Optimal. Leaf size=262 \[ \frac {4}{3} a b^2 e^2 x-\frac {40 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{27 d}-\frac {2 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{27 d}+\frac {4 b^3 e^2 (c+d x) \cosh ^{-1}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d} \]
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Rubi [A]
time = 0.31, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5996, 12,
5883, 5939, 5915, 5879, 75, 102} \begin {gather*} \frac {2 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}+\frac {4}{3} a b^2 e^2 x-\frac {b e^2 \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {2 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}-\frac {2 b^3 e^2 \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1}}{27 d}-\frac {40 b^3 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{27 d}+\frac {4 b^3 e^2 (c+d x) \cosh ^{-1}(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 102
Rule 5879
Rule 5883
Rule 5915
Rule 5939
Rule 5996
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int e^2 x^2 \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int x^2 \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x^3 \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d}+\frac {\left (2 b^2 e^2\right ) \text {Subst}\left (\int x^2 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d}\\ &=\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}+\frac {\left (4 b^2 e^2\right ) \text {Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d}-\frac {\left (2 b^3 e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d}\\ &=\frac {4}{3} a b^2 e^2 x-\frac {2 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{27 d}+\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (2 b^3 e^2\right ) \text {Subst}\left (\int \frac {2 x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{27 d}+\frac {\left (4 b^3 e^2\right ) \text {Subst}\left (\int \cosh ^{-1}(x) \, dx,x,c+d x\right )}{3 d}\\ &=\frac {4}{3} a b^2 e^2 x-\frac {2 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{27 d}+\frac {4 b^3 e^2 (c+d x) \cosh ^{-1}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (4 b^3 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{27 d}-\frac {\left (4 b^3 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d}\\ &=\frac {4}{3} a b^2 e^2 x-\frac {40 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{27 d}-\frac {2 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{27 d}+\frac {4 b^3 e^2 (c+d x) \cosh ^{-1}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 296, normalized size = 1.13 \begin {gather*} \frac {e^2 \left (12 a b^2 (c+d x)+a \left (3 a^2+2 b^2\right ) (c+d x)^3+\frac {1}{3} b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-2 \left (9 a^2+20 b^2\right )-\left (9 a^2+2 b^2\right ) (c+d x)^2\right )-b \left (-12 b^2 (c+d x)-9 a^2 (c+d x)^3-2 b^2 (c+d x)^3+12 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}+6 a b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \cosh ^{-1}(c+d x)-3 b^2 \left (-3 a (c+d x)^3+2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}+b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \cosh ^{-1}(c+d x)^2+3 b^3 (c+d x)^3 \cosh ^{-1}(c+d x)^3\right )}{9 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. \(637\) vs.
\(2(230)=460\).
time = 37.66, size = 638, normalized size = 2.44
method | result | size |
default | \(\frac {e^{2} \left (d x +c \right )^{3} a^{3}}{3 d}+\frac {b^{3} e^{2} \left (9 \mathrm {arccosh}\left (d x +c \right )^{3} x^{3} d^{3}+27 \mathrm {arccosh}\left (d x +c \right )^{3} x^{2} c \,d^{2}-9 \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x^{2} d^{2}+27 \mathrm {arccosh}\left (d x +c \right )^{3} x \,c^{2} d -18 \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x c d +6 \,\mathrm {arccosh}\left (d x +c \right ) x^{3} d^{3}+9 \mathrm {arccosh}\left (d x +c \right )^{3} c^{3}-9 \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, c^{2}+18 \,\mathrm {arccosh}\left (d x +c \right ) x^{2} c \,d^{2}-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x^{2} d^{2}+18 \,\mathrm {arccosh}\left (d x +c \right ) x \,c^{2} d -4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x c d -18 \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c +1}\, \sqrt {d x +c -1}+6 \,\mathrm {arccosh}\left (d x +c \right ) c^{3}-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, c^{2}+36 \,\mathrm {arccosh}\left (d x +c \right ) x d +36 \,\mathrm {arccosh}\left (d x +c \right ) c -40 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{27 d}+\frac {a \,b^{2} e^{2} \left (9 \mathrm {arccosh}\left (d x +c \right )^{2} x^{3} d^{3}+27 \mathrm {arccosh}\left (d x +c \right )^{2} x^{2} c \,d^{2}-6 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x^{2} d^{2}+27 \mathrm {arccosh}\left (d x +c \right )^{2} x \,c^{2} d -12 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x c d +2 d^{3} x^{3}+9 \mathrm {arccosh}\left (d x +c \right )^{2} c^{3}-6 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, c^{2}+6 x^{2} c \,d^{2}+6 c^{2} d x -12 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 c^{3}+12 d x +12 c \right )}{9 d}+\frac {3 a^{2} b \,e^{2} \left (\frac {\left (d x +c \right )^{3} \mathrm {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(638\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1389 vs.
\(2 (222) = 444\).
time = 0.39, size = 1389, normalized size = 5.30 \begin {gather*} \frac {9 \, {\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \sinh \left (1\right )^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + 3 \, {\left ({\left (3 \, a^{3} + 2 \, a b^{2}\right )} d^{3} x^{3} + 3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c d^{2} x^{2} + 3 \, {\left (4 \, a b^{2} + {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{2}\right )} d x\right )} \cosh \left (1\right )^{2} + 9 \, {\left (3 \, {\left (a b^{2} d^{3} x^{3} + 3 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{2} d x + a b^{2} c^{3}\right )} \cosh \left (1\right )^{2} + 6 \, {\left (a b^{2} d^{3} x^{3} + 3 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{2} d x + a b^{2} c^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + 3 \, {\left (a b^{2} d^{3} x^{3} + 3 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{2} d x + a b^{2} c^{3}\right )} \sinh \left (1\right )^{2} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2} + 2 \, b^{3}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2} + 2 \, b^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2} + 2 \, b^{3}\right )} \sinh \left (1\right )^{2}\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 6 \, {\left ({\left (3 \, a^{3} + 2 \, a b^{2}\right )} d^{3} x^{3} + 3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c d^{2} x^{2} + 3 \, {\left (4 \, a b^{2} + {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{2}\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right ) + 3 \, {\left ({\left (3 \, a^{3} + 2 \, a b^{2}\right )} d^{3} x^{3} + 3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c d^{2} x^{2} + 3 \, {\left (4 \, a b^{2} + {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{2}\right )} d x\right )} \sinh \left (1\right )^{2} + 3 \, {\left ({\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} x^{2} + 12 \, b^{3} c + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3} + 3 \, {\left (4 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} d x\right )} \cosh \left (1\right )^{2} + 2 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} x^{2} + 12 \, b^{3} c + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3} + 3 \, {\left (4 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} x^{2} + 12 \, b^{3} c + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3} + 3 \, {\left (4 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} d x\right )} \sinh \left (1\right )^{2} - 6 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (a b^{2} d^{2} x^{2} + 2 \, a b^{2} c d x + a b^{2} c^{2} + 2 \, a b^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (a b^{2} d^{2} x^{2} + 2 \, a b^{2} c d x + a b^{2} c^{2} + 2 \, a b^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (a b^{2} d^{2} x^{2} + 2 \, a b^{2} c d x + a b^{2} c^{2} + 2 \, a b^{2}\right )} \sinh \left (1\right )^{2}\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{2} x^{2} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d x + 18 \, a^{2} b + 40 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{2} x^{2} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d x + 18 \, a^{2} b + 40 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{2} x^{2} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d x + 18 \, a^{2} b + 40 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} \sinh \left (1\right )^{2}\right )}}{27 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1173 vs.
\(2 (252) = 504\).
time = 0.63, size = 1173, normalized size = 4.48 \begin {gather*} \begin {cases} a^{3} c^{2} e^{2} x + a^{3} c d e^{2} x^{2} + \frac {a^{3} d^{2} e^{2} x^{3}}{3} + \frac {a^{2} b c^{3} e^{2} \operatorname {acosh}{\left (c + d x \right )}}{d} + 3 a^{2} b c^{2} e^{2} x \operatorname {acosh}{\left (c + d x \right )} - \frac {a^{2} b c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{3 d} + 3 a^{2} b c d e^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )} - \frac {2 a^{2} b c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{3} + a^{2} b d^{2} e^{2} x^{3} \operatorname {acosh}{\left (c + d x \right )} - \frac {a^{2} b d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{3} - \frac {2 a^{2} b e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{3 d} + \frac {a b^{2} c^{3} e^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{d} + 3 a b^{2} c^{2} e^{2} x \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {2 a b^{2} c^{2} e^{2} x}{3} - \frac {2 a b^{2} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{3 d} + 3 a b^{2} c d e^{2} x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {2 a b^{2} c d e^{2} x^{2}}{3} - \frac {4 a b^{2} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{3} + a b^{2} d^{2} e^{2} x^{3} \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {2 a b^{2} d^{2} e^{2} x^{3}}{9} - \frac {2 a b^{2} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{3} + \frac {4 a b^{2} e^{2} x}{3} - \frac {4 a b^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{3 d} + \frac {b^{3} c^{3} e^{2} \operatorname {acosh}^{3}{\left (c + d x \right )}}{3 d} + \frac {2 b^{3} c^{3} e^{2} \operatorname {acosh}{\left (c + d x \right )}}{9 d} + b^{3} c^{2} e^{2} x \operatorname {acosh}^{3}{\left (c + d x \right )} + \frac {2 b^{3} c^{2} e^{2} x \operatorname {acosh}{\left (c + d x \right )}}{3} - \frac {b^{3} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (c + d x \right )}}{3 d} - \frac {2 b^{3} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{27 d} + b^{3} c d e^{2} x^{2} \operatorname {acosh}^{3}{\left (c + d x \right )} + \frac {2 b^{3} c d e^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}}{3} - \frac {2 b^{3} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (c + d x \right )}}{3} - \frac {4 b^{3} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{27} + \frac {4 b^{3} c e^{2} \operatorname {acosh}{\left (c + d x \right )}}{3 d} + \frac {b^{3} d^{2} e^{2} x^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}{3} + \frac {2 b^{3} d^{2} e^{2} x^{3} \operatorname {acosh}{\left (c + d x \right )}}{9} - \frac {b^{3} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (c + d x \right )}}{3} - \frac {2 b^{3} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{27} + \frac {4 b^{3} e^{2} x \operatorname {acosh}{\left (c + d x \right )}}{3} - \frac {2 b^{3} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (c + d x \right )}}{3 d} - \frac {40 b^{3} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{27 d} & \text {for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname {acosh}{\left (c \right )}\right )^{3} & \text {otherwise} \end {cases} \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 = -0.00 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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