Optimal. Leaf size=182 \[ \frac {\sqrt {1-c x} \text {Int}\left (\frac {1}{x^2 \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b c \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c x \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.57, 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 = {} \[ \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
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 )^2} \, 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 )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x) \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c \sqrt {1-c^2 x^2}\right ) \int \frac {1}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x) \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{b^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x) \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \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 {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{b^2 \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 {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{b^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x) \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2} \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 33.14, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1}}{b^{2} x \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b x \operatorname {arcosh}\left (c x\right ) + a^{2} x}, x\right ) \]
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 = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2}+1}}{x \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]
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 \[ -\frac {{\left ({\left (c^{2} x^{2} - 1\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (c^{3} x^{3} - c x\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}{a b c^{3} x^{3} + \sqrt {c x + 1} \sqrt {c x - 1} a b c^{2} x^{2} - a b c x + {\left (b^{2} c^{3} x^{3} + \sqrt {c x + 1} \sqrt {c x - 1} b^{2} c^{2} x^{2} - b^{2} c x\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} + \int \frac {{\left ({\left (c^{3} x^{3} + 2 \, c x\right )} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} + {\left (2 \, c^{4} x^{4} + c^{2} x^{2} - 1\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (c^{5} x^{5} - c^{3} x^{3}\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}{a b c^{5} x^{6} + {\left (c x + 1\right )} {\left (c x - 1\right )} a b c^{3} x^{4} - 2 \, a b c^{3} x^{4} + a b c x^{2} + 2 \, {\left (a b c^{4} x^{5} - a b c^{2} x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{6} + {\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} c^{3} x^{4} - 2 \, b^{2} c^{3} x^{4} + b^{2} c x^{2} + 2 \, {\left (b^{2} c^{4} x^{5} - b^{2} c^{2} x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}\,{d x} \]
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 \[ \int \frac {\sqrt {1-c^2\,x^2}}{x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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