Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{(d+e x) \left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.03, 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 = {} \[ \int \frac {1}{(d+e x) \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\int \frac {1}{(d+e x) \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 10.17, size = 0, normalized size = 0.00 \[ \int \frac {1}{(d+e x) \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} e x + a^{2} d + {\left (b^{2} e x + b^{2} d\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b e x + a b d\right )} \operatorname {arcosh}\left (c x\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (e x + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e x +d \right ) \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]
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 {c^{3} x^{3} + {\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {c x - 1} - c x}{a b c^{3} e x^{3} + a b c^{3} d x^{2} - a b c e x - a b c d + {\left (a b c^{2} e x^{2} + a b c^{2} d x\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{3} e x^{3} + b^{2} c^{3} d x^{2} - b^{2} c e x - b^{2} c d + {\left (b^{2} c^{2} e x^{2} + b^{2} c^{2} d x\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} + \int \frac {c^{5} d x^{4} - 2 \, c^{3} d x^{2} + {\left (c^{3} d x^{2} + 2 \, c e x + c d\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + {\left (2 \, c^{4} d x^{3} + 2 \, c^{2} e x^{2} - c^{2} d x - e\right )} \sqrt {c x + 1} \sqrt {c x - 1} + c d}{a b c^{5} e^{2} x^{6} + 2 \, a b c^{5} d e x^{5} - 4 \, a b c^{3} d e x^{3} + {\left (c^{5} d^{2} - 2 \, c^{3} e^{2}\right )} a b x^{4} + 2 \, a b c d e x + a b c d^{2} - {\left (2 \, c^{3} d^{2} - c e^{2}\right )} a b x^{2} + {\left (a b c^{3} e^{2} x^{4} + 2 \, a b c^{3} d e x^{3} + a b c^{3} d^{2} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 2 \, {\left (a b c^{4} e^{2} x^{5} + 2 \, a b c^{4} d e x^{4} - 2 \, a b c^{2} d e x^{2} - a b c^{2} d^{2} x + {\left (c^{4} d^{2} - c^{2} e^{2}\right )} a b x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{5} e^{2} x^{6} + 2 \, b^{2} c^{5} d e x^{5} - 4 \, b^{2} c^{3} d e x^{3} + {\left (c^{5} d^{2} - 2 \, c^{3} e^{2}\right )} b^{2} x^{4} + 2 \, b^{2} c d e x + b^{2} c d^{2} - {\left (2 \, c^{3} d^{2} - c e^{2}\right )} b^{2} x^{2} + {\left (b^{2} c^{3} e^{2} x^{4} + 2 \, b^{2} c^{3} d e x^{3} + b^{2} c^{3} d^{2} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 2 \, {\left (b^{2} c^{4} e^{2} x^{5} + 2 \, b^{2} c^{4} d e x^{4} - 2 \, b^{2} c^{2} d e x^{2} - b^{2} c^{2} d^{2} x + {\left (c^{4} d^{2} - c^{2} e^{2}\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 = -0.05 \[ \int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {1}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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