Optimal. Leaf size=155 \[ -\frac {2 a b x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d}-\frac {2 b^2 (1-c x) (c x+1)}{c^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.34, antiderivative size = 163, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5798, 5718, 5654, 74} \[ -\frac {2 a b x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 (1-c x) (c x+1)}{c^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 74
Rule 5654
Rule 5718
Rule 5798
Rubi steps
\begin {align*} \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{c \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b x \sqrt {-1+c x} \sqrt {1+c x}}{c \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \cosh ^{-1}(c x) \, dx}{c \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b x \sqrt {-1+c x} \sqrt {1+c x}}{c \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b x \sqrt {-1+c x} \sqrt {1+c x}}{c \sqrt {d-c^2 d x^2}}-\frac {2 b^2 (1-c x) (1+c x)}{c^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 149, normalized size = 0.96 \[ \frac {\sqrt {d-c^2 d x^2} \left (a^2 \left (1-c^2 x^2\right )+2 b \cosh ^{-1}(c x) \left (-a c^2 x^2+a+b c x \sqrt {c x-1} \sqrt {c x+1}\right )+2 a b c x \sqrt {c x-1} \sqrt {c x+1}-2 b^2 \left (c^2 x^2-1\right )+b^2 \left (1-c^2 x^2\right ) \cosh ^{-1}(c x)^2\right )}{c^2 d (c x-1) (c x+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 218, normalized size = 1.41 \[ \frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} a b c x - {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 2 \, {\left (\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b^{2} c x - {\left (a b c^{2} x^{2} - a b\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left ({\left (a^{2} + 2 \, b^{2}\right )} c^{2} x^{2} - a^{2} - 2 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d x^{2} - c^{2} d} \]
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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.31, size = 314, normalized size = 2.03 \[ -\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (\mathrm {arccosh}\left (c x \right )^{2}-2 \,\mathrm {arccosh}\left (c x \right )+2\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (\mathrm {arccosh}\left (c x \right )^{2}+2 \,\mathrm {arccosh}\left (c x \right )+2\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (-1+\mathrm {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (1+\mathrm {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 145, normalized size = 0.94 \[ 2 \, b^{2} {\left (\frac {\sqrt {-d} x \operatorname {arcosh}\left (c x\right )}{c d} - \frac {\sqrt {c^{2} x^{2} - 1} \sqrt {-d}}{c^{2} d}\right )} + \frac {2 \, a b \sqrt {-d} x}{c d} - \frac {\sqrt {-c^{2} d x^{2} + d} b^{2} \operatorname {arcosh}\left (c x\right )^{2}}{c^{2} d} - \frac {2 \, \sqrt {-c^{2} d x^{2} + d} a b \operatorname {arcosh}\left (c x\right )}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} + d} a^{2}}{c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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