Optimal. Leaf size=211 \[ \frac {\sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{2 b c^3 (1+n) \sqrt {1-c x}}+\frac {2^{-3-n} e^{-\frac {2 a}{b}} \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c x}}-\frac {2^{-3-n} e^{\frac {2 a}{b}} \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c x}} \]
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Rubi [A]
time = 0.20, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5952, 3393,
3388, 2212} \begin {gather*} \frac {2^{-n-3} e^{-\frac {2 a}{b}} \sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c x}}-\frac {2^{-n-3} e^{\frac {2 a}{b}} \sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c x}}+\frac {\sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{2 b c^3 (n+1) \sqrt {1-c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 3393
Rule 5952
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {1-c^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x)^n \cosh ^2(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \left (\frac {1}{2} (a+b x)^n+\frac {1}{2} (a+b x)^n \cosh (2 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{2 b c^3 (1+n) \sqrt {1-c^2 x^2}}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x)^n \cosh (2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{2 b c^3 (1+n) \sqrt {1-c^2 x^2}}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{2 b c^3 (1+n) \sqrt {1-c^2 x^2}}+\frac {2^{-3-n} e^{-\frac {2 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {2^{-3-n} e^{\frac {2 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.56, size = 212, normalized size = 1.00 \begin {gather*} \frac {2^{-3-n} e^{-\frac {2 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (2^{2+n} e^{\frac {2 a}{b}} \left (a+b \cosh ^{-1}(c x)\right ) \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n+b (1+n) \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (1+n,-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-b e^{\frac {4 a}{b}} (1+n) \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{b c^3 (1+n) \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}}{\sqrt {-c^{2} x^{2}+1}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{\sqrt {1-c^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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