Optimal. Leaf size=110 \[ \frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac {b^2 e (c+d x)^2}{4 d} \]
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Rubi [A] time = 0.25, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac {b^2 e (c+d x)^2}{4 d} \]
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 )^2 \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac {(b e) \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 {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 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac {(b e) \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 (b^2 e\right ) \operatorname {Subst}(\int x \, dx,x,c+d x)}{2 d}\\ &=\frac {b^2 e (c+d x)^2}{4 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 )}{2 d}-\frac {e \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 )^2}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 167, normalized size = 1.52 \[ \frac {e \left ((c+d x) \left (2 a^2 (c+d x)-2 a b \sqrt {c+d x-1} \sqrt {c+d x+1}+b^2 (c+d x)\right )-2 a b \log \left (\sqrt {c+d x-1} \sqrt {c+d x+1}+c+d x\right )-2 b (c+d x) \cosh ^{-1}(c+d x) \left (b \sqrt {c+d x-1} \sqrt {c+d x+1}-2 a (c+d x)\right )+b^2 \left (2 c^2+4 c d x+2 d^2 x^2-1\right ) \cosh ^{-1}(c+d x)^2\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 233, normalized size = 2.12 \[ \frac {{\left (2 \, a^{2} + b^{2}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} c d e x + {\left (2 \, b^{2} d^{2} e x^{2} + 4 \, b^{2} c d e x + {\left (2 \, b^{2} c^{2} - b^{2}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 2 \, {\left (2 \, a b d^{2} e x^{2} + 4 \, a b c d e x + {\left (2 \, a b c^{2} - a b\right )} e - {\left (b^{2} d e x + b^{2} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (a b d e x + a b c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{4 \, d} \]
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 \[ \int {\left (d e x + c e\right )} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 334, normalized size = 3.04 \[ \frac {a^{2} e \,x^{2} d}{2}+x \,a^{2} c e +\frac {a^{2} c^{2} e}{2 d}+\frac {d e \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} x^{2}}{2}+e \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} x c +\frac {e \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} c^{2}}{2 d}-\frac {e \,b^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \mathrm {arccosh}\left (d x +c \right ) x}{2}-\frac {e \,b^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \mathrm {arccosh}\left (d x +c \right ) c}{2 d}-\frac {e \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{4 d}+\frac {b^{2} d e \,x^{2}}{4}+\frac {e \,b^{2} x c}{2}+\frac {e \,b^{2} c^{2}}{4 d}+d \,\mathrm {arccosh}\left (d x +c \right ) x^{2} a b e +2 \,\mathrm {arccosh}\left (d x +c \right ) x a b c e +\frac {\mathrm {arccosh}\left (d x +c \right ) a b \,c^{2} e}{d}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x a b e}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, a b c e}{2 d}-\frac {e a b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )}{2 d \sqrt {\left (d x +c \right )^{2}-1}} \]
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 \[ \frac {1}{2} \, a^{2} d e x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c}{d^{3}}\right )}\right )} a b d e + a^{2} c e x + \frac {2 \, {\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} a b c e}{d} + \frac {1}{2} \, {\left (b^{2} d e x^{2} + 2 \, b^{2} c e x\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2} - \int \frac {{\left (b^{2} d^{4} e x^{4} + 4 \, b^{2} c d^{3} e x^{3} + {\left (5 \, c^{2} d^{2} e - d^{2} e\right )} b^{2} x^{2} + 2 \, {\left (c^{3} d e - c d e\right )} b^{2} x + {\left (b^{2} d^{3} e x^{3} + 3 \, b^{2} c d^{2} e x^{2} + 2 \, b^{2} c^{2} d e x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, c^{2} d - d\right )} x - c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.82, size = 335, normalized size = 3.05 \[ \begin {cases} a^{2} c e x + \frac {a^{2} d e x^{2}}{2} + \frac {a b c^{2} e \operatorname {acosh}{\left (c + d x \right )}}{d} + 2 a b c e x \operatorname {acosh}{\left (c + d x \right )} - \frac {a b c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{2 d} + a b d e x^{2} \operatorname {acosh}{\left (c + d x \right )} - \frac {a b e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{2} - \frac {a b e \operatorname {acosh}{\left (c + d x \right )}}{2 d} + \frac {b^{2} c^{2} e \operatorname {acosh}^{2}{\left (c + d x \right )}}{2 d} + b^{2} c e x \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {b^{2} c e x}{2} - \frac {b^{2} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{2 d} + \frac {b^{2} d e x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{2} + \frac {b^{2} d e x^{2}}{4} - \frac {b^{2} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{2} - \frac {b^{2} e \operatorname {acosh}^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {acosh}{\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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