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 = 0.210074, antiderivative size = 80, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, 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 5798
Rule 5718
Rule 8
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*}
Mathematica [A] time = 0.158515, 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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Maple [B] time = 0.15, size = 158, normalized size = 2.2 \begin{align*} -{\frac{a}{{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}+b \left ( -{\frac{-1+{\rm arccosh} \left (cx\right )}{2\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( \sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ) }-{\frac{1+{\rm arccosh} \left (cx\right )}{2\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17786, size = 85, normalized size = 1.18 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08748, size = 236, normalized size = 3.28 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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