\(\int (c (d \sin (e+f x))^p)^n (3+b \sin (e+f x))^2 \, dx\) [833]

Optimal result

Integrand size = 27, antiderivative size = 228 \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^2 \, dx=-\frac {b^2 \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p)}+\frac {\left (b^2 (1+n p)+9 (2+n p)\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (1+n p) (2+n p) \sqrt {\cos ^2(e+f x)}}+\frac {6 b \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (2+n p),\frac {1}{2} (4+n p),\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) \sqrt {\cos ^2(e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2905, 2868, 2722, 3093} \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^2 \, dx=\frac {\left (\frac {a^2}{n p+1}+\frac {b^2}{n p+2}\right ) \sin (e+f x) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+1),\frac {1}{2} (n p+3),\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f \sqrt {\cos ^2(e+f x)}}+\frac {2 a b \sin ^2(e+f x) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+2),\frac {1}{2} (n p+4),\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) \sqrt {\cos ^2(e+f x)}}-\frac {b^2 \sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2)} \]

[In]

Int[(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x])^2,x]

[Out]

-((b^2*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p))) + ((a^2/(1 + n*p) + b^2/(2 + n*p))*C
os[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^
p)^n)/(f*Sqrt[Cos[e + f*x]^2]) + (2*a*b*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e +
f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*Sqrt[Cos[e + f*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 2868

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Dist[2*c*(d/b)
, Int[(b*Sin[e + f*x])^(m + 1), x], x] + Int[(b*Sin[e + f*x])^m*(c^2 + d^2*Sin[e + f*x]^2), x] /; FreeQ[{b, c,
d, e, f, m}, x]

Rule 2905

Int[((c_.)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(p_))^(n_)*((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol]
:> Dist[c^IntPart[n]*((c*(d*Sin[e + f*x])^p)^FracPart[n]/(d*Sin[e + f*x])^(p*FracPart[n])), Int[(a + b*Sin[e
+ f*x])^m*(d*Sin[e + f*x])^(n*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[n]

Rule 3093

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos
[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[(A*(m + 2) + C*(m + 1))/(m + 2), Int[(b*Sin[e +
f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{n p} (a+b \sin (e+f x))^2 \, dx \\ & = \left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{n p} \left (a^2+b^2 \sin ^2(e+f x)\right ) \, dx+\frac {\left (2 a b (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{1+n p} \, dx}{d} \\ & = -\frac {b^2 \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p)}+\frac {2 a b \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (2+n p),\frac {1}{2} (4+n p),\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) \sqrt {\cos ^2(e+f x)}}+\left (\left (a^2+\frac {b^2 (1+n p)}{2+n p}\right ) (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{n p} \, dx \\ & = -\frac {b^2 \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p)}+\frac {\left (a^2+\frac {b^2 (1+n p)}{2+n p}\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (1+n p) \sqrt {\cos ^2(e+f x)}}+\frac {2 a b \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (2+n p),\frac {1}{2} (4+n p),\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) \sqrt {\cos ^2(e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.86 \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^2 \, dx=\frac {\sqrt {\cos ^2(e+f x)} \left (c (d \sin (e+f x))^p\right )^n \left (9 \left (6+5 n p+n^2 p^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right )+b (1+n p) \sin (e+f x) \left (6 (3+n p) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+\frac {n p}{2},2+\frac {n p}{2},\sin ^2(e+f x)\right )+b (2+n p) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (3+n p),\frac {1}{2} (5+n p),\sin ^2(e+f x)\right ) \sin (e+f x)\right )\right ) \tan (e+f x)}{f (1+n p) (2+n p) (3+n p)} \]

[In]

Integrate[(c*(d*Sin[e + f*x])^p)^n*(3 + b*Sin[e + f*x])^2,x]

[Out]

(Sqrt[Cos[e + f*x]^2]*(c*(d*Sin[e + f*x])^p)^n*(9*(6 + 5*n*p + n^2*p^2)*Hypergeometric2F1[1/2, (1 + n*p)/2, (3
+ n*p)/2, Sin[e + f*x]^2] + b*(1 + n*p)*Sin[e + f*x]*(6*(3 + n*p)*Hypergeometric2F1[1/2, 1 + (n*p)/2, 2 + (n*
p)/2, Sin[e + f*x]^2] + b*(2 + n*p)*Hypergeometric2F1[1/2, (3 + n*p)/2, (5 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f
*x]))*Tan[e + f*x])/(f*(1 + n*p)*(2 + n*p)*(3 + n*p))

Maple [F]

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

[In]

int((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^2,x)

[Out]

int((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^2,x)

Fricas [F]

\[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^2 \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n} \,d x } \]

[In]

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

[Out]

integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*((d*sin(f*x + e))^p*c)^n, x)

Sympy [F]

\[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^2 \, dx=\int \left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n} \left (a + b \sin {\left (e + f x \right )}\right )^{2}\, dx \]

[In]

integrate((c*(d*sin(f*x+e))**p)**n*(a+b*sin(f*x+e))**2,x)

[Out]

Integral((c*(d*sin(e + f*x))**p)**n*(a + b*sin(e + f*x))**2, x)

Maxima [F]

\[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^2 \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n} \,d x } \]

[In]

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

[Out]

integrate((b*sin(f*x + e) + a)^2*((d*sin(f*x + e))^p*c)^n, x)

Giac [F]

\[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^2 \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n} \,d x } \]

[In]

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

[Out]

integrate((b*sin(f*x + e) + a)^2*((d*sin(f*x + e))^p*c)^n, x)

Mupad [F(-1)]

Timed out. \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+b \sin (e+f x))^2 \, dx=\int {\left (c\,{\left (d\,\sin \left (e+f\,x\right )\right )}^p\right )}^n\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2 \,d x \]

[In]

int((c*(d*sin(e + f*x))^p)^n*(a + b*sin(e + f*x))^2,x)

[Out]

int((c*(d*sin(e + f*x))^p)^n*(a + b*sin(e + f*x))^2, x)