Optimal. Leaf size=200 \[ \frac {f \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt {1-c x}}+\frac {e^{-\frac {a}{b}} g \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x}}-\frac {e^{a/b} g \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x}} \]
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Rubi [A]
time = 0.41, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5982, 5980,
3398, 3388, 2212} \begin {gather*} \frac {g e^{-\frac {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 {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x}}-\frac {g e^{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 {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x}}+\frac {f \sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt {1-c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 3398
Rule 5980
Rule 5982
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {1-c x} \sqrt {1+c x}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x) \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 {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x)^n (c f+g \cosh (x)) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt {1-c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \left (c f (a+b x)^n+g (a+b x)^n \cosh (x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt {1-c x} \sqrt {1+c x}}\\ &=\frac {f \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt {1-c x} \sqrt {1+c x}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt {1-c x} \sqrt {1+c x}}\\ &=\frac {f \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt {1-c x} \sqrt {1+c x}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {1-c x} \sqrt {1+c x}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int e^x (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {1-c x} \sqrt {1+c x}}\\ &=\frac {f \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt {1-c x} \sqrt {1+c x}}+\frac {e^{-\frac {a}{b}} g \sqrt {1-c^2 x^2} \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 {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x} \sqrt {1+c x}}-\frac {e^{a/b} g \sqrt {1-c^2 x^2} \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 {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 204, normalized size = 1.02 \begin {gather*} \frac {e^{-\frac {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 c e^{a/b} f \left (a+b \cosh ^{-1}(c x)\right ) \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n-b e^{\frac {2 a}{b}} g (1+n) \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {a}{b}+\cosh ^{-1}(c x)\right )+b g (1+n) \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (1+n,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )\right )}{2 b c^2 (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 \[\int \frac {\left (g x +f \right ) \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}}{\sqrt {-c x +1}\, \sqrt {c x +1}}\, dx\]
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 {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n} \left (f + g x\right )}{\sqrt {- c x + 1} \sqrt {c x + 1}}\, 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 {\left (f+g\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{\sqrt {1-c\,x}\,\sqrt {c\,x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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