Optimal. Leaf size=159 \[ \frac {e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}+\frac {b e^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d}-\frac {e^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}+\frac {1}{2} a b e^3 x+\frac {b^2 e^3 (c+d x)^2}{12 d}+\frac {b^2 e^3 \log \left (1-(c+d x)^2\right )}{3 d}+\frac {b^2 e^3 (c+d x) \tanh ^{-1}(c+d x)}{2 d} \]
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Rubi [A] time = 0.25, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6107, 12, 5916, 5980, 266, 43, 5910, 260, 5948} \[ \frac {e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}+\frac {b e^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d}-\frac {e^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}+\frac {1}{2} a b e^3 x+\frac {b^2 e^3 (c+d x)^2}{12 d}+\frac {b^2 e^3 \log \left (1-(c+d x)^2\right )}{3 d}+\frac {b^2 e^3 (c+d x) \tanh ^{-1}(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 260
Rule 266
Rule 5910
Rule 5916
Rule 5948
Rule 5980
Rule 6107
Rubi steps
\begin {align*} \int (c e+d e x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int e^3 x^3 \left (a+b \tanh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \operatorname {Subst}\left (\int x^3 \left (a+b \tanh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}-\frac {\left (b e^3\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}+\frac {\left (b e^3\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{2 d}-\frac {\left (b e^3\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {b e^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d}+\frac {e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}+\frac {\left (b e^3\right ) \operatorname {Subst}\left (\int \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{2 d}-\frac {\left (b e^3\right ) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{2 d}-\frac {\left (b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{1-x^2} \, dx,x,c+d x\right )}{6 d}\\ &=\frac {1}{2} a b e^3 x+\frac {b e^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d}-\frac {e^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}-\frac {\left (b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {x}{1-x} \, dx,x,(c+d x)^2\right )}{12 d}+\frac {\left (b^2 e^3\right ) \operatorname {Subst}\left (\int \tanh ^{-1}(x) \, dx,x,c+d x\right )}{2 d}\\ &=\frac {1}{2} a b e^3 x+\frac {b^2 e^3 (c+d x) \tanh ^{-1}(c+d x)}{2 d}+\frac {b e^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d}-\frac {e^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}-\frac {\left (b^2 e^3\right ) \operatorname {Subst}\left (\int \left (-1+\frac {1}{1-x}\right ) \, dx,x,(c+d x)^2\right )}{12 d}-\frac {\left (b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {1}{2} a b e^3 x+\frac {b^2 e^3 (c+d x)^2}{12 d}+\frac {b^2 e^3 (c+d x) \tanh ^{-1}(c+d x)}{2 d}+\frac {b e^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d}-\frac {e^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}+\frac {b^2 e^3 \log \left (1-(c+d x)^2\right )}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 148, normalized size = 0.93 \[ \frac {e^3 \left (3 a^2 (c+d x)^4+2 a b (c+d x)^3+6 a b (c+d x)+b (3 a+4 b) \log (-c-d x+1)+b (4 b-3 a) \log (c+d x+1)+2 b (c+d x) \tanh ^{-1}(c+d x) \left (3 a (c+d x)^3+b (c+d x)^2+3 b\right )+b^2 (c+d x)^2+3 b^2 \left ((c+d x)^4-1\right ) \tanh ^{-1}(c+d x)^2\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 383, normalized size = 2.41 \[ \frac {12 \, a^{2} d^{4} e^{3} x^{4} + 8 \, {\left (6 \, a^{2} c + a b\right )} d^{3} e^{3} x^{3} + 4 \, {\left (18 \, a^{2} c^{2} + 6 \, a b c + b^{2}\right )} d^{2} e^{3} x^{2} + 8 \, {\left (6 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c + 3 \, a b\right )} d e^{3} x + 4 \, {\left (3 \, a b c^{4} + b^{2} c^{3} + 3 \, b^{2} c - 3 \, a b + 4 \, b^{2}\right )} e^{3} \log \left (d x + c + 1\right ) - 4 \, {\left (3 \, a b c^{4} + b^{2} c^{3} + 3 \, b^{2} c - 3 \, a b - 4 \, b^{2}\right )} e^{3} \log \left (d x + c - 1\right ) + 3 \, {\left (b^{2} d^{4} e^{3} x^{4} + 4 \, b^{2} c d^{3} e^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 4 \, b^{2} c^{3} d e^{3} x + {\left (b^{2} c^{4} - b^{2}\right )} e^{3}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2} + 4 \, {\left (3 \, a b d^{4} e^{3} x^{4} + {\left (12 \, a b c + b^{2}\right )} d^{3} e^{3} x^{3} + 3 \, {\left (6 \, a b c^{2} + b^{2} c\right )} d^{2} e^{3} x^{2} + 3 \, {\left (4 \, a b c^{3} + b^{2} c^{2} + b^{2}\right )} d e^{3} x\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 791, normalized size = 4.97 \[ \frac {{\left (\frac {3 \, {\left (d x + c + 1\right )}^{3} b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2}}{{\left (d x + c - 1\right )}^{3}} + \frac {3 \, {\left (d x + c + 1\right )} b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2}}{d x + c - 1} - \frac {4 \, {\left (d x + c + 1\right )}^{4} b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1} + 1\right )}{{\left (d x + c - 1\right )}^{4}} + \frac {16 \, {\left (d x + c + 1\right )}^{3} b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1} + 1\right )}{{\left (d x + c - 1\right )}^{3}} - \frac {24 \, {\left (d x + c + 1\right )}^{2} b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1} + 1\right )}{{\left (d x + c - 1\right )}^{2}} + \frac {16 \, {\left (d x + c + 1\right )} b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1} + 1\right )}{d x + c - 1} - 4 \, b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1} + 1\right ) + \frac {12 \, {\left (d x + c + 1\right )}^{3} a b e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{{\left (d x + c - 1\right )}^{3}} + \frac {12 \, {\left (d x + c + 1\right )} a b e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d x + c - 1} + \frac {4 \, {\left (d x + c + 1\right )}^{4} b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{{\left (d x + c - 1\right )}^{4}} - \frac {10 \, {\left (d x + c + 1\right )}^{3} b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{{\left (d x + c - 1\right )}^{3}} + \frac {12 \, {\left (d x + c + 1\right )}^{2} b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{{\left (d x + c - 1\right )}^{2}} - \frac {6 \, {\left (d x + c + 1\right )} b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d x + c - 1} + \frac {12 \, {\left (d x + c + 1\right )}^{3} a^{2} e^{3}}{{\left (d x + c - 1\right )}^{3}} + \frac {12 \, {\left (d x + c + 1\right )} a^{2} e^{3}}{d x + c - 1} + \frac {12 \, {\left (d x + c + 1\right )}^{3} a b e^{3}}{{\left (d x + c - 1\right )}^{3}} - \frac {24 \, {\left (d x + c + 1\right )}^{2} a b e^{3}}{{\left (d x + c - 1\right )}^{2}} + \frac {20 \, {\left (d x + c + 1\right )} a b e^{3}}{d x + c - 1} - 8 \, a b e^{3} + \frac {2 \, {\left (d x + c + 1\right )}^{3} b^{2} e^{3}}{{\left (d x + c - 1\right )}^{3}} - \frac {4 \, {\left (d x + c + 1\right )}^{2} b^{2} e^{3}}{{\left (d x + c - 1\right )}^{2}} + \frac {2 \, {\left (d x + c + 1\right )} b^{2} e^{3}}{d x + c - 1}\right )} {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )}}{12 \, {\left (\frac {{\left (d x + c + 1\right )}^{4} d^{2}}{{\left (d x + c - 1\right )}^{4}} - \frac {4 \, {\left (d x + c + 1\right )}^{3} d^{2}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )}^{2} d^{2}}{{\left (d x + c - 1\right )}^{2}} - \frac {4 \, {\left (d x + c + 1\right )} d^{2}}{d x + c - 1} + d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 732, normalized size = 4.60 \[ \frac {a b \,e^{3} x}{2}+\frac {d^{3} x^{4} a^{2} e^{3}}{4}+\frac {\arctanh \left (d x +c \right ) x \,b^{2} e^{3}}{2}+2 d^{2} \arctanh \left (d x +c \right ) x^{3} a b c \,e^{3}+3 d \arctanh \left (d x +c \right ) x^{2} a b \,c^{2} e^{3}+\frac {e^{3} b^{2} \ln \left (d x +c -1\right )^{2}}{16 d}+\frac {e^{3} b^{2} \ln \left (d x +c +1\right )}{3 d}+\frac {e^{3} b^{2} \ln \left (d x +c -1\right )}{3 d}+\frac {e^{3} b^{2} \ln \left (d x +c +1\right )^{2}}{16 d}+\frac {e^{3} b^{2} c^{2}}{12 d}+\frac {a^{2} c^{4} e^{3}}{4 d}+2 \arctanh \left (d x +c \right ) x a b \,c^{3} e^{3}+\frac {d \arctanh \left (d x +c \right ) x^{2} b^{2} c \,e^{3}}{2}+\frac {3 d \arctanh \left (d x +c \right )^{2} x^{2} b^{2} c^{2} e^{3}}{2}+\frac {\arctanh \left (d x +c \right ) a b \,c^{4} e^{3}}{2 d}+d^{2} \arctanh \left (d x +c \right )^{2} x^{3} b^{2} c \,e^{3}+\frac {d \,x^{2} a b c \,e^{3}}{2}+\frac {e^{3} b^{2} x c}{6}+x \,a^{2} c^{3} e^{3}+\frac {d \,e^{3} b^{2} x^{2}}{12}+\frac {x a b \,c^{2} e^{3}}{2}+\frac {3 d \,x^{2} a^{2} c^{2} e^{3}}{2}+d^{2} x^{3} a^{2} c \,e^{3}+\frac {e^{3} a b \ln \left (d x +c -1\right )}{4 d}-\frac {e^{3} a b \ln \left (d x +c +1\right )}{4 d}-\frac {e^{3} b^{2} \ln \left (d x +c -1\right ) \ln \left (\frac {1}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {e^{3} 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 )}{8 d}-\frac {e^{3} b^{2} \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (d x +c +1\right )}{8 d}+\frac {d^{2} x^{3} a b \,e^{3}}{6}+\frac {\arctanh \left (d x +c \right ) x \,b^{2} c^{2} e^{3}}{2}+\arctanh \left (d x +c \right )^{2} x \,b^{2} c^{3} e^{3}+\frac {e^{3} b^{2} \arctanh \left (d x +c \right ) \ln \left (d x +c -1\right )}{4 d}+\frac {\arctanh \left (d x +c \right ) b^{2} c^{3} e^{3}}{6 d}+\frac {\arctanh \left (d x +c \right ) b^{2} c \,e^{3}}{2 d}+\frac {d^{2} \arctanh \left (d x +c \right ) x^{3} b^{2} e^{3}}{6}+\frac {d^{3} \arctanh \left (d x +c \right )^{2} x^{4} b^{2} e^{3}}{4}-\frac {e^{3} b^{2} \arctanh \left (d x +c \right ) \ln \left (d x +c +1\right )}{4 d}+\frac {\arctanh \left (d x +c \right )^{2} b^{2} c^{4} e^{3}}{4 d}+\frac {a b \,c^{3} e^{3}}{6 d}+\frac {d^{3} \arctanh \left (d x +c \right ) x^{4} a b \,e^{3}}{2}+\frac {a b c \,e^{3}}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 827, normalized size = 5.20 \[ \frac {1}{4} \, a^{2} d^{3} e^{3} x^{4} + a^{2} c d^{2} e^{3} x^{3} + \frac {3}{2} \, a^{2} c^{2} d e^{3} x^{2} + \frac {3}{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^{2} d e^{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 c d^{2} e^{3} + \frac {1}{12} \, {\left (6 \, x^{4} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, {\left (d^{2} x^{3} - 3 \, c d x^{2} + 3 \, {\left (3 \, c^{2} + 1\right )} x\right )}}{d^{4}} - \frac {3 \, {\left (c^{4} + 4 \, c^{3} + 6 \, c^{2} + 4 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{5}} + \frac {3 \, {\left (c^{4} - 4 \, c^{3} + 6 \, c^{2} - 4 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{5}}\right )}\right )} a b d^{3} e^{3} + a^{2} c^{3} e^{3} 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^{3} e^{3}}{d} + \frac {4 \, b^{2} d^{2} e^{3} x^{2} + 8 \, b^{2} c d e^{3} x + 3 \, {\left (b^{2} d^{4} e^{3} x^{4} + 4 \, b^{2} c d^{3} e^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 4 \, b^{2} c^{3} d e^{3} x + {\left (c^{4} e^{3} - e^{3}\right )} b^{2}\right )} \log \left (d x + c + 1\right )^{2} + 3 \, {\left (b^{2} d^{4} e^{3} x^{4} + 4 \, b^{2} c d^{3} e^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 4 \, b^{2} c^{3} d e^{3} x + {\left (c^{4} e^{3} - e^{3}\right )} b^{2}\right )} \log \left (-d x - c + 1\right )^{2} + 4 \, {\left (b^{2} d^{3} e^{3} x^{3} + 3 \, b^{2} c d^{2} e^{3} x^{2} + 3 \, {\left (c^{2} d e^{3} + d e^{3}\right )} b^{2} x + {\left (c^{3} e^{3} + 3 \, c e^{3} + 4 \, e^{3}\right )} b^{2}\right )} \log \left (d x + c + 1\right ) - 2 \, {\left (2 \, b^{2} d^{3} e^{3} x^{3} + 6 \, b^{2} c d^{2} e^{3} x^{2} + 6 \, {\left (c^{2} d e^{3} + d e^{3}\right )} b^{2} x + 2 \, {\left (c^{3} e^{3} + 3 \, c e^{3} - 4 \, e^{3}\right )} b^{2} + 3 \, {\left (b^{2} d^{4} e^{3} x^{4} + 4 \, b^{2} c d^{3} e^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 4 \, b^{2} c^{3} d e^{3} x + {\left (c^{4} e^{3} - e^{3}\right )} b^{2}\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 1730, normalized size = 10.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.07, size = 581, normalized size = 3.65 \[ \begin {cases} a^{2} c^{3} e^{3} x + \frac {3 a^{2} c^{2} d e^{3} x^{2}}{2} + a^{2} c d^{2} e^{3} x^{3} + \frac {a^{2} d^{3} e^{3} x^{4}}{4} + \frac {a b c^{4} e^{3} \operatorname {atanh}{\left (c + d x \right )}}{2 d} + 2 a b c^{3} e^{3} x \operatorname {atanh}{\left (c + d x \right )} + 3 a b c^{2} d e^{3} x^{2} \operatorname {atanh}{\left (c + d x \right )} + \frac {a b c^{2} e^{3} x}{2} + 2 a b c d^{2} e^{3} x^{3} \operatorname {atanh}{\left (c + d x \right )} + \frac {a b c d e^{3} x^{2}}{2} + \frac {a b d^{3} e^{3} x^{4} \operatorname {atanh}{\left (c + d x \right )}}{2} + \frac {a b d^{2} e^{3} x^{3}}{6} + \frac {a b e^{3} x}{2} - \frac {a b e^{3} \operatorname {atanh}{\left (c + d x \right )}}{2 d} + \frac {b^{2} c^{4} e^{3} \operatorname {atanh}^{2}{\left (c + d x \right )}}{4 d} + b^{2} c^{3} e^{3} x \operatorname {atanh}^{2}{\left (c + d x \right )} + \frac {b^{2} c^{3} e^{3} \operatorname {atanh}{\left (c + d x \right )}}{6 d} + \frac {3 b^{2} c^{2} d e^{3} x^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}}{2} + \frac {b^{2} c^{2} e^{3} x \operatorname {atanh}{\left (c + d x \right )}}{2} + b^{2} c d^{2} e^{3} x^{3} \operatorname {atanh}^{2}{\left (c + d x \right )} + \frac {b^{2} c d e^{3} x^{2} \operatorname {atanh}{\left (c + d x \right )}}{2} + \frac {b^{2} c e^{3} x}{6} + \frac {b^{2} c e^{3} \operatorname {atanh}{\left (c + d x \right )}}{2 d} + \frac {b^{2} d^{3} e^{3} x^{4} \operatorname {atanh}^{2}{\left (c + d x \right )}}{4} + \frac {b^{2} d^{2} e^{3} x^{3} \operatorname {atanh}{\left (c + d x \right )}}{6} + \frac {b^{2} d e^{3} x^{2}}{12} + \frac {b^{2} e^{3} x \operatorname {atanh}{\left (c + d x \right )}}{2} + \frac {2 b^{2} e^{3} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{3 d} - \frac {b^{2} e^{3} \operatorname {atanh}^{2}{\left (c + d x \right )}}{4 d} - \frac {2 b^{2} e^{3} \operatorname {atanh}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\c^{3} e^{3} x \left (a + b \operatorname {atanh}{\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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