Optimal. Leaf size=286 \[ \frac {5 a b d^3 x}{2 c}+\frac {13 b^2 d^3 x}{10 c}+\frac {1}{4} b^2 d^3 x^2+\frac {1}{30} b^2 c d^3 x^3-\frac {13 b^2 d^3 \tanh ^{-1}(c x)}{10 c^2}+\frac {5 b^2 d^3 x \tanh ^{-1}(c x)}{2 c}+\frac {6}{5} b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} b c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {12 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^2}+\frac {3 b^2 d^3 \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {6 b^2 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{5 c^2} \]
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Rubi [A]
time = 0.43, antiderivative size = 286, normalized size of antiderivative = 1.00, number
of steps used = 38, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules
used = {6087, 6065, 6021, 266, 6037, 327, 212, 272, 45, 1600, 6055, 2449, 2352, 308}
\begin {gather*} \frac {1}{10} b c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {d^3 (c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}-\frac {12 b d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {1}{2} b c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {6}{5} b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {5 a b d^3 x}{2 c}-\frac {6 b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{5 c^2}+\frac {3 b^2 d^3 \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {13 b^2 d^3 \tanh ^{-1}(c x)}{10 c^2}+\frac {1}{30} b^2 c d^3 x^3+\frac {13 b^2 d^3 x}{10 c}+\frac {5 b^2 d^3 x \tanh ^{-1}(c x)}{2 c}+\frac {1}{4} b^2 d^3 x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 212
Rule 266
Rule 272
Rule 308
Rule 327
Rule 1600
Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6055
Rule 6065
Rule 6087
Rubi steps
\begin {align*} \int x (d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (-\frac {(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {(d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c d}\right ) \, dx\\ &=-\frac {\int (d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c}+\frac {\int (d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c d}\\ &=-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {(2 b) \int \left (-15 d^5 \left (a+b \tanh ^{-1}(c x)\right )-11 c d^5 x \left (a+b \tanh ^{-1}(c x)\right )-5 c^2 d^5 x^2 \left (a+b \tanh ^{-1}(c x)\right )-c^3 d^5 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {16 \left (d^5+c d^5 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2}\right ) \, dx}{5 c d^2}+\frac {b \int \left (-7 d^4 \left (a+b \tanh ^{-1}(c x)\right )-4 c d^4 x \left (a+b \tanh ^{-1}(c x)\right )-c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {8 \left (d^4+c d^4 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2}\right ) \, dx}{2 c d}\\ &=-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {(32 b) \int \frac {\left (d^5+c d^5 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c d^2}+\frac {(4 b) \int \frac {\left (d^4+c d^4 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c d}-\left (2 b d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\frac {1}{5} \left (22 b d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {\left (7 b d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c}+\frac {\left (6 b d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c}-\frac {1}{2} \left (b c d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (2 b c d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\frac {1}{5} \left (2 b c^2 d^3\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=\frac {5 a b d^3 x}{2 c}+\frac {6}{5} b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} b c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {(32 b) \int \frac {a+b \tanh ^{-1}(c x)}{\frac {1}{d^5}-\frac {c x}{d^5}} \, dx}{5 c d^2}+\frac {(4 b) \int \frac {a+b \tanh ^{-1}(c x)}{\frac {1}{d^4}-\frac {c x}{d^4}} \, dx}{c d}-\frac {\left (7 b^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx}{2 c}+\frac {\left (6 b^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx}{c}+\left (b^2 c d^3\right ) \int \frac {x^2}{1-c^2 x^2} \, dx-\frac {1}{5} \left (11 b^2 c d^3\right ) \int \frac {x^2}{1-c^2 x^2} \, dx+\frac {1}{6} \left (b^2 c^2 d^3\right ) \int \frac {x^3}{1-c^2 x^2} \, dx-\frac {1}{3} \left (2 b^2 c^2 d^3\right ) \int \frac {x^3}{1-c^2 x^2} \, dx-\frac {1}{10} \left (b^2 c^3 d^3\right ) \int \frac {x^4}{1-c^2 x^2} \, dx\\ &=\frac {5 a b d^3 x}{2 c}+\frac {6 b^2 d^3 x}{5 c}+\frac {5 b^2 d^3 x \tanh ^{-1}(c x)}{2 c}+\frac {6}{5} b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} b c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {12 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^2}+\frac {1}{2} \left (7 b^2 d^3\right ) \int \frac {x}{1-c^2 x^2} \, dx-\left (6 b^2 d^3\right ) \int \frac {x}{1-c^2 x^2} \, dx+\frac {\left (b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{c}-\frac {\left (11 b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{5 c}-\frac {\left (4 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c}+\frac {\left (32 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{5 c}+\frac {1}{12} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {1}{3} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (b^2 c^3 d^3\right ) \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 {5 a b d^3 x}{2 c}+\frac {13 b^2 d^3 x}{10 c}+\frac {1}{30} b^2 c d^3 x^3-\frac {6 b^2 d^3 \tanh ^{-1}(c x)}{5 c^2}+\frac {5 b^2 d^3 x \tanh ^{-1}(c x)}{2 c}+\frac {6}{5} b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} b c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {12 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^2}+\frac {5 b^2 d^3 \log \left (1-c^2 x^2\right )}{4 c^2}+\frac {\left (4 b^2 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c^2}-\frac {\left (32 b^2 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{5 c^2}-\frac {\left (b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{10 c}+\frac {1}{12} \left (b^2 c^2 d^3\right ) \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 )-\frac {1}{3} \left (b^2 c^2 d^3\right ) \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 )\\ &=\frac {5 a b d^3 x}{2 c}+\frac {13 b^2 d^3 x}{10 c}+\frac {1}{4} b^2 d^3 x^2+\frac {1}{30} b^2 c d^3 x^3-\frac {13 b^2 d^3 \tanh ^{-1}(c x)}{10 c^2}+\frac {5 b^2 d^3 x \tanh ^{-1}(c x)}{2 c}+\frac {6}{5} b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} b c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {12 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^2}+\frac {3 b^2 d^3 \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {6 b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{5 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.70, size = 325, normalized size = 1.14 \begin {gather*} \frac {d^3 \left (-18 a b-15 b^2+150 a b c x+78 b^2 c x+30 a^2 c^2 x^2+72 a b c^2 x^2+15 b^2 c^2 x^2+60 a^2 c^3 x^3+30 a b c^3 x^3+2 b^2 c^3 x^3+45 a^2 c^4 x^4+6 a b c^4 x^4+12 a^2 c^5 x^5+3 b^2 \left (-49+10 c^2 x^2+20 c^3 x^3+15 c^4 x^4+4 c^5 x^5\right ) \tanh ^{-1}(c x)^2+6 b \tanh ^{-1}(c x) \left (a c^2 x^2 \left (10+20 c x+15 c^2 x^2+4 c^3 x^3\right )+b \left (-13+25 c x+12 c^2 x^2+5 c^3 x^3+c^4 x^4\right )-24 b \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+75 a b \log (1-c x)-75 a b \log (1+c x)+90 b^2 \log \left (1-c^2 x^2\right )+72 a b \log \left (-1+c^2 x^2\right )+72 b^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )}{60 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(545\) vs.
\(2(258)=516\).
time = 0.40, size = 546, normalized size = 1.91 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 780 vs.
\(2 (255) = 510\).
time = 0.49, size = 780, normalized size = 2.73 \begin {gather*} \frac {1}{5} \, a^{2} c^{3} d^{3} x^{5} + \frac {3}{4} \, a^{2} c^{2} d^{3} x^{4} + \frac {1}{10} \, {\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 )} a b c^{3} d^{3} + a^{2} c d^{3} x^{3} + \frac {1}{2} \, b^{2} d^{3} x^{2} \operatorname {artanh}\left (c x\right )^{2} + \frac {1}{4} \, {\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 )} a b c^{2} d^{3} + {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a b c d^{3} + \frac {1}{2} \, a^{2} d^{3} x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} a b d^{3} + \frac {1}{8} \, {\left (4 \, c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )} \operatorname {artanh}\left (c x\right ) - \frac {2 \, {\left (\log \left (c x - 1\right ) - 2\right )} \log \left (c x + 1\right ) - \log \left (c x + 1\right )^{2} - \log \left (c x - 1\right )^{2} - 4 \, \log \left (c x - 1\right )}{c^{2}}\right )} b^{2} d^{3} + \frac {6 \, {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} d^{3}}{5 \, c^{2}} + \frac {7 \, b^{2} d^{3} \log \left (c x + 1\right )}{20 \, c^{2}} + \frac {33 \, b^{2} d^{3} \log \left (c x - 1\right )}{20 \, c^{2}} + \frac {8 \, b^{2} c^{3} d^{3} x^{3} + 60 \, b^{2} c^{2} d^{3} x^{2} + 312 \, b^{2} c d^{3} x + 3 \, {\left (4 \, b^{2} c^{5} d^{3} x^{5} + 15 \, b^{2} c^{4} d^{3} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 9 \, b^{2} d^{3}\right )} \log \left (c x + 1\right )^{2} + 3 \, {\left (4 \, b^{2} c^{5} d^{3} x^{5} + 15 \, b^{2} c^{4} d^{3} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} - 39 \, b^{2} d^{3}\right )} \log \left (-c x + 1\right )^{2} + 12 \, {\left (b^{2} c^{4} d^{3} x^{4} + 5 \, b^{2} c^{3} d^{3} x^{3} + 12 \, b^{2} c^{2} d^{3} x^{2} + 15 \, b^{2} c d^{3} x\right )} \log \left (c x + 1\right ) - 6 \, {\left (2 \, b^{2} c^{4} d^{3} x^{4} + 10 \, b^{2} c^{3} d^{3} x^{3} + 24 \, b^{2} c^{2} d^{3} x^{2} + 30 \, b^{2} c d^{3} x + {\left (4 \, b^{2} c^{5} d^{3} x^{5} + 15 \, b^{2} c^{4} d^{3} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 9 \, b^{2} d^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{240 \, c^{2}} \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*} d^{3} \left (\int a^{2} x\, dx + \int 3 a^{2} c x^{2}\, dx + \int 3 a^{2} c^{2} x^{3}\, dx + \int a^{2} c^{3} x^{4}\, dx + \int b^{2} x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x \operatorname {atanh}{\left (c x \right )}\, dx + \int 3 b^{2} c x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{2} x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{4} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{4} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \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 x\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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