Optimal. Leaf size=259 \[ \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{n+1}}{2 b c (n+1) \sqrt {1-c^2 x^2}}-\frac {i 2^{-n-3} e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i 2^{-n-3} e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.29, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4663, 4661, 3312, 3307, 2181} \[ -\frac {i 2^{-n-3} e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i 2^{-n-3} e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{n+1}}{2 b c (n+1) \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rule 3312
Rule 4661
Rule 4663
Rubi steps
\begin {align*} \int \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx &=\frac {\sqrt {d-c^2 d x^2} \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-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 \cos ^2(x) \, dx,x,\sin ^{-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 \cos (2 x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-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 \cos (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c \sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-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 i x} (a+b x)^n \, dx,x,\sin ^{-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 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{4 c \sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{2 b c (1+n) \sqrt {1-c^2 x^2}}-\frac {i 2^{-3-n} e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i 2^{-3-n} e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.83, size = 182, normalized size = 0.70 \[ \frac {d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {4 a+4 b \sin ^{-1}(c x)}{b n+b}-i 2^{-n} e^{-\frac {2 i a}{b}} \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+i 2^{-n} e^{\frac {2 i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{8 c \sqrt {d \left (1-c^2 x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}, x\right ) \]
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 = 0.26, size = 0, normalized size = 0.00 \[ \int \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{n}\, dx \]
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 \[ \int \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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 \[ \int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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