Optimal. Leaf size=186 \[ \frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}-\frac {b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{8 d}-\frac {3 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )}{16 d}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{32 d}+\frac {b^2 e^3 (c+d x)^4}{32 d}+\frac {3 b^2 e^3 (c+d x)^2}{32 d} \]
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Rubi [A] time = 0.44, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5866, 12, 5662, 5759, 5676, 30} \[ \frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}-\frac {b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{8 d}-\frac {3 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )}{16 d}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{32 d}+\frac {b^2 e^3 (c+d x)^4}{32 d}+\frac {3 b^2 e^3 (c+d x)^2}{32 d} \]
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)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int e^3 x^3 \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \operatorname {Subst}\left (\int x^3 \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}-\frac {\left (b e^3\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (a+b \cosh ^{-1}(x)\right )}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{8 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}-\frac {\left (3 b e^3\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 )}{8 d}+\frac {\left (b^2 e^3\right ) \operatorname {Subst}\left (\int x^3 \, dx,x,c+d x\right )}{8 d}\\ &=\frac {b^2 e^3 (c+d x)^4}{32 d}-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{16 d}-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{8 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}-\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{16 d}+\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}(\int x \, dx,x,c+d x)}{16 d}\\ &=\frac {3 b^2 e^3 (c+d x)^2}{32 d}+\frac {b^2 e^3 (c+d x)^4}{32 d}-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{16 d}-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{8 d}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 212, normalized size = 1.14 \[ \frac {e^3 \left (\left (8 a^2+b^2\right ) (c+d x)^4+2 a b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (-2 (c+d x)^2-3\right ) (c+d x)-6 a b \log \left (\sqrt {c+d x-1} \sqrt {c+d x+1}+c+d x\right )+2 b (c+d x) \cosh ^{-1}(c+d x) \left (8 a (c+d x)^3-2 b \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2-3 b \sqrt {c+d x-1} \sqrt {c+d x+1}\right )+3 b^2 (c+d x)^2+b^2 \left (8 (c+d x)^4-3\right ) \cosh ^{-1}(c+d x)^2\right )}{32 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 481, normalized size = 2.59 \[ \frac {{\left (8 \, a^{2} + b^{2}\right )} d^{4} e^{3} x^{4} + 4 \, {\left (8 \, a^{2} + b^{2}\right )} c d^{3} e^{3} x^{3} + 3 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{2} + b^{2}\right )} d^{2} e^{3} x^{2} + 2 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{3} + 3 \, b^{2} c\right )} d e^{3} x + {\left (8 \, b^{2} d^{4} e^{3} x^{4} + 32 \, b^{2} c d^{3} e^{3} x^{3} + 48 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 32 \, b^{2} c^{3} d e^{3} x + {\left (8 \, b^{2} c^{4} - 3 \, b^{2}\right )} e^{3}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 2 \, {\left (8 \, a b d^{4} e^{3} x^{4} + 32 \, a b c d^{3} e^{3} x^{3} + 48 \, a b c^{2} d^{2} e^{3} x^{2} + 32 \, a b c^{3} d e^{3} x + {\left (8 \, a b c^{4} - 3 \, a b\right )} e^{3} - {\left (2 \, b^{2} d^{3} e^{3} x^{3} + 6 \, b^{2} c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, b^{2} c^{2} + b^{2}\right )} d e^{3} x + {\left (2 \, b^{2} c^{3} + 3 \, b^{2} c\right )} e^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (2 \, a b d^{3} e^{3} x^{3} + 6 \, a b c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, a b c^{2} + a b\right )} d e^{3} x + {\left (2 \, a b c^{3} + 3 \, a b c\right )} e^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{32 \, d} \]
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 \[ \int {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 822, normalized size = 4.42 \[ \frac {d^{3} x^{4} a^{2} e^{3}}{4}-\frac {e^{3} b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, c^{3}}{8 d}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x a b \,c^{2} e^{3}}{8}-\frac {3 e^{3} b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x \,c^{2}}{8}+2 d^{2} \mathrm {arccosh}\left (d x +c \right ) x^{3} a b c \,e^{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, a b \,c^{3} e^{3}}{8 d}+3 d \,\mathrm {arccosh}\left (d x +c \right ) x^{2} a b \,c^{2} e^{3}-\frac {d^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x^{3} a b \,e^{3}}{8}-\frac {d^{2} e^{3} b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x^{3}}{8}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, a b c \,e^{3}}{16 d}-\frac {3 e^{3} b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, c}{16 d}+\frac {e^{3} b^{2} c^{4}}{32 d}+\frac {3 e^{3} b^{2} c^{2}}{32 d}+\frac {a^{2} c^{4} e^{3}}{4 d}+\frac {d^{3} e^{3} b^{2} x^{4}}{32}-\frac {3 d \,e^{3} b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x^{2} c}{8}-\frac {3 d \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x^{2} a b c \,e^{3}}{8}-\frac {3 e^{3} a 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 )}{16 d \sqrt {\left (d x +c \right )^{2}-1}}+\frac {3 d \,e^{3} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} x^{2} c^{2}}{2}+d^{2} e^{3} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} x^{3} c +\frac {3 e^{3} b^{2} x c}{16}+\frac {e^{3} b^{2} x \,c^{3}}{8}+x \,a^{2} c^{3} e^{3}+\frac {3 d \,e^{3} b^{2} x^{2}}{32}+\frac {d^{3} e^{3} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} x^{4}}{4}+e^{3} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} x \,c^{3}+\frac {d^{2} e^{3} b^{2} x^{3} c}{8}+\frac {3 d \,e^{3} b^{2} x^{2} c^{2}}{16}+\frac {3 d \,x^{2} a^{2} c^{2} e^{3}}{2}+\frac {e^{3} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} c^{4}}{4 d}-\frac {3 e^{3} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{32 d}+d^{2} x^{3} a^{2} c \,e^{3}+\frac {d^{3} \mathrm {arccosh}\left (d x +c \right ) x^{4} a b \,e^{3}}{2}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x a b \,e^{3}}{16}-\frac {3 e^{3} b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x}{16}+2 \,\mathrm {arccosh}\left (d x +c \right ) x a b \,c^{3} e^{3}+\frac {\mathrm {arccosh}\left (d x +c \right ) a b \,c^{4} e^{3}}{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 \[ \text {result too large to display} \]
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 \[ \int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.32, size = 916, normalized size = 4.92 \[ \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 {acosh}{\left (c + d x \right )}}{2 d} + 2 a b c^{3} e^{3} x \operatorname {acosh}{\left (c + d x \right )} - \frac {a b c^{3} e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{8 d} + 3 a b c^{2} d e^{3} x^{2} \operatorname {acosh}{\left (c + d x \right )} - \frac {3 a b c^{2} e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{8} + 2 a b c d^{2} e^{3} x^{3} \operatorname {acosh}{\left (c + d x \right )} - \frac {3 a b c d e^{3} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{8} - \frac {3 a b c e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{16 d} + \frac {a b d^{3} e^{3} x^{4} \operatorname {acosh}{\left (c + d x \right )}}{2} - \frac {a b d^{2} e^{3} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{8} - \frac {3 a b e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{16} - \frac {3 a b e^{3} \operatorname {acosh}{\left (c + d x \right )}}{16 d} + \frac {b^{2} c^{4} e^{3} \operatorname {acosh}^{2}{\left (c + d x \right )}}{4 d} + b^{2} c^{3} e^{3} x \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {b^{2} c^{3} e^{3} x}{8} - \frac {b^{2} c^{3} e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{8 d} + \frac {3 b^{2} c^{2} d e^{3} x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{2} + \frac {3 b^{2} c^{2} d e^{3} x^{2}}{16} - \frac {3 b^{2} c^{2} e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{8} + b^{2} c d^{2} e^{3} x^{3} \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {b^{2} c d^{2} e^{3} x^{3}}{8} - \frac {3 b^{2} c d e^{3} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{8} + \frac {3 b^{2} c e^{3} x}{16} - \frac {3 b^{2} c e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{16 d} + \frac {b^{2} d^{3} e^{3} x^{4} \operatorname {acosh}^{2}{\left (c + d x \right )}}{4} + \frac {b^{2} d^{3} e^{3} x^{4}}{32} - \frac {b^{2} d^{2} e^{3} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{8} + \frac {3 b^{2} d e^{3} x^{2}}{32} - \frac {3 b^{2} e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{16} - \frac {3 b^{2} e^{3} \operatorname {acosh}^{2}{\left (c + d x \right )}}{32 d} & \text {for}\: d \neq 0 \\c^{3} e^{3} x \left (a + b \operatorname {acosh}{\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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