Optimal. Leaf size=259 \[ \frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^{1+n}}{2 b c (1+n) \sqrt {1-c^2 x^2}}-\frac {i 2^{-3-n} e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,-\frac {2 i (a+b \text {ArcSin}(c x))}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i 2^{-3-n} e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,\frac {2 i (a+b \text {ArcSin}(c x))}{b}\right )}{c \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4753, 3393,
3388, 2212} \begin {gather*} -\frac {i 2^{-n-3} e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {2 i (a+b \text {ArcSin}(c x))}{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} (a+b \text {ArcSin}(c x))^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {2 i (a+b \text {ArcSin}(c x))}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^{n+1}}{2 b c (n+1) \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 3393
Rule 4753
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} \text {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} \text {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} \text {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} \text {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} \text {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 = 0.52, size = 182, normalized size = 0.70 \begin {gather*} \frac {d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {4 a+4 b \text {ArcSin}(c x)}{b+b n}-i 2^{-n} e^{-\frac {2 i a}{b}} \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,-\frac {2 i (a+b \text {ArcSin}(c x))}{b}\right )+i 2^{-n} e^{\frac {2 i a}{b}} \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,\frac {2 i (a+b \text {ArcSin}(c x))}{b}\right )\right )}{8 c \sqrt {d \left (1-c^2 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.19, 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 \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 \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{n}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,\sqrt {d-c^2\,d\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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