Optimal. Leaf size=72 \[ -\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}-\frac {b x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.21, antiderivative size = 80, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5798, 5718, 8} \[ -\frac {(1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {b x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 5718
Rule 5798
Rubi steps
\begin {align*} \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\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 )}{\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 )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int 1 \, dx}{c \sqrt {d-c^2 d x^2}}\\ &=-\frac {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 )}{c^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 85, normalized size = 1.18 \[ \frac {\sqrt {d-c^2 d x^2} \left (-a c^2 x^2+a+\left (b-b c^2 x^2\right ) \cosh ^{-1}(c x)+b c x \sqrt {c x-1} \sqrt {c x+1}\right )}{c^2 d (c x-1) (c x+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 117, normalized size = 1.62 \[ \frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b c x - {\left (b c^{2} x^{2} - b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (a c^{2} x^{2} - a\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d x^{2} - c^{2} d} \]
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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.25, size = 158, normalized size = 2.19 \[ -\frac {a \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (-1+\mathrm {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (1+\mathrm {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 63, normalized size = 0.88 \[ \frac {b \sqrt {-d} x}{c d} - \frac {\sqrt {-c^{2} d x^{2} + d} b \operatorname {arcosh}\left (c x\right )}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} + d} a}{c^{2} d} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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