Optimal. Leaf size=118 \[ \frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac{b c \log (x) \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.367616, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {5798, 5738, 29, 5676} \[ \frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac{b c \log (x) \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5738
Rule 29
Rule 5676
Rubi steps
\begin{align*} \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx &=\frac{\sqrt{d-c^2 d x^2} \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c \sqrt{d-c^2 d x^2} \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.432655, size = 137, normalized size = 1.16 \[ -\frac{a \sqrt{d-c^2 d x^2}}{x}+a c \sqrt{d} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+\frac{1}{2} b c \sqrt{d-c^2 d x^2} \left (\frac{2 \log (c x)+\cosh ^{-1}(c x)^2}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{2 \cosh ^{-1}(c x)}{c x}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.241, size = 286, normalized size = 2.4 \begin{align*} -{\frac{a}{dx} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-a{c}^{2}x\sqrt{-{c}^{2}d{x}^{2}+d}-{a{c}^{2}d\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+{\frac{b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}c}{2}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{b{\rm arccosh} \left (cx\right )c\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{b{\rm arccosh} \left (cx\right )x{c}^{2}}{ \left ( cx+1 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b{\rm arccosh} \left (cx\right )}{ \left ( cx+1 \right ) \left ( cx-1 \right ) x}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{bc\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ){\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{2}}, x\right ) \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{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}{x^{2}}\, 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{\sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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