Optimal. Leaf size=57 \[ \frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.19, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5713, 5676} \[ \frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 5676
Rule 5713
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {d-c^2 d x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 57, normalized size = 1.00 \[ \frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 221, normalized size = 3.88 \[ \frac {{\left (\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} a\right )} \cosh \left (n \log \left (b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + a\right )\right ) + {\left (\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} a\right )} \sinh \left (n \log \left (b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + a\right )\right )}{b c d n + b c d - {\left (b c^{3} d n + b c^{3} d\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 54, normalized size = 0.95 \[ \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{1+n} \sqrt {c x -1}\, \sqrt {c x +1}}{b \left (1+n \right ) c \sqrt {-\left (c x -1\right ) \left (c x +1\right ) d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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