Optimal. Leaf size=159 \[ \frac {1}{2} a b e^3 x+\frac {b^2 e^3 (c+d x)^2}{12 d}+\frac {b^2 e^3 (c+d x) \tanh ^{-1}(c+d x)}{2 d}+\frac {b e^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d}-\frac {e^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d}+\frac {b^2 e^3 \log \left (1-(c+d x)^2\right )}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, 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 = {6242, 12,
6037, 6127, 272, 45, 6021, 266, 6095} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 266
Rule 272
Rule 6021
Rule 6037
Rule 6095
Rule 6127
Rule 6242
Rubi steps
\begin {align*} \int (c e+d e x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{2 d}-\frac {\left (b e^3\right ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {x}{1-x} \, dx,x,(c+d x)^2\right )}{12 d}+\frac {\left (b^2 e^3\right ) \text {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 ) \text {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 ) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 148, normalized size = 0.93 \begin {gather*} \frac {e^3 \left (6 a b (c+d x)+b^2 (c+d x)^2+2 a b (c+d x)^3+3 a^2 (c+d x)^4+2 b (c+d x) \left (3 b+b (c+d x)^2+3 a (c+d x)^3\right ) \tanh ^{-1}(c+d x)+3 b^2 \left (-1+(c+d x)^4\right ) \tanh ^{-1}(c+d x)^2+b (3 a+4 b) \log (1-c-d x)+b (-3 a+4 b) \log (1+c+d x)\right )}{12 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(356\) vs.
\(2(145)=290\).
time = 2.70, size = 357, normalized size = 2.25 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 755 vs.
\(2 (138) = 276\).
time = 0.46, size = 755, normalized size = 4.75 \begin {gather*} \frac {1}{4} \, a^{2} d^{3} x^{4} e^{3} + a^{2} c d^{2} x^{3} e^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} e^{3} + \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} x e^{3} + \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} x^{2} e^{3} + 8 \, b^{2} c d x e^{3} + 3 \, {\left (b^{2} d^{4} x^{4} e^{3} + 4 \, b^{2} c d^{3} x^{3} e^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} e^{3} + 4 \, b^{2} c^{3} d x e^{3} + {\left (c^{4} - 1\right )} b^{2} e^{3}\right )} \log \left (d x + c + 1\right )^{2} + 3 \, {\left (b^{2} d^{4} x^{4} e^{3} + 4 \, b^{2} c d^{3} x^{3} e^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} e^{3} + 4 \, b^{2} c^{3} d x e^{3} + {\left (c^{4} - 1\right )} b^{2} e^{3}\right )} \log \left (-d x - c + 1\right )^{2} + 4 \, {\left (b^{2} d^{3} x^{3} e^{3} + 3 \, b^{2} c d^{2} x^{2} e^{3} + 3 \, {\left (c^{2} d + d\right )} b^{2} x e^{3} + {\left (c^{3} + 3 \, c + 4\right )} b^{2} e^{3}\right )} \log \left (d x + c + 1\right ) - 2 \, {\left (2 \, b^{2} d^{3} x^{3} e^{3} + 6 \, b^{2} c d^{2} x^{2} e^{3} + 6 \, {\left (c^{2} d + d\right )} b^{2} x e^{3} + 2 \, {\left (c^{3} + 3 \, c - 4\right )} b^{2} e^{3} + 3 \, {\left (b^{2} d^{4} x^{4} e^{3} + 4 \, b^{2} c d^{3} x^{3} e^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} e^{3} + 4 \, b^{2} c^{3} d x e^{3} + {\left (c^{4} - 1\right )} b^{2} e^{3}\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )}{48 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1288 vs.
\(2 (138) = 276\).
time = 0.38, size = 1288, normalized size = 8.10 \begin {gather*} \frac {4 \, {\left (3 \, a^{2} d^{4} x^{4} + 2 \, {\left (6 \, a^{2} c + a b\right )} d^{3} x^{3} + {\left (18 \, a^{2} c^{2} + 6 \, a b c + b^{2}\right )} d^{2} x^{2} + 2 \, {\left (6 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c + 3 \, a b\right )} d x\right )} \cosh \left (1\right )^{3} + 12 \, {\left (3 \, a^{2} d^{4} x^{4} + 2 \, {\left (6 \, a^{2} c + a b\right )} d^{3} x^{3} + {\left (18 \, a^{2} c^{2} + 6 \, a b c + b^{2}\right )} d^{2} x^{2} + 2 \, {\left (6 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c + 3 \, a b\right )} d x\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 12 \, {\left (3 \, a^{2} d^{4} x^{4} + 2 \, {\left (6 \, a^{2} c + a b\right )} d^{3} x^{3} + {\left (18 \, a^{2} c^{2} + 6 \, a b c + b^{2}\right )} d^{2} x^{2} + 2 \, {\left (6 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c + 3 \, a b\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + 4 \, {\left (3 \, a^{2} d^{4} x^{4} + 2 \, {\left (6 \, a^{2} c + a b\right )} d^{3} x^{3} + {\left (18 \, a^{2} c^{2} + 6 \, a b c + b^{2}\right )} d^{2} x^{2} + 2 \, {\left (6 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c + 3 \, a b\right )} d x\right )} \sinh \left (1\right )^{3} + 3 \, {\left ({\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - b^{2}\right )} \cosh \left (1\right )^{3} + 3 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - b^{2}\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - b^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - b^{2}\right )} \sinh \left (1\right )^{3}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2} + 4 \, {\left ({\left (3 \, a b c^{4} + b^{2} c^{3} + 3 \, b^{2} c - 3 \, a b + 4 \, b^{2}\right )} \cosh \left (1\right )^{3} + 3 \, {\left (3 \, a b c^{4} + b^{2} c^{3} + 3 \, b^{2} c - 3 \, a b + 4 \, b^{2}\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (3 \, a b c^{4} + b^{2} c^{3} + 3 \, b^{2} c - 3 \, a b + 4 \, b^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (3 \, a b c^{4} + b^{2} c^{3} + 3 \, b^{2} c - 3 \, a b + 4 \, b^{2}\right )} \sinh \left (1\right )^{3}\right )} \log \left (d x + c + 1\right ) - 4 \, {\left ({\left (3 \, a b c^{4} + b^{2} c^{3} + 3 \, b^{2} c - 3 \, a b - 4 \, b^{2}\right )} \cosh \left (1\right )^{3} + 3 \, {\left (3 \, a b c^{4} + b^{2} c^{3} + 3 \, b^{2} c - 3 \, a b - 4 \, b^{2}\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (3 \, a b c^{4} + b^{2} c^{3} + 3 \, b^{2} c - 3 \, a b - 4 \, b^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (3 \, a b c^{4} + b^{2} c^{3} + 3 \, b^{2} c - 3 \, a b - 4 \, b^{2}\right )} \sinh \left (1\right )^{3}\right )} \log \left (d x + c - 1\right ) + 4 \, {\left ({\left (3 \, a b d^{4} x^{4} + {\left (12 \, a b c + b^{2}\right )} d^{3} x^{3} + 3 \, {\left (6 \, a b c^{2} + b^{2} c\right )} d^{2} x^{2} + 3 \, {\left (4 \, a b c^{3} + b^{2} c^{2} + b^{2}\right )} d x\right )} \cosh \left (1\right )^{3} + 3 \, {\left (3 \, a b d^{4} x^{4} + {\left (12 \, a b c + b^{2}\right )} d^{3} x^{3} + 3 \, {\left (6 \, a b c^{2} + b^{2} c\right )} d^{2} x^{2} + 3 \, {\left (4 \, a b c^{3} + b^{2} c^{2} + b^{2}\right )} d x\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (3 \, a b d^{4} x^{4} + {\left (12 \, a b c + b^{2}\right )} d^{3} x^{3} + 3 \, {\left (6 \, a b c^{2} + b^{2} c\right )} d^{2} x^{2} + 3 \, {\left (4 \, a b c^{3} + b^{2} c^{2} + b^{2}\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (3 \, a b d^{4} x^{4} + {\left (12 \, a b c + b^{2}\right )} d^{3} x^{3} + 3 \, {\left (6 \, a b c^{2} + b^{2} c\right )} d^{2} x^{2} + 3 \, {\left (4 \, a b c^{3} + b^{2} c^{2} + b^{2}\right )} d x\right )} \sinh \left (1\right )^{3}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{48 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 581 vs.
\(2 (138) = 276\).
time = 2.38, size = 581, normalized size = 3.65 \begin {gather*} \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}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 733 vs.
\(2 (145) = 290\).
time = 0.46, size = 733, normalized size = 4.61 \begin {gather*} -\frac {1}{12} \, {\left (\frac {4 \, b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1} + 1\right )}{d^{2}} - \frac {4 \, b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d^{2}} - \frac {3 \, {\left (\frac {{\left (d x + c + 1\right )}^{3} b^{2} e^{3}}{{\left (d x + c - 1\right )}^{3}} + \frac {{\left (d x + c + 1\right )} b^{2} e^{3}}{d x + c - 1}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2}}{\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}} - \frac {2 \, {\left (\frac {6 \, {\left (d x + c + 1\right )}^{3} a b e^{3}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )} a b e^{3}}{d x + c - 1} + \frac {3 \, {\left (d x + c + 1\right )}^{3} b^{2} e^{3}}{{\left (d x + c - 1\right )}^{3}} - \frac {6 \, {\left (d x + c + 1\right )}^{2} b^{2} e^{3}}{{\left (d x + c - 1\right )}^{2}} + \frac {5 \, {\left (d x + c + 1\right )} b^{2} e^{3}}{d x + c - 1} - 2 \, b^{2} e^{3}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{\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}} - \frac {2 \, {\left (\frac {6 \, {\left (d x + c + 1\right )}^{3} a^{2} e^{3}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )} a^{2} e^{3}}{d x + c - 1} + \frac {6 \, {\left (d x + c + 1\right )}^{3} a b e^{3}}{{\left (d x + c - 1\right )}^{3}} - \frac {12 \, {\left (d x + c + 1\right )}^{2} a b e^{3}}{{\left (d x + c - 1\right )}^{2}} + \frac {10 \, {\left (d x + c + 1\right )} a b e^{3}}{d x + c - 1} - 4 \, a b e^{3} + \frac {{\left (d x + c + 1\right )}^{3} b^{2} e^{3}}{{\left (d x + c - 1\right )}^{3}} - \frac {2 \, {\left (d x + c + 1\right )}^{2} b^{2} e^{3}}{{\left (d x + c - 1\right )}^{2}} + \frac {{\left (d x + c + 1\right )} b^{2} e^{3}}{d x + c - 1}\right )}}{\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 )} {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.12, size = 1730, normalized size = 10.88 \begin {gather*} {\ln \left (1-d\,x-c\right )}^2\,\left (\frac {b^2\,c^3\,e^3\,x}{4}-\frac {b^2\,e^3-b^2\,c^4\,e^3}{16\,d}+\frac {b^2\,d^3\,e^3\,x^4}{16}+\frac {3\,b^2\,c^2\,d\,e^3\,x^2}{8}+\frac {b^2\,c\,d^2\,e^3\,x^3}{4}\right )+x\,\left (\frac {c\,e^3\,\left (20\,a^2\,c^2-6\,a^2+6\,a\,b\,c+b^2\right )}{2}+\frac {\left (6\,c^2-6\right )\,\left (2\,a^2\,c\,d^2\,e^3-\frac {a\,d^2\,e^3\,\left (b+10\,a\,c\right )}{2}\right )}{6\,d^2}-\frac {2\,c\,\left (\frac {2\,c\,\left (2\,a^2\,c\,d^2\,e^3-\frac {a\,d^2\,e^3\,\left (b+10\,a\,c\right )}{2}\right )}{d}+\frac {d\,e^3\,\left (60\,a^2\,c^2-6\,a^2+12\,a\,b\,c+b^2\right )}{6}-\frac {a^2\,d\,e^3\,\left (6\,c^2-6\right )}{6}\right )}{d}\right )-\ln \left (1-d\,x-c\right )\,\left (\ln \left (c+d\,x+1\right )\,\left (\frac {b^2\,c^3\,e^3\,x}{2}-\frac {\frac {b^2\,e^3}{2}-\frac {b^2\,c^4\,e^3}{2}}{4\,d}+\frac {b^2\,d^3\,e^3\,x^4}{8}+\frac {3\,b^2\,c^2\,d\,e^3\,x^2}{4}+\frac {b^2\,c\,d^2\,e^3\,x^3}{2}\right )+\frac {x^2\,\left (\frac {\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )\,\left (32\,b^2\,c\,d^4\,e^3-8\,b^2\,d^3\,e^3\,\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )+8\,b^2\,d^4\,e^3\,\left (c-1\right )\right )}{d^2}-48\,b^2\,c^2\,d^3\,e^3+8\,b^2\,d^3\,e^3\,\left (c-1\right )\,\left (c+1\right )-32\,b^2\,c\,d^3\,e^3\,\left (c-1\right )\right )}{128\,d^2}-\frac {x^2\,\left (\frac {\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )\,\left (32\,b\,d^4\,e^3\,\left (8\,a\,c-2\,a+b\,c\right )-8\,b\,d^3\,e^3\,\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )\,\left (8\,a+b\right )+8\,b\,d^4\,e^3\,\left (8\,a+b\right )\,\left (c+1\right )\right )}{d^2}-48\,b\,c\,d^3\,e^3\,\left (8\,a\,c-4\,a+b\,c\right )-32\,b\,d^3\,e^3\,\left (c+1\right )\,\left (8\,a\,c-2\,a+b\,c\right )+8\,b\,d^3\,e^3\,\left (8\,a+b\right )\,\left (c-1\right )\,\left (c+1\right )\right )}{128\,d^2}+\frac {x^3\,\left (32\,b\,d^4\,e^3\,\left (8\,a\,c-2\,a+b\,c\right )-8\,b\,d^3\,e^3\,\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )\,\left (8\,a+b\right )+8\,b\,d^4\,e^3\,\left (8\,a+b\right )\,\left (c+1\right )\right )}{192\,d^2}-\frac {x^3\,\left (32\,b^2\,c\,d^4\,e^3-8\,b^2\,d^3\,e^3\,\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )+8\,b^2\,d^4\,e^3\,\left (c-1\right )\right )}{192\,d^2}+\frac {x\,\left (\frac {\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )\,\left (\frac {\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )\,\left (32\,b\,d^4\,e^3\,\left (8\,a\,c-2\,a+b\,c\right )-8\,b\,d^3\,e^3\,\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )\,\left (8\,a+b\right )+8\,b\,d^4\,e^3\,\left (8\,a+b\right )\,\left (c+1\right )\right )}{d^2}-48\,b\,c\,d^3\,e^3\,\left (8\,a\,c-4\,a+b\,c\right )-32\,b\,d^3\,e^3\,\left (c+1\right )\,\left (8\,a\,c-2\,a+b\,c\right )+8\,b\,d^3\,e^3\,\left (8\,a+b\right )\,\left (c-1\right )\,\left (c+1\right )\right )}{d^2}-\frac {\left (c-1\right )\,\left (c+1\right )\,\left (32\,b\,d^4\,e^3\,\left (8\,a\,c-2\,a+b\,c\right )-8\,b\,d^3\,e^3\,\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )\,\left (8\,a+b\right )+8\,b\,d^4\,e^3\,\left (8\,a+b\right )\,\left (c+1\right )\right )}{d^2}+32\,b\,c^2\,d^2\,e^3\,\left (8\,a\,c-6\,a+b\,c\right )+48\,b\,c\,d^2\,e^3\,\left (c+1\right )\,\left (8\,a\,c-4\,a+b\,c\right )\right )}{64\,d^2}-\frac {x\,\left (\frac {\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )\,\left (\frac {\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )\,\left (32\,b^2\,c\,d^4\,e^3-8\,b^2\,d^3\,e^3\,\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )+8\,b^2\,d^4\,e^3\,\left (c-1\right )\right )}{d^2}-48\,b^2\,c^2\,d^3\,e^3+8\,b^2\,d^3\,e^3\,\left (c-1\right )\,\left (c+1\right )-32\,b^2\,c\,d^3\,e^3\,\left (c-1\right )\right )}{d^2}+32\,b^2\,c^3\,d^2\,e^3-\frac {\left (c-1\right )\,\left (c+1\right )\,\left (32\,b^2\,c\,d^4\,e^3-8\,b^2\,d^3\,e^3\,\left (d\,\left (c-1\right )+d\,\left (c+1\right )\right )+8\,b^2\,d^4\,e^3\,\left (c-1\right )\right )}{d^2}+48\,b^2\,c^2\,d^2\,e^3\,\left (c-1\right )\right )}{64\,d^2}-\frac {b^2\,d^3\,e^3\,x^4}{32}+\frac {b\,d^3\,e^3\,x^4\,\left (8\,a+b\right )}{32}\right )+x^2\,\left (\frac {c\,\left (2\,a^2\,c\,d^2\,e^3-\frac {a\,d^2\,e^3\,\left (b+10\,a\,c\right )}{2}\right )}{d}+\frac {d\,e^3\,\left (60\,a^2\,c^2-6\,a^2+12\,a\,b\,c+b^2\right )}{12}-\frac {a^2\,d\,e^3\,\left (6\,c^2-6\right )}{12}\right )-x^3\,\left (\frac {2\,a^2\,c\,d^2\,e^3}{3}-\frac {a\,d^2\,e^3\,\left (b+10\,a\,c\right )}{6}\right )+{\ln \left (c+d\,x+1\right )}^2\,\left (\frac {b^2\,c^3\,e^3\,x}{4}-\frac {b^2\,e^3-b^2\,c^4\,e^3}{16\,d}+\frac {b^2\,d^3\,e^3\,x^4}{16}+\frac {3\,b^2\,c^2\,d\,e^3\,x^2}{8}+\frac {b^2\,c\,d^2\,e^3\,x^3}{4}\right )-\frac {\ln \left (c+d\,x-1\right )\,\left (b^2\,c^3\,e^3+3\,b^2\,c\,e^3-4\,b^2\,e^3+3\,a\,b\,c^4\,e^3-3\,a\,b\,e^3\right )}{12\,d}+\frac {\ln \left (c+d\,x+1\right )\,\left (b^2\,c^3\,e^3+3\,b^2\,c\,e^3+4\,b^2\,e^3+3\,a\,b\,c^4\,e^3-3\,a\,b\,e^3\right )}{12\,d}+d\,\ln \left (c+d\,x+1\right )\,\left (x^2\,\left (\frac {b^2\,c\,e^3}{4}+\frac {3\,a\,b\,c^2\,e^3}{2}\right )+x^3\,\left (\frac {d\,b^2\,e^3}{12}+a\,c\,d\,b\,e^3\right )+\frac {x\,\left (b^2\,c^2\,e^3+b^2\,e^3+4\,a\,b\,c^3\,e^3\right )}{4\,d}+\frac {a\,b\,d^2\,e^3\,x^4}{4}\right )+\frac {a^2\,d^3\,e^3\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________