Optimal. Leaf size=260 \[ \frac {g e^{-\frac {a}{b}} \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}-\frac {g e^{a/b} \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {f \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}} \]
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Rubi [A] time = 0.69, antiderivative size = 248, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {5837, 5832, 3317, 3307, 2181} \[ \frac {g e^{-\frac {a}{b}} \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}-\frac {g e^{a/b} \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {f \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}} \]
Warning: Unable to verify antiderivative.
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Rule 2181
Rule 3307
Rule 3317
Rule 5832
Rule 5837
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} 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 {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ &=\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int (a+b x)^n (c f+g \cosh (x)) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ &=\frac {\sqrt {1-c^2 x^2} \operatorname {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 {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ &=\frac {f \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ &=\frac {f \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int e^x (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ &=\frac {f \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} 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 {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} 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 {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ \end {align*}
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Mathematica [A] time = 2.17, size = 219, normalized size = 0.84 \[ \frac {e^{-\frac {a}{b}} \sqrt {\frac {c x-1}{c x+1}} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (-2 c f e^{a/b} \left (a+b \cosh ^{-1}(c x)\right ) \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n-b g (n+1) \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )+b g (n+1) e^{\frac {2 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )}{2 b c^2 \text {d1} \text {d2} (n+1) (c x-1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.17, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {c d_{1} x + d_{1}} \sqrt {-c d_{2} x + d_{2}} {\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{c^{2} d_{1} d_{2} x^{2} - d_{1} d_{2}}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.17, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right ) \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}}{\sqrt {c \mathit {d1} x +\mathit {d1}}\, \sqrt {-c \mathit {d2} x +\mathit {d2}}}\, dx \]
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 \[ \int \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt {c d_{1} x + d_{1}} \sqrt {-c d_{2} x + d_{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (f+g\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{\sqrt {d_{1}+c\,d_{1}\,x}\,\sqrt {d_{2}-c\,d_{2}\,x}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n} \left (f + g x\right )}{\sqrt {d_{1} \left (c x + 1\right )} \sqrt {- d_{2} \left (c x - 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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