\(\int (c \sin ^2(a+b x))^p \, dx\) [26]

Optimal result

Integrand size = 12, antiderivative size = 77 \[ \int \left (c \sin ^2(a+b x)\right )^p \, dx=\frac {\cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+2 p),\frac {1}{2} (3+2 p),\sin ^2(a+b x)\right ) \sin (a+b x) \left (c \sin ^2(a+b x)\right )^p}{b (1+2 p) \sqrt {\cos ^2(a+b x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3286, 2722} \[ \int \left (c \sin ^2(a+b x)\right )^p \, dx=\frac {\sin (a+b x) \cos (a+b x) \left (c \sin ^2(a+b x)\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (2 p+1),\frac {1}{2} (2 p+3),\sin ^2(a+b x)\right )}{b (2 p+1) \sqrt {\cos ^2(a+b x)}} \]

[In]

Int[(c*Sin[a + b*x]^2)^p,x]

[Out]

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

Rule 3286

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \left (c \sin ^2(a+b x)\right )^p \, dx=\frac {\sqrt {\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}+p,\frac {3}{2}+p,\sin ^2(a+b x)\right ) \left (c \sin ^2(a+b x)\right )^p \tan (a+b x)}{b+2 b p} \]

[In]

Integrate[(c*Sin[a + b*x]^2)^p,x]

[Out]

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

Maple [F]

\[\int {\left (c \left (\sin ^{2}\left (b x +a \right )\right )\right )}^{p}d x\]

[In]

int((c*sin(b*x+a)^2)^p,x)

[Out]

int((c*sin(b*x+a)^2)^p,x)

Fricas [F]

\[ \int \left (c \sin ^2(a+b x)\right )^p \, dx=\int { \left (c \sin \left (b x + a\right )^{2}\right )^{p} \,d x } \]

[In]

integrate((c*sin(b*x+a)^2)^p,x, algorithm="fricas")

[Out]

integral((-c*cos(b*x + a)^2 + c)^p, x)

Sympy [F]

\[ \int \left (c \sin ^2(a+b x)\right )^p \, dx=\int \left (c \sin ^{2}{\left (a + b x \right )}\right )^{p}\, dx \]

[In]

integrate((c*sin(b*x+a)**2)**p,x)

[Out]

Integral((c*sin(a + b*x)**2)**p, x)

Maxima [F]

\[ \int \left (c \sin ^2(a+b x)\right )^p \, dx=\int { \left (c \sin \left (b x + a\right )^{2}\right )^{p} \,d x } \]

[In]

integrate((c*sin(b*x+a)^2)^p,x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a)^2)^p, x)

Giac [F]

\[ \int \left (c \sin ^2(a+b x)\right )^p \, dx=\int { \left (c \sin \left (b x + a\right )^{2}\right )^{p} \,d x } \]

[In]

integrate((c*sin(b*x+a)^2)^p,x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a)^2)^p, x)

Mupad [F(-1)]

Timed out. \[ \int \left (c \sin ^2(a+b x)\right )^p \, dx=\int {\left (c\,{\sin \left (a+b\,x\right )}^2\right )}^p \,d x \]

[In]

int((c*sin(a + b*x)^2)^p,x)

[Out]

int((c*sin(a + b*x)^2)^p, x)