Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\sqrt {d+e x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.05, 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 = {} \[ \int \frac {\sqrt {d+e x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\int \frac {\sqrt {d+e x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 28.23, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d+e x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d}}{b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x^{2} + d}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \,x^{2}+d}}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]
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 \[ -\frac {{\left (c^{3} x^{3} + {\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {c x - 1} - c x\right )} \sqrt {e x^{2} + d}}{a b c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} a b c^{2} x - a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} + \int \frac {{\left (2 \, c^{5} e x^{6} + {\left (c^{5} d - 4 \, c^{3} e\right )} x^{4} + {\left (2 \, c^{3} e x^{4} + c^{3} d x^{2} + c d\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} - 2 \, {\left (c^{3} d - c e\right )} x^{2} + {\left (4 \, c^{4} e x^{5} + 2 \, {\left (c^{4} d - 2 \, c^{2} e\right )} x^{3} - {\left (c^{2} d - e\right )} x\right )} \sqrt {c x + 1} \sqrt {c x - 1} + c d\right )} \sqrt {e x^{2} + d}}{a b c^{5} e x^{6} + {\left (c^{5} d - 2 \, c^{3} e\right )} a b x^{4} - {\left (2 \, c^{3} d - c e\right )} a b x^{2} + a b c d + {\left (a b c^{3} e x^{4} + a b c^{3} d x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 2 \, {\left (a b c^{4} e x^{5} - a b c^{2} d x + {\left (c^{4} d - c^{2} e\right )} a b x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{5} e x^{6} + {\left (c^{5} d - 2 \, c^{3} e\right )} b^{2} x^{4} - {\left (2 \, c^{3} d - c e\right )} b^{2} x^{2} + b^{2} c d + {\left (b^{2} c^{3} e x^{4} + b^{2} c^{3} d x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 2 \, {\left (b^{2} c^{4} e x^{5} - b^{2} c^{2} d x + {\left (c^{4} d - c^{2} e\right )} b^{2} x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {e\,x^2+d}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {\sqrt {d + e x^{2}}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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