Integrand size = 23, antiderivative size = 124 \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac {(a+b (3+2 p)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b \sin ^2(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p}}{b f (3+2 p)} \]
-sin(f*x+e)*(a+b*sin(f*x+e)^2)^(p+1)/b/f/(3+2*p)+(a+b*(3+2*p))*hypergeom([ 1/2, -p],[3/2],-b*sin(f*x+e)^2/a)*sin(f*x+e)*(a+b*sin(f*x+e)^2)^p/b/f/(3+2 *p)/((1+b*sin(f*x+e)^2/a)^p)
Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.97 \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p} \left (-\left ((a+b (3+2 p)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b \sin ^2(e+f x)}{a}\right )\right )+\left (a+b \sin ^2(e+f x)\right ) \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^p\right )}{b f (3+2 p)} \]
-((Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p*(-((a + b*(3 + 2*p))*Hypergeometr ic2F1[1/2, -p, 3/2, -((b*Sin[e + f*x]^2)/a)]) + (a + b*Sin[e + f*x]^2)*(1 + (b*Sin[e + f*x]^2)/a)^p))/(b*f*(3 + 2*p)*(1 + (b*Sin[e + f*x]^2)/a)^p))
Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 3669, 299, 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.
\(\displaystyle \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos (e+f x)^3 \left (a+b \sin (e+f x)^2\right )^pdx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \frac {\int \left (1-\sin ^2(e+f x)\right ) \left (b \sin ^2(e+f x)+a\right )^pd\sin (e+f x)}{f}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {\left (\frac {a}{2 b p+3 b}+1\right ) \int \left (b \sin ^2(e+f x)+a\right )^pd\sin (e+f x)-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{p+1}}{b (2 p+3)}}{f}\) |
\(\Big \downarrow \) 238 |
\(\displaystyle \frac {\left (\frac {a}{2 b p+3 b}+1\right ) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^{-p} \int \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^pd\sin (e+f x)-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{p+1}}{b (2 p+3)}}{f}\) |
\(\Big \downarrow \) 237 |
\(\displaystyle \frac {\left (\frac {a}{2 b p+3 b}+1\right ) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b \sin ^2(e+f x)}{a}\right )-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{p+1}}{b (2 p+3)}}{f}\) |
(-((Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(1 + p))/(b*(3 + 2*p))) + ((1 + a/ (3*b + 2*b*p))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sin[e + f*x]^2)/a)]*Si n[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(1 + (b*Sin[e + f*x]^2)/a)^p)/f
3.4.74.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f S ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] /ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
\[\int \left (\cos ^{3}\left (f x +e \right )\right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{p}d x\]
\[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{3} \,d x } \]
Timed out. \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\text {Timed out} \]
\[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{3} \,d x } \]
\[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{3} \,d x } \]
Timed out. \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int {\cos \left (e+f\,x\right )}^3\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p \,d x \]