Integrand size = 21, antiderivative size = 74 \[ \int \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=-\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {b \cos ^2(e+f x)}{a+b}\right )}{f} \] Output:
-cos(f*x+e)*(a+b-b*cos(f*x+e)^2)^p*hypergeom([1/2, -p],[3/2],b*cos(f*x+e)^ 2/(a+b))/f/((1-b*cos(f*x+e)^2/(a+b))^p)
Time = 0.42 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=-\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {b \cos ^2(e+f x)}{a+b}\right )}{f} \] Input:
Integrate[Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p,x]
Output:
-((Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/ 2, (b*Cos[e + f*x]^2)/(a + b)])/(f*(1 - (b*Cos[e + f*x]^2)/(a + b))^p))
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3665, 238, 237}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (e+f x) \left (a+b \sin (e+f x)^2\right )^pdx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle -\frac {\int \left (-b \cos ^2(e+f x)+a+b\right )^pd\cos (e+f x)}{f}\) |
\(\Big \downarrow \) 238 |
\(\displaystyle -\frac {\left (a-b \cos ^2(e+f x)+b\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p} \int \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^pd\cos (e+f x)}{f}\) |
\(\Big \downarrow \) 237 |
\(\displaystyle -\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {b \cos ^2(e+f x)}{a+b}\right )}{f}\) |
Input:
Int[Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p,x]
Output:
-((Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/ 2, (b*Cos[e + f*x]^2)/(a + b)])/(f*(1 - (b*Cos[e + f*x]^2)/(a + b))^p))
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] && !IntegerQ[2*p ] && GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2) ^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(1 + b*(x^2/a))^p, x], x] / ; FreeQ[{a, b, p}, x] && !IntegerQ[2*p] && !GtQ[a, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
\[\int \sin \left (f x +e \right ) \left (a +b \sin \left (f x +e \right )^{2}\right )^{p}d x\]
Input:
int(sin(f*x+e)*(a+b*sin(f*x+e)^2)^p,x)
Output:
int(sin(f*x+e)*(a+b*sin(f*x+e)^2)^p,x)
\[ \int \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right ) \,d x } \] Input:
integrate(sin(f*x+e)*(a+b*sin(f*x+e)^2)^p,x, algorithm="fricas")
Output:
integral((-b*cos(f*x + e)^2 + a + b)^p*sin(f*x + e), x)
Timed out. \[ \int \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\text {Timed out} \] Input:
integrate(sin(f*x+e)*(a+b*sin(f*x+e)**2)**p,x)
Output:
Timed out
\[ \int \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right ) \,d x } \] Input:
integrate(sin(f*x+e)*(a+b*sin(f*x+e)^2)^p,x, algorithm="maxima")
Output:
integrate((b*sin(f*x + e)^2 + a)^p*sin(f*x + e), x)
\[ \int \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right ) \,d x } \] Input:
integrate(sin(f*x+e)*(a+b*sin(f*x+e)^2)^p,x, algorithm="giac")
Output:
integrate((b*sin(f*x + e)^2 + a)^p*sin(f*x + e), x)
Timed out. \[ \int \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int \sin \left (e+f\,x\right )\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p \,d x \] Input:
int(sin(e + f*x)*(a + b*sin(e + f*x)^2)^p,x)
Output:
int(sin(e + f*x)*(a + b*sin(e + f*x)^2)^p, x)
\[ \int \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int \left (\sin \left (f x +e \right )^{2} b +a \right )^{p} \sin \left (f x +e \right )d x \] Input:
int(sin(f*x+e)*(a+b*sin(f*x+e)^2)^p,x)
Output:
int((sin(e + f*x)**2*b + a)**p*sin(e + f*x),x)