Optimal. Leaf size=179 \[ \frac {1}{3} b^2 e^2 x-\frac {b^2 e^2 \tanh ^{-1}(c+d x)}{3 d}+\frac {b e^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac {e^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}-\frac {2 b e^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d}-\frac {b^2 e^2 \text {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{3 d} \]
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Rubi [A]
time = 0.17, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6242, 12,
6037, 6127, 327, 212, 6131, 6055, 2449, 2352} \begin {gather*} \frac {e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}+\frac {b e^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac {e^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}-\frac {2 b e^2 \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}-\frac {b^2 e^2 \text {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right )}{3 d}-\frac {b^2 e^2 \tanh ^{-1}(c+d x)}{3 d}+\frac {1}{3} b^2 e^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 327
Rule 2352
Rule 2449
Rule 6037
Rule 6055
Rule 6127
Rule 6131
Rule 6242
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int e^2 x^2 \left (a+b \tanh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int x^2 \left (a+b \tanh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {x^3 \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}+\frac {\left (2 b e^2\right ) \text {Subst}\left (\int x \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d}\\ &=\frac {b e^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac {e^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{3 d}-\frac {\left (b^2 e^2\right ) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,c+d x\right )}{3 d}\\ &=\frac {1}{3} b^2 e^2 x+\frac {b e^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac {e^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}-\frac {2 b e^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d}-\frac {\left (b^2 e^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,c+d x\right )}{3 d}+\frac {\left (2 b^2 e^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d}\\ &=\frac {1}{3} b^2 e^2 x-\frac {b^2 e^2 \tanh ^{-1}(c+d x)}{3 d}+\frac {b e^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac {e^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}-\frac {2 b e^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d}-\frac {\left (2 b^2 e^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c-d x}\right )}{3 d}\\ &=\frac {1}{3} b^2 e^2 x-\frac {b^2 e^2 \tanh ^{-1}(c+d x)}{3 d}+\frac {b e^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac {e^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}-\frac {2 b e^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d}-\frac {b^2 e^2 \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 150, normalized size = 0.84 \begin {gather*} \frac {e^2 \left (a^2 (c+d x)^3+a b \left ((c+d x)^2+2 (c+d x)^3 \tanh ^{-1}(c+d x)+\log \left (-1+(c+d x)^2\right )\right )+b^2 \left (c+d x-\tanh ^{-1}(c+d x)+(c+d x)^2 \tanh ^{-1}(c+d x)-\tanh ^{-1}(c+d x)^2+(c+d x)^3 \tanh ^{-1}(c+d x)^2-2 \tanh ^{-1}(c+d x) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )+\text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )\right )}{3 d} \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. \(341\) vs.
\(2(167)=334\).
time = 4.12, size = 342, 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. 572 vs.
\(2 (150) = 300\).
time = 0.45, size = 572, normalized size = 3.20 \begin {gather*} \frac {1}{3} \, a^{2} d^{2} x^{3} e^{2} + a^{2} c d x^{2} e^{2} + {\left (2 \, x^{2} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} a b c d e^{2} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} + \frac {{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac {{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} a b d^{2} e^{2} + a^{2} c^{2} x e^{2} + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} a b c^{2} e^{2}}{d} + \frac {{\left (c^{2} - 1\right )} b^{2} e^{2} \log \left (d x + c + 1\right )}{6 \, d} - \frac {{\left (c^{2} - 1\right )} b^{2} e^{2} \log \left (d x + c - 1\right )}{6 \, d} + \frac {{\left (\log \left (d x + c + 1\right ) \log \left (-\frac {1}{2} \, d x - \frac {1}{2} \, c + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, d x + \frac {1}{2} \, c + \frac {1}{2}\right )\right )} b^{2} e^{2}}{3 \, d} + \frac {4 \, b^{2} d x e^{2} + {\left (b^{2} d^{3} x^{3} e^{2} + 3 \, b^{2} c d^{2} x^{2} e^{2} + 3 \, b^{2} c^{2} d x e^{2} + {\left (c^{3} + 1\right )} b^{2} e^{2}\right )} \log \left (d x + c + 1\right )^{2} + {\left (b^{2} d^{3} x^{3} e^{2} + 3 \, b^{2} c d^{2} x^{2} e^{2} + 3 \, b^{2} c^{2} d x e^{2} + {\left (c^{3} - 1\right )} b^{2} e^{2}\right )} \log \left (-d x - c + 1\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} e^{2} + 2 \, b^{2} c d x e^{2}\right )} \log \left (d x + c + 1\right ) - 2 \, {\left (b^{2} d^{2} x^{2} e^{2} + 2 \, b^{2} c d x e^{2} + {\left (b^{2} d^{3} x^{3} e^{2} + 3 \, b^{2} c d^{2} x^{2} e^{2} + 3 \, b^{2} c^{2} d x e^{2} + {\left (c^{3} + 1\right )} b^{2} e^{2}\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )}{12 \, d} \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*} e^{2} \left (\int a^{2} c^{2}\, dx + \int a^{2} d^{2} x^{2}\, dx + \int b^{2} c^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{2} \operatorname {atanh}{\left (c + d x \right )}\, dx + \int 2 a^{2} c d x\, dx + \int b^{2} d^{2} x^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{2} x^{2} \operatorname {atanh}{\left (c + d x \right )}\, dx + \int 2 b^{2} c d x \operatorname {atanh}^{2}{\left (c + d x \right )}\, dx + \int 4 a b c d x \operatorname {atanh}{\left (c + d 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 = -0.01 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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