Optimal. Leaf size=188 \[ \frac {b^2}{2 c^2 d^2 (1+c x)}-\frac {b^2 \tanh ^{-1}(c x)}{2 c^2 d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 c^2 d^2} \]
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Rubi [A]
time = 0.25, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6087, 6065,
6063, 641, 46, 213, 6095, 6055, 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^2 d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (c x+1)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (c x+1)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}-\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 c^2 d^2}+\frac {b^2}{2 c^2 d^2 (c x+1)}-\frac {b^2 \tanh ^{-1}(c x)}{2 c^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 213
Rule 641
Rule 6055
Rule 6063
Rule 6065
Rule 6087
Rule 6095
Rule 6203
Rule 6745
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{(d+c d x)^2} \, dx &=\int \left (-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c d^2 (1+c x)^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c d^2 (1+c x)}\right ) \, dx\\ &=-\frac {\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx}{c d^2}+\frac {\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1+c x} \, dx}{c d^2}\\ &=\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^2 d^2}-\frac {(2 b) \int \left (\frac {a+b \tanh ^{-1}(c x)}{2 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c d^2}+\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 d^2}\\ &=\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^2 d^2}-\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c d^2}+\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{c d^2}-\frac {b^2 \int \frac {\text {Li}_2\left (1-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c d^2}\\ &=\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 c^2 d^2}-\frac {b^2 \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c d^2}\\ &=\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 c^2 d^2}-\frac {b^2 \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{c d^2}\\ &=\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 c^2 d^2}-\frac {b^2 \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c d^2}\\ &=\frac {b^2}{2 c^2 d^2 (1+c x)}+\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 c^2 d^2}+\frac {b^2 \int \frac {1}{-1+c^2 x^2} \, dx}{2 c d^2}\\ &=\frac {b^2}{2 c^2 d^2 (1+c x)}-\frac {b^2 \tanh ^{-1}(c x)}{2 c^2 d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 c^2 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 233, normalized size = 1.24 \begin {gather*} \frac {\frac {4 a^2}{1+c x}+4 a^2 \log (1+c x)+2 a b \left (\cosh \left (2 \tanh ^{-1}(c x)\right )+2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+2 \tanh ^{-1}(c x) \left (\cosh \left (2 \tanh ^{-1}(c x)\right )-2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )\right )+b^2 \left (\cosh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x) \cosh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x)^2 \cosh \left (2 \tanh ^{-1}(c x)\right )-4 \tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+4 \tanh ^{-1}(c x) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+2 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )-2 \tanh ^{-1}(c x) \sinh \left (2 \tanh ^{-1}(c x)\right )-2 \tanh ^{-1}(c x)^2 \sinh \left (2 \tanh ^{-1}(c x)\right )\right )}{4 c^2 d^2} \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 = 6.67, size = 946, normalized size = 5.03
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(946\) |
default | \(\text {Expression too large to display}\) | \(946\) |
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}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {b^{2} x \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {2 a b x \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx}{d^{2}} \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 = -0.01 \begin {gather*} \int \frac {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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