Optimal. Leaf size=301 \[ \frac{i c e^{-\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^2}-\frac{2^{-n-3} e^{-\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^2}-\frac{i c e^{\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^2}-\frac{2^{-n-3} e^{\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^2} \]
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Rubi [A] time = 0.517349, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {4805, 4747, 6741, 12, 6742, 3307, 2181, 4406, 3308} \[ \frac{i c e^{-\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^2}-\frac{2^{-n-3} e^{-\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^2}-\frac{i c e^{\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^2}-\frac{2^{-n-3} e^{\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^2} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 4747
Rule 6741
Rule 12
Rule 6742
Rule 3307
Rule 2181
Rule 4406
Rule 3308
Rubi steps
\begin{align*} \int x \left (a+b \sin ^{-1}(c+d x)\right )^n \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{c}{d}+\frac{x}{d}\right ) \left (a+b \sin ^{-1}(x)\right )^n \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \cos (x) \left (-\frac{c}{d}+\frac{\sin (x)}{d}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^n \cos (x) (-c+\sin (x))}{d} \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \cos (x) (-c+\sin (x)) \, dx,x,\sin ^{-1}(c+d x)\right )}{d^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (-c (a+b x)^n \cos (x)+(a+b x)^n \cos (x) \sin (x)\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d^2}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \cos (x) \sin (x) \, dx,x,\sin ^{-1}(c+d x)\right )}{d^2}-\frac{c \operatorname{Subst}\left (\int (a+b x)^n \cos (x) \, dx,x,\sin ^{-1}(c+d x)\right )}{d^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{2} (a+b x)^n \sin (2 x) \, dx,x,\sin ^{-1}(c+d x)\right )}{d^2}-\frac{c \operatorname{Subst}\left (\int e^{-i x} (a+b x)^n \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d^2}-\frac{c \operatorname{Subst}\left (\int e^{i x} (a+b x)^n \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d^2}\\ &=\frac{i c 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^2}-\frac{i c 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^2}+\frac{\operatorname{Subst}\left (\int (a+b x)^n \sin (2 x) \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d^2}\\ &=\frac{i c 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^2}-\frac{i c 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^2}+\frac{i \operatorname{Subst}\left (\int e^{-2 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d^2}-\frac{i \operatorname{Subst}\left (\int e^{2 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d^2}\\ &=\frac{i c 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^2}-\frac{i c 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^2}-\frac{2^{-3-n} e^{-\frac{2 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{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^2}-\frac{2^{-3-n} e^{\frac{2 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{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.220187, size = 269, normalized size = 0.89 \[ -\frac{i 2^{-n-3} e^{-\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}\right )^{-n} \left (c 2^{n+2} e^{\frac{3 i a}{b}} \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-c 2^{n+2} e^{\frac{i a}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-i \left (e^{\frac{4 i a}{b}} \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \text{Gamma}\left (n+1,\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \text{Gamma}\left (n+1,-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{d^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.266, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\arcsin \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x + c\right ) + a\right )}^{n} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \arcsin \left (d x + c\right ) + a\right )}^{n} x, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \operatorname{asin}{\left (c + d x \right )}\right )^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x + c\right ) + a\right )}^{n} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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