\(\int (1+\sin (c+d x))^n \, dx\) [147]

Optimal result

Integrand size = 10, antiderivative size = 58 \[ \int (1+\sin (c+d x))^n \, dx=-\frac {2^{\frac {1}{2}+n} \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{d \sqrt {1+\sin (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, 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 = {2730} \[ \int (1+\sin (c+d x))^n \, dx=-\frac {2^{n+\frac {1}{2}} \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{d \sqrt {\sin (c+d x)+1}} \]

[In]

Int[(1 + Sin[c + d*x])^n,x]

[Out]

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

Rule 2730

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/
(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; Free
Q[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.79 \[ \int (1+\sin (c+d x))^n \, dx=\frac {2^n B_{\frac {1}{2} (1+\sin (c+d x))}\left (\frac {1}{2}+n,\frac {1}{2}\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x)}{d} \]

[In]

Integrate[(1 + Sin[c + d*x])^n,x]

[Out]

(2^n*Beta[(1 + Sin[c + d*x])/2, 1/2 + n, 1/2]*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x])/d

Maple [F]

\[\int \left (1+\sin \left (d x +c \right )\right )^{n}d x\]

[In]

int((1+sin(d*x+c))^n,x)

[Out]

int((1+sin(d*x+c))^n,x)

Fricas [F]

\[ \int (1+\sin (c+d x))^n \, dx=\int { {\left (\sin \left (d x + c\right ) + 1\right )}^{n} \,d x } \]

[In]

integrate((1+sin(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((sin(d*x + c) + 1)^n, x)

Sympy [F]

\[ \int (1+\sin (c+d x))^n \, dx=\int \left (\sin {\left (c + d x \right )} + 1\right )^{n}\, dx \]

[In]

integrate((1+sin(d*x+c))**n,x)

[Out]

Integral((sin(c + d*x) + 1)**n, x)

Maxima [F]

\[ \int (1+\sin (c+d x))^n \, dx=\int { {\left (\sin \left (d x + c\right ) + 1\right )}^{n} \,d x } \]

[In]

integrate((1+sin(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((sin(d*x + c) + 1)^n, x)

Giac [F]

\[ \int (1+\sin (c+d x))^n \, dx=\int { {\left (\sin \left (d x + c\right ) + 1\right )}^{n} \,d x } \]

[In]

integrate((1+sin(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((sin(d*x + c) + 1)^n, x)

Mupad [F(-1)]

Timed out. \[ \int (1+\sin (c+d x))^n \, dx=\int {\left (\sin \left (c+d\,x\right )+1\right )}^n \,d x \]

[In]

int((sin(c + d*x) + 1)^n,x)

[Out]

int((sin(c + d*x) + 1)^n, x)