Optimal. Leaf size=117 \[ \text {Int}\left (\frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )},x\right )-\frac {\sqrt {c x-1} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b \sqrt {1-c x}}+\frac {\sqrt {c x-1} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b \sqrt {1-c x}} \]
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Rubi [A] time = 1.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \cosh ^{-1}(c x)\right )} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {\sqrt {-1+c x} \sqrt {1+c x}}{x \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (-\frac {1}{x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {c^2 x}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 1.25, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1}}{b x \operatorname {arcosh}\left (c x\right ) + a x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2}+1}}{x \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {1-c^2\,x^2}}{x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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