Optimal. Leaf size=175 \[ \frac {3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {3 b e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac {3 b^3 e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}{8 d}-\frac {3 b^3 e \cosh ^{-1}(c+d x)}{8 d} \]
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Rubi [A] time = 0.36, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5866, 12, 5662, 5759, 5676, 90, 52} \[ \frac {3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {3 b e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac {3 b^3 e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}{8 d}-\frac {3 b^3 e \cosh ^{-1}(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 52
Rule 90
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 )^3 \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d}-\frac {(3 b e) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac {3 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d}-\frac {(3 b e) \operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{4 d}+\frac {\left (3 b^2 e\right ) \operatorname {Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{2 d}\\ &=\frac {3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {3 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d}-\frac {\left (3 b^3 e\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac {3 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{8 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {3 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d}-\frac {\left (3 b^3 e\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{8 d}\\ &=-\frac {3 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{8 d}-\frac {3 b^3 e \cosh ^{-1}(c+d x)}{8 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {3 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 244, normalized size = 1.39 \[ \frac {e \left (2 a \left (2 a^2+3 b^2\right ) (c+d x)^2-3 b \left (2 a^2+b^2\right ) \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}-3 b \left (2 a^2+b^2\right ) \log \left (\sqrt {c+d x-1} \sqrt {c+d x+1}+c+d x\right )-6 b (c+d x) \cosh ^{-1}(c+d x) \left (-2 a^2 (c+d x)+2 a b \sqrt {c+d x-1} \sqrt {c+d x+1}-b^2 (c+d x)\right )+6 b^2 \cosh ^{-1}(c+d x)^2 \left (2 a (c+d x)^2-a-b \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)\right )+2 b^3 \left (2 (c+d x)^2-1\right ) \cosh ^{-1}(c+d x)^3\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 395, normalized size = 2.26 \[ \frac {2 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c d e x + 2 \, {\left (2 \, b^{3} d^{2} e x^{2} + 4 \, b^{3} c d e x + {\left (2 \, b^{3} c^{2} - b^{3}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + 6 \, {\left (2 \, a b^{2} d^{2} e x^{2} + 4 \, a b^{2} c d e x + {\left (2 \, a b^{2} c^{2} - a b^{2}\right )} e - {\left (b^{3} d e x + b^{3} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 3 \, {\left (2 \, {\left (2 \, a^{2} b + b^{3}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{2} b + b^{3}\right )} c d e x - {\left (2 \, a^{2} b + b^{3} - 2 \, {\left (2 \, a^{2} b + b^{3}\right )} c^{2}\right )} e - 4 \, {\left (a b^{2} d e x + a b^{2} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 3 \, {\left ({\left (2 \, a^{2} b + b^{3}\right )} d e x + {\left (2 \, a^{2} b + b^{3}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{8 \, d} \]
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 \[ \int {\left (d e x + c e\right )} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 605, normalized size = 3.46 \[ \frac {3 \,\mathrm {arccosh}\left (d x +c \right ) a^{2} b \,c^{2} e}{2 d}+\frac {3 e a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} c^{2}}{2 d}-\frac {3 e \,b^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, c}{8 d}+\frac {3 d \,\mathrm {arccosh}\left (d x +c \right ) x^{2} a^{2} b e}{2}+3 e a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} x c +3 \,\mathrm {arccosh}\left (d x +c \right ) x \,a^{2} b c e -\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x \,a^{2} b e}{4}+\frac {3 d e a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} x^{2}}{2}-\frac {3 e \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x}{4}-\frac {3 e a \,b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x}{2}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, a^{2} b c e}{4 d}-\frac {3 e \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, c}{4 d}+\frac {a^{3} c^{2} e}{2 d}+\frac {d \,x^{2} a^{3} e}{2}-\frac {e \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}}{4 d}+x \,a^{3} c e -\frac {3 e a \,b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, c}{2 d}-\frac {3 e \,a^{2} 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 )}{4 d \sqrt {\left (d x +c \right )^{2}-1}}-\frac {3 b^{3} e \,\mathrm {arccosh}\left (d x +c \right )}{8 d}+\frac {e \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3} c^{2}}{2 d}+\frac {3 e \,b^{3} \mathrm {arccosh}\left (d x +c \right ) c^{2}}{4 d}+\frac {3 d e a \,b^{2} x^{2}}{4}+\frac {3 e \,b^{3} \mathrm {arccosh}\left (d x +c \right ) x c}{2}+\frac {3 e a \,b^{2} c^{2}}{4 d}-\frac {3 e \,b^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x}{8}+e \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3} x c +\frac {d e \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3} x^{2}}{2}+\frac {3 d e \,b^{3} \mathrm {arccosh}\left (d x +c \right ) x^{2}}{4}-\frac {3 e a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{4 d}+\frac {3 e a \,b^{2} x c}{2} \]
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 \[ \frac {1}{2} \, a^{3} d e x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c}{d^{3}}\right )}\right )} a^{2} b d e + a^{3} c e x + \frac {3 \, {\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} a^{2} b c e}{d} + \frac {1}{2} \, {\left (b^{3} d e x^{2} + 2 \, b^{3} c e x\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{3} + \int \frac {3 \, {\left ({\left (2 \, a b^{2} d^{4} e - b^{3} d^{4} e\right )} x^{4} + 2 \, {\left (c^{4} e - c^{2} e\right )} a b^{2} + 4 \, {\left (2 \, a b^{2} c d^{3} e - b^{3} c d^{3} e\right )} x^{3} + {\left (2 \, {\left (6 \, c^{2} d^{2} e - d^{2} e\right )} a b^{2} - {\left (5 \, c^{2} d^{2} e - d^{2} e\right )} b^{3}\right )} x^{2} + {\left (2 \, {\left (c^{3} e - c e\right )} a b^{2} + {\left (2 \, a b^{2} d^{3} e - b^{3} d^{3} e\right )} x^{3} + 3 \, {\left (2 \, a b^{2} c d^{2} e - b^{3} c d^{2} e\right )} x^{2} - 2 \, {\left (b^{3} c^{2} d e - {\left (3 \, c^{2} d e - d e\right )} a b^{2}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + 2 \, {\left (2 \, {\left (2 \, c^{3} d e - c d e\right )} a b^{2} - {\left (c^{3} d e - c d e\right )} b^{3}\right )} x\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2}}{2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, c^{2} d - d\right )} x - c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.00, size = 685, normalized size = 3.91 \[ \begin {cases} a^{3} c e x + \frac {a^{3} d e x^{2}}{2} + \frac {3 a^{2} b c^{2} e \operatorname {acosh}{\left (c + d x \right )}}{2 d} + 3 a^{2} b c e x \operatorname {acosh}{\left (c + d x \right )} - \frac {3 a^{2} b c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{4 d} + \frac {3 a^{2} b d e x^{2} \operatorname {acosh}{\left (c + d x \right )}}{2} - \frac {3 a^{2} b e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{4} - \frac {3 a^{2} b e \operatorname {acosh}{\left (c + d x \right )}}{4 d} + \frac {3 a b^{2} c^{2} e \operatorname {acosh}^{2}{\left (c + d x \right )}}{2 d} + 3 a b^{2} c e x \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {3 a b^{2} c e x}{2} - \frac {3 a b^{2} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{2 d} + \frac {3 a b^{2} d e x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{2} + \frac {3 a b^{2} d e x^{2}}{4} - \frac {3 a b^{2} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{2} - \frac {3 a b^{2} e \operatorname {acosh}^{2}{\left (c + d x \right )}}{4 d} + \frac {b^{3} c^{2} e \operatorname {acosh}^{3}{\left (c + d x \right )}}{2 d} + \frac {3 b^{3} c^{2} e \operatorname {acosh}{\left (c + d x \right )}}{4 d} + b^{3} c e x \operatorname {acosh}^{3}{\left (c + d x \right )} + \frac {3 b^{3} c e x \operatorname {acosh}{\left (c + d x \right )}}{2} - \frac {3 b^{3} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (c + d x \right )}}{4 d} - \frac {3 b^{3} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{8 d} + \frac {b^{3} d e x^{2} \operatorname {acosh}^{3}{\left (c + d x \right )}}{2} + \frac {3 b^{3} d e x^{2} \operatorname {acosh}{\left (c + d x \right )}}{4} - \frac {3 b^{3} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (c + d x \right )}}{4} - \frac {3 b^{3} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{8} - \frac {b^{3} e \operatorname {acosh}^{3}{\left (c + d x \right )}}{4 d} - \frac {3 b^{3} e \operatorname {acosh}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {acosh}{\relax (c )}\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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