Optimal. Leaf size=209 \[ -\frac {3 b^3 e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac {3 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}-\frac {b e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 d}+\frac {3 b^4 e (c+d x)^2}{4 d} \]
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Rubi [A] time = 0.56, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5866, 12, 5662, 5759, 5676, 30} \[ -\frac {3 b^3 e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac {3 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}-\frac {b e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 d}+\frac {3 b^4 e (c+d x)^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 5662
Rule 5676
Rule 5759
Rule 5866
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \cosh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d}-\frac {(2 b e) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}+\frac {\left (3 b^2 e\right ) \operatorname {Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d}-\frac {\left (3 b^3 e\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \cosh ^{-1}(x)\right )}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d}-\frac {\left (3 b^3 e\right ) \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{2 d}+\frac {\left (3 b^4 e\right ) \operatorname {Subst}(\int x \, dx,x,c+d x)}{2 d}\\ &=\frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}-\frac {3 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 360, normalized size = 1.72 \[ \frac {e \left (-2 a b \left (2 a^2+3 b^2\right ) \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}-2 a b \left (2 a^2+3 b^2\right ) \log \left (\sqrt {c+d x-1} \sqrt {c+d x+1}+c+d x\right )+3 b^2 \cosh ^{-1}(c+d x)^2 \left (4 a^2 (c+d x)^2-2 a^2-4 a b \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}+2 b^2 (c+d x)^2-b^2\right )+\left (2 a^4+6 a^2 b^2+3 b^4\right ) (c+d x)^2-2 b (c+d x) \cosh ^{-1}(c+d x) \left (-4 a^3 (c+d x)+6 a^2 b \sqrt {c+d x-1} \sqrt {c+d x+1}-6 a b^2 (c+d x)+3 b^3 \sqrt {c+d x-1} \sqrt {c+d x+1}\right )+4 b^3 \cosh ^{-1}(c+d x)^3 \left (2 a (c+d x)^2-a-b \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)\right )+b^4 \left (2 (c+d x)^2-1\right ) \cosh ^{-1}(c+d x)^4\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 579, normalized size = 2.77 \[ \frac {{\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d e x + {\left (2 \, b^{4} d^{2} e x^{2} + 4 \, b^{4} c d e x + {\left (2 \, b^{4} c^{2} - b^{4}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{4} + 4 \, {\left (2 \, a b^{3} d^{2} e x^{2} + 4 \, a b^{3} c d e x + {\left (2 \, a b^{3} c^{2} - a b^{3}\right )} e - {\left (b^{4} d e x + b^{4} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + 3 \, {\left (2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c d e x - {\left (2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c^{2}\right )} e - 4 \, {\left (a b^{3} d e x + a b^{3} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 2 \, {\left (2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c d e x - {\left (2 \, a^{3} b + 3 \, a b^{3} - 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c^{2}\right )} e - 3 \, {\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} d e x + {\left (2 \, a^{2} b^{2} + b^{4}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d e x + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{4 \, d} \]
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 \[ \int {\left (d e x + c e\right )} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 933, normalized size = 4.46 \[ -\frac {3 e a \,b^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x}{2}+4 \,\mathrm {arccosh}\left (d x +c \right ) x \,a^{3} b c e +3 d e a \,b^{3} \mathrm {arccosh}\left (d x +c \right ) x^{2}+6 e \,a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} x c +2 d e a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3} x^{2}+4 e a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3} x c +6 e a \,b^{3} \mathrm {arccosh}\left (d x +c \right ) x c -e \,b^{4} \mathrm {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x -\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x \,a^{3} b e +\frac {2 \,\mathrm {arccosh}\left (d x +c \right ) a^{3} b \,c^{2} e}{d}+\frac {3 e \,a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} c^{2}}{d}+\frac {a^{4} c^{2} e}{2 d}-\frac {e \,b^{4} \mathrm {arccosh}\left (d x +c \right )^{4}}{4 d}-\frac {3 e \,b^{4} \mathrm {arccosh}\left (d x +c \right )^{2}}{4 d}+\frac {3 d e \,b^{4} x^{2}}{4}+\frac {d \,x^{2} a^{4} e}{2}+\frac {3 e \,b^{4} x c}{2}+x \,a^{4} c e +\frac {3 e \,b^{4} c^{2}}{4 d}+2 d \,\mathrm {arccosh}\left (d x +c \right ) x^{2} a^{3} b e +3 d e \,a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} x^{2}+\frac {2 e a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3} c^{2}}{d}+\frac {3 e a \,b^{3} \mathrm {arccosh}\left (d x +c \right ) c^{2}}{d}-\frac {3 e \,b^{4} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x}{2}-\frac {e \,a^{3} b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )}{d \sqrt {\left (d x +c \right )^{2}-1}}-\frac {3 e \,a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, c}{d}-\frac {3 e a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, c}{d}-3 e a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x -\frac {3 e \,b^{4} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, c}{2 d}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, a^{3} b c e}{d}-\frac {3 e a \,b^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, c}{2 d}-\frac {e \,b^{4} \mathrm {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, c}{d}-3 e \,a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x +\frac {3 d e \,a^{2} b^{2} x^{2}}{2}+3 e \,b^{4} \mathrm {arccosh}\left (d x +c \right )^{2} x c +e \,b^{4} \mathrm {arccosh}\left (d x +c \right )^{4} x c +\frac {d e \,b^{4} \mathrm {arccosh}\left (d x +c \right )^{4} x^{2}}{2}+\frac {3 d e \,b^{4} \mathrm {arccosh}\left (d x +c \right )^{2} x^{2}}{2}-\frac {3 e \,a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{2 d}-\frac {e a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}}{d}-\frac {3 e a \,b^{3} \mathrm {arccosh}\left (d x +c \right )}{2 d}+\frac {e \,b^{4} \mathrm {arccosh}\left (d x +c \right )^{4} c^{2}}{2 d}+\frac {3 e \,b^{4} \mathrm {arccosh}\left (d x +c \right )^{2} c^{2}}{2 d}+3 e \,a^{2} b^{2} x c +\frac {3 e \,a^{2} b^{2} c^{2}}{2 d} \]
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 \[ \frac {1}{2} \, a^{4} d e x^{2} + {\left (2 \, x^{2} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c}{d^{3}}\right )}\right )} a^{3} b d e + a^{4} c e x + \frac {4 \, {\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} a^{3} b c e}{d} + \frac {1}{2} \, {\left (b^{4} d e x^{2} + 2 \, b^{4} c e x\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{4} + \int \frac {2 \, {\left ({\left (2 \, {\left (c^{4} e - c^{2} e\right )} a b^{3} + {\left (2 \, a b^{3} d^{4} e - b^{4} d^{4} e\right )} x^{4} + 4 \, {\left (2 \, a b^{3} c d^{3} e - b^{4} c d^{3} e\right )} x^{3} + {\left (2 \, {\left (6 \, c^{2} d^{2} e - d^{2} e\right )} a b^{3} - {\left (5 \, c^{2} d^{2} e - d^{2} e\right )} b^{4}\right )} x^{2} + {\left (2 \, {\left (c^{3} e - c e\right )} a b^{3} + {\left (2 \, a b^{3} d^{3} e - b^{4} d^{3} e\right )} x^{3} + 3 \, {\left (2 \, a b^{3} c d^{2} e - b^{4} c d^{2} e\right )} x^{2} - 2 \, {\left (b^{4} c^{2} d e - {\left (3 \, c^{2} d e - d e\right )} a b^{3}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + 2 \, {\left (2 \, {\left (2 \, c^{3} d e - c d e\right )} a b^{3} - {\left (c^{3} d e - c d e\right )} b^{4}\right )} x\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{3} + 3 \, {\left (a^{2} b^{2} d^{4} e x^{4} + 4 \, a^{2} b^{2} c d^{3} e x^{3} + {\left (6 \, c^{2} d^{2} e - d^{2} e\right )} a^{2} b^{2} x^{2} + 2 \, {\left (2 \, c^{3} d e - c d e\right )} a^{2} b^{2} x + {\left (c^{4} e - c^{2} e\right )} a^{2} b^{2} + {\left (a^{2} b^{2} d^{3} e x^{3} + 3 \, a^{2} b^{2} c d^{2} e x^{2} + {\left (3 \, c^{2} d e - d e\right )} a^{2} b^{2} x + {\left (c^{3} e - c e\right )} a^{2} b^{2}\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2}\right )}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, c^{2} d - d\right )} x - c}\,{d x} \]
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 \[ \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.52, size = 1027, normalized size = 4.91 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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