Optimal. Leaf size=27 \[ \frac {\text {Int}\left (\frac {1}{(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2},x\right )}{e} \]
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Rubi [A] time = 0.06, 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 = {} \[ \int \frac {1}{(c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{e x \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d e}\\ \end {align*}
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Mathematica [A] time = 7.77, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} d e x + a^{2} c e + {\left (b^{2} d e x + b^{2} c e\right )} \operatorname {arcosh}\left (d x + c\right )^{2} + 2 \, {\left (a b d e x + a b c e\right )} \operatorname {arcosh}\left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d e x +c e \right ) \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}\, dx \]
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 {d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, c^{2} d - d\right )} x - c}{a b d^{4} e x^{3} + 3 \, a b c d^{3} e x^{2} + {\left (3 \, c^{2} d^{2} e - d^{2} e\right )} a b x + {\left (c^{3} d e - c d e\right )} a b + {\left (a b d^{3} e x^{2} + 2 \, a b c d^{2} e x + a b c^{2} d e\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (b^{2} d^{4} e x^{3} + 3 \, b^{2} c d^{3} e x^{2} + {\left (3 \, c^{2} d^{2} e - d^{2} e\right )} b^{2} x + {\left (c^{3} d e - c d e\right )} b^{2} + {\left (b^{2} d^{3} e x^{2} + 2 \, b^{2} c d^{2} e x + b^{2} c^{2} d e\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )} + \int \frac {2 \, {\left (d x + c + 1\right )} {\left (d x + c\right )} {\left (d x + c - 1\right )} + {\left (2 \, d^{2} x^{2} + 4 \, c d x + 2 \, c^{2} - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}}{a b d^{6} e x^{6} + 6 \, a b c d^{5} e x^{5} + {\left (15 \, c^{2} d^{4} e - 2 \, d^{4} e\right )} a b x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e - 2 \, c d^{3} e\right )} a b x^{3} + {\left (15 \, c^{4} d^{2} e - 12 \, c^{2} d^{2} e + d^{2} e\right )} a b x^{2} + 2 \, {\left (3 \, c^{5} d e - 4 \, c^{3} d e + c d e\right )} a b x + {\left (a b d^{4} e x^{4} + 4 \, a b c d^{3} e x^{3} + 6 \, a b c^{2} d^{2} e x^{2} + 4 \, a b c^{3} d e x + a b c^{4} e\right )} {\left (d x + c + 1\right )} {\left (d x + c - 1\right )} + {\left (c^{6} e - 2 \, c^{4} e + c^{2} e\right )} a b + 2 \, {\left (a b d^{5} e x^{5} + 5 \, a b c d^{4} e x^{4} + {\left (10 \, c^{2} d^{3} e - d^{3} e\right )} a b x^{3} + {\left (10 \, c^{3} d^{2} e - 3 \, c d^{2} e\right )} a b x^{2} + {\left (5 \, c^{4} d e - 3 \, c^{2} d e\right )} a b x + {\left (c^{5} e - c^{3} e\right )} a b\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (b^{2} d^{6} e x^{6} + 6 \, b^{2} c d^{5} e x^{5} + {\left (15 \, c^{2} d^{4} e - 2 \, d^{4} e\right )} b^{2} x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e - 2 \, c d^{3} e\right )} b^{2} x^{3} + {\left (15 \, c^{4} d^{2} e - 12 \, c^{2} d^{2} e + d^{2} e\right )} b^{2} x^{2} + 2 \, {\left (3 \, c^{5} d e - 4 \, c^{3} d e + c d e\right )} b^{2} x + {\left (b^{2} d^{4} e x^{4} + 4 \, b^{2} c d^{3} e x^{3} + 6 \, b^{2} c^{2} d^{2} e x^{2} + 4 \, b^{2} c^{3} d e x + b^{2} c^{4} e\right )} {\left (d x + c + 1\right )} {\left (d x + c - 1\right )} + {\left (c^{6} e - 2 \, c^{4} e + c^{2} e\right )} b^{2} + 2 \, {\left (b^{2} d^{5} e x^{5} + 5 \, b^{2} c d^{4} e x^{4} + {\left (10 \, c^{2} d^{3} e - d^{3} e\right )} b^{2} x^{3} + {\left (10 \, c^{3} d^{2} e - 3 \, c d^{2} e\right )} b^{2} x^{2} + {\left (5 \, c^{4} d e - 3 \, c^{2} d e\right )} b^{2} x + {\left (c^{5} e - c^{3} e\right )} b^{2}\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\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 \[ \int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2} \,d x \]
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 \[ \frac {\int \frac {1}{a^{2} c + a^{2} d x + 2 a b c \operatorname {acosh}{\left (c + d x \right )} + 2 a b d x \operatorname {acosh}{\left (c + d x \right )} + b^{2} c \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{2} d x \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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