Optimal. Leaf size=92 \[ \frac {f m \sqrt {c x-1} \text {Int}\left (\frac {(f x)^{m-1}}{a+b \cosh ^{-1}(c x)},x\right )}{b c \sqrt {1-c x}}-\frac {\sqrt {c x-1} (f x)^m}{b c \sqrt {1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.52, 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 = {} \[ \int \frac {(f x)^m}{\sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(f x)^m}{\sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f x)^m}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {(f x)^m \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (f m \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f x)^{-1+m}}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 0, normalized size = 0.00 \[ \int \frac {(f x)^m}{\sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1} \left (f x\right )^{m}}{a^{2} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} - a^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \operatorname {arcosh}\left (c x\right )}, x\right ) \]
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 \[ \int \frac {\left (f x\right )^{m}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x \right )^{m}}{\sqrt {-c^{2} x^{2}+1}\, \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]
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 \[ -\frac {{\left (c^{2} f^{m} x^{2} - f^{m}\right )} \sqrt {c x + 1} \sqrt {c x - 1} x^{m} + {\left (c^{3} f^{m} x^{3} - c f^{m} x\right )} x^{m}}{{\left ({\left (c x + 1\right )} \sqrt {c x - 1} b^{2} c^{2} x + {\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )} \sqrt {c x - 1} a b c^{2} x + {\left (a b c^{3} x^{2} - a b c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}} + \int \frac {{\left (c^{3} f^{m} m x^{3} - c f^{m} {\left (m - 1\right )} x\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} x^{m} + {\left (2 \, c^{4} f^{m} m x^{4} - 3 \, c^{2} f^{m} m x^{2} + f^{m} m\right )} \sqrt {c x + 1} \sqrt {c x - 1} x^{m} + {\left (c^{5} f^{m} m x^{5} - c^{3} f^{m} {\left (2 \, m + 1\right )} x^{3} + c f^{m} {\left (m + 1\right )} x\right )} x^{m}}{{\left ({\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} b^{2} c^{3} x^{3} + 2 \, {\left (b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2}\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{5} - 2 \, b^{2} c^{3} x^{3} + b^{2} c x\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} a b c^{3} x^{3} + 2 \, {\left (a b c^{4} x^{4} - a b c^{2} x^{2}\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (a b c^{5} x^{5} - 2 \, a b c^{3} x^{3} + a b c x\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}\,{d x} \]
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 \[ \int \frac {{\left (f\,x\right )}^m}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \]
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 \[ \int \frac {\left (f x\right )^{m}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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