\(\int (c \sin (a+b x))^n \, dx\) [40]

Optimal result

Integrand size = 10, antiderivative size = 68 \[ \int (c \sin (a+b x))^n \, dx=\frac {\cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+n}}{b c (1+n) \sqrt {\cos ^2(a+b x)}} \]

[Out]

cos(b*x+a)*hypergeom([1/2, 1/2+1/2*n],[3/2+1/2*n],sin(b*x+a)^2)*(c*sin(b*x+a))^(1+n)/b/c/(1+n)/(cos(b*x+a)^2)^
(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2722} \[ \int (c \sin (a+b x))^n \, dx=\frac {\cos (a+b x) (c \sin (a+b x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(a+b x)\right )}{b c (n+1) \sqrt {\cos ^2(a+b x)}} \]

[In]

Int[(c*Sin[a + b*x])^n,x]

[Out]

(Cos[a + b*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + n))/(b*c*(1 +
n)*Sqrt[Cos[a + b*x]^2])

Rule 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \frac {\cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+n}}{b c (1+n) \sqrt {\cos ^2(a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93 \[ \int (c \sin (a+b x))^n \, dx=\frac {\sqrt {\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^n \tan (a+b x)}{b (1+n)} \]

[In]

Integrate[(c*Sin[a + b*x])^n,x]

[Out]

(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^n*Tan[a +
b*x])/(b*(1 + n))

Maple [F]

\[\int \left (c \sin \left (b x +a \right )\right )^{n}d x\]

[In]

int((c*sin(b*x+a))^n,x)

[Out]

int((c*sin(b*x+a))^n,x)

Fricas [F]

\[ \int (c \sin (a+b x))^n \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{n} \,d x } \]

[In]

integrate((c*sin(b*x+a))^n,x, algorithm="fricas")

[Out]

integral((c*sin(b*x + a))^n, x)

Sympy [F]

\[ \int (c \sin (a+b x))^n \, dx=\int \left (c \sin {\left (a + b x \right )}\right )^{n}\, dx \]

[In]

integrate((c*sin(b*x+a))**n,x)

[Out]

Integral((c*sin(a + b*x))**n, x)

Maxima [F]

\[ \int (c \sin (a+b x))^n \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{n} \,d x } \]

[In]

integrate((c*sin(b*x+a))^n,x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^n, x)

Giac [F]

\[ \int (c \sin (a+b x))^n \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{n} \,d x } \]

[In]

integrate((c*sin(b*x+a))^n,x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a))^n, x)

Mupad [F(-1)]

Timed out. \[ \int (c \sin (a+b x))^n \, dx=\int {\left (c\,\sin \left (a+b\,x\right )\right )}^n \,d x \]

[In]

int((c*sin(a + b*x))^n,x)

[Out]

int((c*sin(a + b*x))^n, x)