Optimal. Leaf size=280 \[ \frac {3 a b d^2 x}{2 c}+\frac {2 b^2 d^2 x}{3 c}+\frac {1}{12} b^2 d^2 x^2-\frac {2 b^2 d^2 \tanh ^{-1}(c x)}{3 c^2}+\frac {3 b^2 d^2 x \tanh ^{-1}(c x)}{2 c}+\frac {2}{3} b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} b c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{12 c^2}+\frac {1}{2} d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {2}{3} c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c^2 d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {4 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^2}+\frac {5 b^2 d^2 \log \left (1-c^2 x^2\right )}{6 c^2}-\frac {2 b^2 d^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^2} \]
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Rubi [A]
time = 0.47, antiderivative size = 280, normalized size of antiderivative = 1.00, number
of steps used = 28, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules
used = {6087, 6037, 6127, 6021, 266, 6095, 327, 212, 6131, 6055, 2449, 2352, 272, 45}
\begin {gather*} \frac {1}{4} c^2 d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{12 c^2}-\frac {4 b d^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^2}+\frac {2}{3} c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{6} b c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {2}{3} b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3 a b d^2 x}{2 c}-\frac {2 b^2 d^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^2}+\frac {5 b^2 d^2 \log \left (1-c^2 x^2\right )}{6 c^2}-\frac {2 b^2 d^2 \tanh ^{-1}(c x)}{3 c^2}+\frac {2 b^2 d^2 x}{3 c}+\frac {3 b^2 d^2 x \tanh ^{-1}(c x)}{2 c}+\frac {1}{12} b^2 d^2 x^2 \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 6021
Rule 6037
Rule 6055
Rule 6087
Rule 6095
Rule 6127
Rule 6131
Rubi steps
\begin {align*} \int x (d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x \left (a+b \tanh ^{-1}(c x)\right )^2+2 c d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+c^2 d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (2 c d^2\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (c^2 d^2\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{2} d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {2}{3} c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c^2 d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\left (b c d^2\right ) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{3} \left (4 b c^2 d^2\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{2} \left (b c^3 d^2\right ) \int \frac {x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{2} d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {2}{3} c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c^2 d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} \left (4 b d^2\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{3} \left (4 b d^2\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac {\left (b d^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c}-\frac {\left (b d^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c}+\frac {1}{2} \left (b c d^2\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{2} \left (b c d^2\right ) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {a b d^2 x}{c}+\frac {2}{3} b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} b c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac {1}{2} d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {2}{3} c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c^2 d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {\left (b d^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c}-\frac {\left (b d^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{2 c}-\frac {\left (4 b d^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c}+\frac {\left (b^2 d^2\right ) \int \tanh ^{-1}(c x) \, dx}{c}-\frac {1}{3} \left (2 b^2 c d^2\right ) \int \frac {x^2}{1-c^2 x^2} \, dx-\frac {1}{6} \left (b^2 c^2 d^2\right ) \int \frac {x^3}{1-c^2 x^2} \, dx\\ &=\frac {3 a b d^2 x}{2 c}+\frac {2 b^2 d^2 x}{3 c}+\frac {b^2 d^2 x \tanh ^{-1}(c x)}{c}+\frac {2}{3} b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} b c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{12 c^2}+\frac {1}{2} d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {2}{3} c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c^2 d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {4 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^2}-\left (b^2 d^2\right ) \int \frac {x}{1-c^2 x^2} \, dx+\frac {\left (b^2 d^2\right ) \int \tanh ^{-1}(c x) \, dx}{2 c}-\frac {\left (2 b^2 d^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 c}+\frac {\left (4 b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{3 c}-\frac {1}{12} \left (b^2 c^2 d^2\right ) \text {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )\\ &=\frac {3 a b d^2 x}{2 c}+\frac {2 b^2 d^2 x}{3 c}-\frac {2 b^2 d^2 \tanh ^{-1}(c x)}{3 c^2}+\frac {3 b^2 d^2 x \tanh ^{-1}(c x)}{2 c}+\frac {2}{3} b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} b c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{12 c^2}+\frac {1}{2} d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {2}{3} c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c^2 d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {4 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^2}+\frac {b^2 d^2 \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {1}{2} \left (b^2 d^2\right ) \int \frac {x}{1-c^2 x^2} \, dx-\frac {\left (4 b^2 d^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{3 c^2}-\frac {1}{12} \left (b^2 c^2 d^2\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 {3 a b d^2 x}{2 c}+\frac {2 b^2 d^2 x}{3 c}+\frac {1}{12} b^2 d^2 x^2-\frac {2 b^2 d^2 \tanh ^{-1}(c x)}{3 c^2}+\frac {3 b^2 d^2 x \tanh ^{-1}(c x)}{2 c}+\frac {2}{3} b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} b c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{12 c^2}+\frac {1}{2} d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {2}{3} c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c^2 d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {4 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^2}+\frac {5 b^2 d^2 \log \left (1-c^2 x^2\right )}{6 c^2}-\frac {2 b^2 d^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 263, normalized size = 0.94 \begin {gather*} \frac {d^2 \left (-b^2+18 a b c x+8 b^2 c x+6 a^2 c^2 x^2+8 a b c^2 x^2+b^2 c^2 x^2+8 a^2 c^3 x^3+2 a b c^3 x^3+3 a^2 c^4 x^4+b^2 \left (-17+6 c^2 x^2+8 c^3 x^3+3 c^4 x^4\right ) \tanh ^{-1}(c x)^2+2 b \tanh ^{-1}(c x) \left (a c^2 x^2 \left (6+8 c x+3 c^2 x^2\right )+b \left (-4+9 c x+4 c^2 x^2+c^3 x^3\right )-8 b \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+9 a b \log (1-c x)-9 a b \log (1+c x)+10 b^2 \log \left (1-c^2 x^2\right )+8 a b \log \left (-1+c^2 x^2\right )+8 b^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )}{12 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 458, normalized size = 1.64 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. 610 vs.
\(2 (249) = 498\).
time = 0.48, size = 610, normalized size = 2.18 \begin {gather*} \frac {1}{4} \, a^{2} c^{2} d^{2} x^{4} + \frac {2}{3} \, a^{2} c d^{2} x^{3} + \frac {1}{2} \, b^{2} d^{2} x^{2} \operatorname {artanh}\left (c x\right )^{2} + \frac {1}{12} \, {\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^{2} + \frac {2}{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^{2} + \frac {1}{2} \, a^{2} d^{2} 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^{2} + \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^{2} + \frac {2 \, {\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^{2}}{3 \, c^{2}} + \frac {2 \, b^{2} d^{2} \log \left (c x - 1\right )}{3 \, c^{2}} + \frac {4 \, b^{2} c^{2} d^{2} x^{2} + 32 \, b^{2} c d^{2} x + {\left (3 \, b^{2} c^{4} d^{2} x^{4} + 8 \, b^{2} c^{3} d^{2} x^{3} + 5 \, b^{2} d^{2}\right )} \log \left (c x + 1\right )^{2} + {\left (3 \, b^{2} c^{4} d^{2} x^{4} + 8 \, b^{2} c^{3} d^{2} x^{3} - 11 \, b^{2} d^{2}\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (b^{2} c^{3} d^{2} x^{3} + 4 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (2 \, b^{2} c^{3} d^{2} x^{3} + 8 \, b^{2} c^{2} d^{2} x^{2} + 6 \, b^{2} c d^{2} x + {\left (3 \, b^{2} c^{4} d^{2} x^{4} + 8 \, b^{2} c^{3} d^{2} x^{3} + 5 \, b^{2} d^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{48 \, 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^{2} \left (\int a^{2} x\, dx + \int 2 a^{2} c x^{2}\, dx + \int a^{2} c^{2} x^{3}\, 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 2 b^{2} c x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{2} x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 4 a b c x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{2} x^{3} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 761 vs.
\(2 (249) = 498\).
time = 1.59, size = 761, normalized size = 2.72 \begin {gather*} \frac {2}{45} \, {\left (\frac {30 \, {\left (c x + 1\right )}^{3} b^{2} d^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (\frac {{\left (c x + 1\right )}^{6} c^{5}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{5}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{5}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{5}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{5}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{5}}{c x - 1} + c^{5}\right )} {\left (c x - 1\right )}^{3}} + \frac {2 \, {\left (\frac {60 \, {\left (c x + 1\right )}^{3} a b d^{2}}{{\left (c x - 1\right )}^{3}} + \frac {10 \, {\left (c x + 1\right )}^{3} b^{2} d^{2}}{{\left (c x - 1\right )}^{3}} - \frac {15 \, {\left (c x + 1\right )}^{2} b^{2} d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {6 \, {\left (c x + 1\right )} b^{2} d^{2}}{c x - 1} - b^{2} d^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{6} c^{5}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{5}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{5}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{5}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{5}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{5}}{c x - 1} + c^{5}} + \frac {\frac {120 \, {\left (c x + 1\right )}^{3} a^{2} d^{2}}{{\left (c x - 1\right )}^{3}} + \frac {40 \, {\left (c x + 1\right )}^{3} a b d^{2}}{{\left (c x - 1\right )}^{3}} - \frac {60 \, {\left (c x + 1\right )}^{2} a b d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {24 \, {\left (c x + 1\right )} a b d^{2}}{c x - 1} - 4 \, a b d^{2} - \frac {2 \, {\left (c x + 1\right )}^{5} b^{2} d^{2}}{{\left (c x - 1\right )}^{5}} + \frac {11 \, {\left (c x + 1\right )}^{4} b^{2} d^{2}}{{\left (c x - 1\right )}^{4}} - \frac {18 \, {\left (c x + 1\right )}^{3} b^{2} d^{2}}{{\left (c x - 1\right )}^{3}} + \frac {11 \, {\left (c x + 1\right )}^{2} b^{2} d^{2}}{{\left (c x - 1\right )}^{2}} - \frac {2 \, {\left (c x + 1\right )} b^{2} d^{2}}{c x - 1}}{\frac {{\left (c x + 1\right )}^{6} c^{5}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{5}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{5}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{5}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{5}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{5}}{c x - 1} + c^{5}} - \frac {2 \, b^{2} d^{2} \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c^{5}} + \frac {2 \, b^{2} d^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )}{c^{5}}\right )} c^{2} \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 )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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