Optimal. Leaf size=301 \[ \frac {i c e^{-\frac {i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^2}-\frac {2^{-n-3} e^{-\frac {2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^2}-\frac {i c e^{\frac {i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^2}-\frac {2^{-n-3} e^{\frac {2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.52, antiderivative size = 301, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 12
Rule 2181
Rule 3307
Rule 3308
Rule 4406
Rule 4747
Rule 4805
Rule 6741
Rule 6742
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 = 0.27, size = 269, normalized size = 0.89 \[ -\frac {i 2^{-n-3} e^{-\frac {2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}\right )^{-n} \left (c 2^{n+2} e^{\frac {3 i a}{b}} \left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \Gamma \left (n+1,\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-c 2^{n+2} e^{\frac {i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-i \left (e^{\frac {4 i a}{b}} \left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \Gamma \left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\left (\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \arcsin \left (d x + c\right ) + a\right )}^{n} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arcsin \left (d x + c\right ) + a\right )}^{n} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.50, size = 0, normalized size = 0.00 \[ \int x \left (a +b \arcsin \left (d x +c \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arcsin \left (d x + c\right ) + a\right )}^{n} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________