Optimal. Leaf size=92 \[ -\frac {c d \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b (1+n) \sqrt {d-c^2 d x^2}}+d \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt {d-c^2 d x^2}},x\right ) \]
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Rubi [A]
time = 0.26, 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 = {}
\begin {gather*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x^2} \, dx &=\frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n}{x^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {d-c^2 d x^2} \int \left (\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b (1+n) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n} \sqrt {-c^{2} d \,x^{2}+d}}{x^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,\sqrt {d-c^2\,d\,x^2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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