Optimal. Leaf size=64 \[ -\frac {2 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}+2 b^2 x \]
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Rubi [A] time = 0.12, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5864, 5654, 5718, 8} \[ -\frac {2 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}+2 b^2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 5654
Rule 5718
Rule 5864
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}(x)\right )}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}+\frac {\left (2 b^2\right ) \operatorname {Subst}(\int 1 \, dx,x,c+d x)}{d}\\ &=2 b^2 x-\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 105, normalized size = 1.64 \[ \frac {a^2 (c+d x)-2 a b \sqrt {c+d x-1} \sqrt {c+d x+1}-2 b \cosh ^{-1}(c+d x) \left (b \sqrt {c+d x-1} \sqrt {c+d x+1}-a (c+d x)\right )+2 b^2 (c+d x)+b^2 (c+d x) \cosh ^{-1}(c+d x)^2}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 141, normalized size = 2.20 \[ \frac {{\left (a^{2} + 2 \, b^{2}\right )} d x + {\left (b^{2} d x + b^{2} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} a b + 2 \, {\left (a b d x + a b c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} b^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )}{d} \]
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 \[ \int {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 100, normalized size = 1.56 \[ \frac {a^{2} \left (d x +c \right )+b^{2} \left (\left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )^{2}-2 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 d x +2 c \right )+2 a b \left (\left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d} \]
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 \[ {\left (x \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2} - \int \frac {2 \, {\left (d^{3} x^{3} + 2 \, c d^{2} x^{2} + {\left (d^{2} x^{2} + c d x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (c^{2} d - d\right )} x\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, c^{2} d - d\right )} x - c}\,{d x}\right )} b^{2} + a^{2} x + \frac {2 \, {\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} a b}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 143, normalized size = 2.23 \[ \begin {cases} a^{2} x + \frac {2 a b c \operatorname {acosh}{\left (c + d x \right )}}{d} + 2 a b x \operatorname {acosh}{\left (c + d x \right )} - \frac {2 a b \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{d} + \frac {b^{2} c \operatorname {acosh}^{2}{\left (c + d x \right )}}{d} + b^{2} x \operatorname {acosh}^{2}{\left (c + d x \right )} + 2 b^{2} x - \frac {2 b^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {acosh}{\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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