Optimal. Leaf size=179 \[ \frac {e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}+\frac {b e^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac {e^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}-\frac {2 b e^2 \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}-\frac {b^2 e^2 \text {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right )}{3 d}-\frac {b^2 e^2 \tanh ^{-1}(c+d x)}{3 d}+\frac {1}{3} b^2 e^2 x \]
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Rubi [A] time = 0.24, 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 = {6107, 12, 5916, 5980, 321, 206, 5984, 5918, 2402, 2315} \[ -\frac {b^2 e^2 \text {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}+\frac {b e^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac {e^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}-\frac {2 b e^2 \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}-\frac {b^2 e^2 \tanh ^{-1}(c+d x)}{3 d}+\frac {1}{3} b^2 e^2 x \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 321
Rule 2315
Rule 2402
Rule 5916
Rule 5918
Rule 5980
Rule 5984
Rule 6107
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,c+d x\right )}{3 d}+\frac {\left (2 b^2 e^2\right ) \operatorname {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 ) \operatorname {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 = 0.47, size = 150, normalized size = 0.84 \[ \frac {e^2 \left (a^2 (c+d x)^3+a b \left ((c+d x)^2+\log \left ((c+d x)^2-1\right )+2 (c+d x)^3 \tanh ^{-1}(c+d x)\right )+b^2 \left (\text {Li}_2\left (-e^{-2 \tanh ^{-1}(c+d x)}\right )+(c+d x)^3 \tanh ^{-1}(c+d x)^2+(c+d x)^2 \tanh ^{-1}(c+d x)-\tanh ^{-1}(c+d x)^2-\tanh ^{-1}(c+d x)-2 \tanh ^{-1}(c+d x) \log \left (e^{-2 \tanh ^{-1}(c+d x)}+1\right )+c+d x\right )\right )}{3 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} d^{2} e^{2} x^{2} + 2 \, a^{2} c d e^{2} x + a^{2} c^{2} e^{2} + {\left (b^{2} d^{2} e^{2} x^{2} + 2 \, b^{2} c d e^{2} x + b^{2} c^{2} e^{2}\right )} \operatorname {artanh}\left (d x + c\right )^{2} + 2 \, {\left (a b d^{2} e^{2} x^{2} + 2 \, a b c d e^{2} x + a b c^{2} e^{2}\right )} \operatorname {artanh}\left (d x + c\right ), x\right ) \]
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 \[ \int {\left (d e x + c e\right )}^{2} {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 583, normalized size = 3.26 \[ -\frac {e^{2} b^{2} \ln \left (d x +c +1\right )^{2}}{12 d}+\frac {d^{2} x^{3} a^{2} e^{2}}{3}+x \,a^{2} c^{2} e^{2}-\frac {e^{2} b^{2} \dilog \left (\frac {1}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{3 d}-\frac {e^{2} b^{2} \ln \left (d x +c +1\right )}{6 d}+\frac {e^{2} b^{2} \ln \left (d x +c -1\right )}{6 d}+\frac {e^{2} b^{2} \ln \left (d x +c -1\right )^{2}}{12 d}+\frac {b^{2} c \,e^{2}}{3 d}+\frac {a^{2} c^{3} e^{2}}{3 d}+d \arctanh \left (d x +c \right )^{2} x^{2} b^{2} c \,e^{2}+2 \arctanh \left (d x +c \right ) x a b \,c^{2} e^{2}+\frac {2 \arctanh \left (d x +c \right ) a b \,c^{3} e^{2}}{3 d}+\frac {2 d^{2} \arctanh \left (d x +c \right ) x^{3} a b \,e^{2}}{3}+2 d \arctanh \left (d x +c \right ) x^{2} a b c \,e^{2}+\frac {a b \,c^{2} e^{2}}{3 d}+\frac {2 x a b c \,e^{2}}{3}+\frac {b^{2} e^{2} x}{3}+d \,x^{2} a^{2} c \,e^{2}+\frac {d \,x^{2} a b \,e^{2}}{3}-\frac {e^{2} b^{2} \ln \left (d x +c -1\right ) \ln \left (\frac {1}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{6 d}+\frac {e^{2} b^{2} \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (d x +c +1\right )}{6 d}-\frac {e^{2} b^{2} \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{6 d}+\frac {e^{2} a b \ln \left (d x +c +1\right )}{3 d}+\frac {d^{2} \arctanh \left (d x +c \right )^{2} x^{3} b^{2} e^{2}}{3}+\frac {e^{2} b^{2} \arctanh \left (d x +c \right ) \ln \left (d x +c -1\right )}{3 d}+\frac {2 \arctanh \left (d x +c \right ) x \,b^{2} c \,e^{2}}{3}+\frac {e^{2} a b \ln \left (d x +c -1\right )}{3 d}+\arctanh \left (d x +c \right )^{2} x \,b^{2} c^{2} e^{2}+\frac {d \arctanh \left (d x +c \right ) x^{2} b^{2} e^{2}}{3}+\frac {\arctanh \left (d x +c \right )^{2} b^{2} c^{3} e^{2}}{3 d}+\frac {\arctanh \left (d x +c \right ) b^{2} c^{2} e^{2}}{3 d}+\frac {e^{2} b^{2} \arctanh \left (d x +c \right ) \ln \left (d x +c +1\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.63, size = 619, normalized size = 3.46 \[ \frac {1}{3} \, a^{2} d^{2} e^{2} x^{3} + a^{2} c d e^{2} x^{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} e^{2} x + \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 (\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 {{\left (c^{2} e^{2} - e^{2}\right )} b^{2} \log \left (d x + c + 1\right )}{6 \, d} - \frac {{\left (c^{2} e^{2} - e^{2}\right )} b^{2} \log \left (d x + c - 1\right )}{6 \, d} + \frac {4 \, b^{2} d e^{2} x + {\left (b^{2} d^{3} e^{2} x^{3} + 3 \, b^{2} c d^{2} e^{2} x^{2} + 3 \, b^{2} c^{2} d e^{2} x + {\left (c^{3} e^{2} + e^{2}\right )} b^{2}\right )} \log \left (d x + c + 1\right )^{2} + {\left (b^{2} d^{3} e^{2} x^{3} + 3 \, b^{2} c d^{2} e^{2} x^{2} + 3 \, b^{2} c^{2} d e^{2} x + {\left (c^{3} e^{2} - e^{2}\right )} b^{2}\right )} \log \left (-d x - c + 1\right )^{2} + 2 \, {\left (b^{2} d^{2} e^{2} x^{2} + 2 \, b^{2} c d e^{2} x\right )} \log \left (d x + c + 1\right ) - 2 \, {\left (b^{2} d^{2} e^{2} x^{2} + 2 \, b^{2} c d e^{2} x + {\left (b^{2} d^{3} e^{2} x^{3} + 3 \, b^{2} c d^{2} e^{2} x^{2} + 3 \, b^{2} c^{2} d e^{2} x + {\left (c^{3} e^{2} + e^{2}\right )} b^{2}\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )}{12 \, d} \]
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 \[ \int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2 \,d x \]
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 \[ e^{2} \left (\int a^{2} c^{2}\, dx + \int a^{2} d^{2} x^{2}\, dx + \int b^{2} c^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{2} \operatorname {atanh}{\left (c + d x \right )}\, dx + \int 2 a^{2} c d x\, dx + \int b^{2} d^{2} x^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{2} x^{2} \operatorname {atanh}{\left (c + d x \right )}\, dx + \int 2 b^{2} c d x \operatorname {atanh}^{2}{\left (c + d x \right )}\, dx + \int 4 a b c d x \operatorname {atanh}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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