (b \cosh \left (1\right ) + b \sinh \left (1\right )\right )} \log \left (-c^{2} x^{2} + 1\right )^{2} + 30 \, {\left (b \cosh \left (1\right ) + b \sinh \left (1\right )\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} - {\left (48 \, a c^{5} x^{5} + 27 \, b c^{4} x^{4} + 80 \, a c^{3} x^{3} + 154 \, b c^{2} x^{2} + 240 \, a c x\right )} \cosh \left (1\right ) + 2 \, {\left (30 \, b d + {\left (60 \, a c^{5} x^{5} + 15 \, b c^{4} x^{4} + 30 \, b c^{2} x^{2} - 137 \, b\right )} \cosh \left (1\right ) + {\left (60 \, a c^{5} x^{5} + 15 \, b c^{4} x^{4} + 30 \, b c^{2} x^{2} - 137 \, b\right )} \sinh \left (1\right )\right )} \log \left (-c^{2} x^{2} + 1\right ) + 4 \, {\left (15 \, b c^{5} d x^{5} - 2 \, {\left (3 \, b c^{5} x^{5} + 5 \, b c^{3} x^{3} + 15 \, b c x - 15 \, a\right )} \cosh \left (1\right ) + 15 \, {\left (b c^{5} x^{5} \cosh \left (1\right ) + b c^{5} x^{5} \sinh \left (1\right )\right )} \log \left (-c^{2} x^{2} + 1\right ) - 2 \, {\left (3 \, b c^{5} x^{5} + 5 \, b c^{3} x^{3} + 15 \, b c x - 15 \, a\right )} \sinh \left (1\right )\right )} \log \left (-\frac {c x + 1}{c x - 1}\right ) - {\left (48 \, a c^{5} x^{5} + 27 \, b c^{4} x^{4} + 80 \, a c^{3} x^{3} + 154 \, b c^{2} x^{2} + 240 \, a c x\right )} \sinh \left (1\right )}{600 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.39, size = 338, normalized size = 1.07 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 313, normalized size = 0.99 \begin {gather*} -\frac {1}{10} \, b e x^{5} \log \left (-c x + 1\right )^{2} + \frac {1}{25} \, {\left (5 \, a d - 2 \, a e\right )} x^{5} + \frac {{\left (10 \, b d - 9 \, b e\right )} x^{4}}{200 \, c} - \frac {2 \, a e x^{3}}{15 \, c^{2}} + \frac {1}{10} \, {\left (b e x^{5} + \frac {b e}{c^{5}}\right )} \log \left (c x + 1\right )^{2} + \frac {1}{300} \, {\left (6 \, {\left (5 \, b d + 10 \, a e - 2 \, b e\right )} x^{5} + \frac {15 \, b e x^{4}}{c} - \frac {20 \, b e x^{3}}{c^{2}} + \frac {30 \, b e x^{2}}{c^{3}} - \frac {60 \, b e x}{c^{4}}\right )} \log \left (c x + 1\right ) - \frac {1}{300} \, {\left (6 \, {\left (5 \, b d - 10 \, a e - 2 \, b e\right )} x^{5} - \frac {15 \, b e x^{4}}{c} - \frac {20 \, b e x^{3}}{c^{2}} - \frac {30 \, b e x^{2}}{c^{3}} - \frac {60 \, b e x}{c^{4}} - \frac {60 \, b e \log \left (c x - 1\right )}{c^{5}}\right )} \log \left (-c x + 1\right ) + \frac {{\left (30 \, b d - 77 \, b e\right )} x^{2}}{300 \, c^{3}} - \frac {2 \, a e x}{5 \, c^{4}} - \frac {b e \log \left (c x - 1\right )^{2}}{10 \, c^{5}} + \frac {{\left (30 \, b d + 60 \, a e - 137 \, b e\right )} \log \left (c x + 1\right )}{300 \, c^{5}} + \frac {{\left (30 \, b d - 60 \, a e - 137 \, b e\right )} \log \left (c x - 1\right )}{300 \, c^{5}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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