Optimal. Leaf size=147 \[ \frac {i e^{\frac {i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d}-\frac {i e^{-\frac {i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d} \]
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Rubi [A] time = 0.13, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4803, 4623, 3307, 2181} \[ \frac {i e^{\frac {i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d}-\frac {i e^{-\frac {i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rule 4623
Rule 4803
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}(c+d x)\right )^n \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^n \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int x^n \cos \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{b d}\\ &=\frac {\operatorname {Subst}\left (\int e^{-i \left (\frac {a}{b}-\frac {x}{b}\right )} x^n \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{2 b d}+\frac {\operatorname {Subst}\left (\int e^{i \left (\frac {a}{b}-\frac {x}{b}\right )} x^n \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{2 b d}\\ &=-\frac {i e^{-\frac {i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d}+\frac {i e^{\frac {i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 129, normalized size = 0.88 \[ -\frac {i e^{-\frac {i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-e^{\frac {2 i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \arcsin \left (d x + c\right ) + a\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arcsin \left (d x + c\right ) + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (a +b \arcsin \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (b \arcsin \left (d x + c\right ) + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^n \,d x \]
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 \[ \int \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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