Optimal. Leaf size=235 \[ \frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^{n+1}}{2 b c (n+1) \sqrt {c^2 x^2+1}}+\frac {2^{-n-3} e^{-\frac {2 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {c^2 x^2+1}}-\frac {2^{-n-3} e^{\frac {2 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.30, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5702, 5699, 3312, 3307, 2181} \[ \frac {2^{-n-3} e^{-\frac {2 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {c^2 x^2+1}}-\frac {2^{-n-3} e^{\frac {2 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^{n+1}}{2 b c (n+1) \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rule 3312
Rule 5699
Rule 5702
Rubi steps
\begin {align*} \int \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx &=\frac {\sqrt {d+c^2 d x^2} \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\sqrt {d+c^2 d x^2} \operatorname {Subst}\left (\int (a+b x)^n \cosh ^2(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt {1+c^2 x^2}}\\ &=\frac {\sqrt {d+c^2 d x^2} \operatorname {Subst}\left (\int \left (\frac {1}{2} (a+b x)^n+\frac {1}{2} (a+b x)^n \cosh (2 x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt {1+c^2 x^2}}\\ &=\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{2 b c (1+n) \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \operatorname {Subst}\left (\int (a+b x)^n \cosh (2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c \sqrt {1+c^2 x^2}}\\ &=\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{2 b c (1+n) \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \operatorname {Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{4 c \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \operatorname {Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{4 c \sqrt {1+c^2 x^2}}\\ &=\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{2 b c (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-3-n} e^{-\frac {2 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1+c^2 x^2}}-\frac {2^{-3-n} e^{\frac {2 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 160, normalized size = 0.68 \[ \frac {d \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {4 a+4 b \sinh ^{-1}(c x)}{b n+b}-2^{-n} e^{\frac {2 a}{b}} \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^{-n} \Gamma \left (n+1,\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+2^{-n} e^{-\frac {2 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{8 c \sqrt {d \left (c^2 x^2+1\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}, x\right ) \]
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 = 0.22, size = 0, normalized size = 0.00 \[ \int \left (a +b \arcsinh \left (c x \right )\right )^{n} \sqrt {c^{2} d \,x^{2}+d}\, dx \]
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 \[ \int \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,\sqrt {d\,c^2\,x^2+d} \,d x \]
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 \[ \int \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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