Optimal. Leaf size=247 \[ \frac {a b x}{c^2 d}+\frac {b^2 x \tanh ^{-1}(c x)}{c^2 d}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d}-\frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}+\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^3 d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^3 d}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^3 d}+\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3 d}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^3 d}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 c^3 d} \]
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Rubi [A]
time = 0.39, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {6077, 6037,
6127, 6021, 266, 6095, 6131, 6055, 2449, 2352, 6203, 6745} \begin {gather*} \frac {b \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d}+\frac {2 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d}-\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d}+\frac {a b x}{c^2 d}-\frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^3 d}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 c^3 d}+\frac {b^2 x \tanh ^{-1}(c x)}{c^2 d}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6055
Rule 6077
Rule 6095
Rule 6127
Rule 6131
Rule 6203
Rule 6745
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d+c d x} \, dx &=-\frac {\int \frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{d+c d x} \, dx}{c}+\frac {\int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c d}\\ &=\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}+\frac {\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d+c d x} \, dx}{c^2}-\frac {b \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{d}-\frac {\int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c^2 d}\\ &=-\frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^3 d}+\frac {b \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^2 d}-\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^2 d}+\frac {(2 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^2 d}+\frac {(2 b) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c d}\\ &=\frac {a b x}{c^2 d}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d}-\frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^3 d}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^3 d}+\frac {(2 b) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c^2 d}+\frac {b^2 \int \tanh ^{-1}(c x) \, dx}{c^2 d}-\frac {b^2 \int \frac {\text {Li}_2\left (1-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^2 d}\\ &=\frac {a b x}{c^2 d}+\frac {b^2 x \tanh ^{-1}(c x)}{c^2 d}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d}-\frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}+\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^3 d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^3 d}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^3 d}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 c^3 d}-\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^2 d}-\frac {b^2 \int \frac {x}{1-c^2 x^2} \, dx}{c d}\\ &=\frac {a b x}{c^2 d}+\frac {b^2 x \tanh ^{-1}(c x)}{c^2 d}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d}-\frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}+\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^3 d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^3 d}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^3 d}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^3 d}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 c^3 d}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c^3 d}\\ &=\frac {a b x}{c^2 d}+\frac {b^2 x \tanh ^{-1}(c x)}{c^2 d}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d}-\frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d}+\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^3 d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^3 d}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^3 d}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^3 d}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^3 d}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 c^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 260, normalized size = 1.05 \begin {gather*} \frac {-2 a^2 c x+2 a b c x+a^2 c^2 x^2-2 a b \tanh ^{-1}(c x)-4 a b c x \tanh ^{-1}(c x)+2 b^2 c x \tanh ^{-1}(c x)+2 a b c^2 x^2 \tanh ^{-1}(c x)+b^2 \tanh ^{-1}(c x)^2-2 b^2 c x \tanh ^{-1}(c x)^2+b^2 c^2 x^2 \tanh ^{-1}(c x)^2-4 a b \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+4 b^2 \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-2 b^2 \tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+2 a^2 \log (1+c x)-2 a b \log \left (1-c^2 x^2\right )+b^2 \log \left (1-c^2 x^2\right )+2 b \left (a-b+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+b^2 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )}{2 c^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 7.85, size = 1099, normalized size = 4.45
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1099\) |
default | \(\text {Expression too large to display}\) | \(1099\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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*} \frac {\int \frac {a^{2} x^{2}}{c x + 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c x + 1}\, dx + \int \frac {2 a b x^{2} \operatorname {atanh}{\left (c x \right )}}{c x + 1}\, dx}{d} \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 \frac {x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{d+c\,d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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