Optimal. Leaf size=450 \[ -\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{8 b c (n+1) \sqrt {c x-1} \sqrt {c x+1}}-\frac {d 2^{-2 (n+3)} e^{-\frac {4 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {d 2^{-n-3} e^{-\frac {2 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d 2^{-n-3} e^{\frac {2 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {d 2^{-2 (n+3)} e^{\frac {4 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.56, antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5713, 5701, 3312, 3307, 2181} \[ -\frac {d 2^{-2 (n+3)} e^{-\frac {4 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {d 2^{-n-3} e^{-\frac {2 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d 2^{-n-3} e^{\frac {2 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {d 2^{-2 (n+3)} e^{\frac {4 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{8 b c (n+1) \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rule 3312
Rule 5701
Rule 5713
Rubi steps
\begin {align*} \int \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^n \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^n \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \sinh ^4(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {3}{8} (a+b x)^n-\frac {1}{2} (a+b x)^n \cosh (2 x)+\frac {1}{8} (a+b x)^n \cosh (4 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b c (1+n) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (4 x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b c (1+n) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-4 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{4 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{16 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b c (1+n) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4^{-3-n} d e^{-\frac {4 a}{b}} \sqrt {d-c^2 d 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 {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-3-n} d e^{-\frac {2 a}{b}} \sqrt {d-c^2 d 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 {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2^{-3-n} d e^{\frac {2 a}{b}} \sqrt {d-c^2 d 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 {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4^{-3-n} d e^{\frac {4 a}{b}} \sqrt {d-c^2 d 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 {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 2.21, size = 384, normalized size = 0.85 \[ \frac {d^2 4^{-n-3} e^{-\frac {4 a}{b}} \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-2 n} \left (3\ 2^{2 n+3} e^{\frac {4 a}{b}} \left (a+b \cosh ^{-1}(c x)\right ) \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n}+b \left (-2^{n+3}\right ) (n+1) e^{\frac {2 a}{b}} \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b 2^{n+3} (n+1) e^{\frac {6 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (n+1,\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-b (n+1) e^{\frac {8 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{2 n} \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b (n+1) \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^{2 n} \Gamma \left (n+1,-\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{b c (n+1) \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}, x\right ) \]
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 = 0.24, size = 0, normalized size = 0.00 \[ \int \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}\, dx \]
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 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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