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

Optimal result

Integrand size = 27, antiderivative size = 323 \[ \int \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))^3 \, dx=-\frac {a b^2 (7+2 n p) \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) (3+n p)}+\frac {a \left (3 b^2 (1+n p)+a^2 (2+n p)\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (1+n p) (2+n p) \sqrt {\cos ^2(e+f x)}}+\frac {b \left (b^2 (2+n p)+3 a^2 (3+n p)\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (2+n p),\frac {1}{2} (4+n p),\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) (3+n p) \sqrt {\cos ^2(e+f x)}}-\frac {b^2 \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))}{f (3+n p)} \] Output:

-a*b^2*(2*n*p+7)*cos(f*x+e)*sin(f*x+e)*(c*(d*sin(f*x+e))^p)^n/f/(n*p+2)/(n 
*p+3)+a*(3*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)+b*(b^2*(n*p+2)+3*a^2*(n*p+3))*cos(f*x+e)*hype 
rgeom([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)/(n*p+3)/(cos(f*x+e)^2)^(1/2)-b^2*cos(f*x+e)*sin(f*x+e 
)*(c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))/f/(n*p+3)
 

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.77 \[ \int \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))^3 \, dx=\frac {\left (c (d \sin (e+f x))^p\right )^n \left (-\frac {a b^2 (7+2 n p) \sin (2 (e+f x))}{4+2 n p}-\frac {1}{2} b^2 (a+b \sin (e+f x)) \sin (2 (e+f x))+\frac {a (3+n p) \left (3 b^2 (1+n p)+a^2 (2+n p)\right ) \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \tan (e+f x)}{(1+n p) (2+n p)}+\frac {b \left (b^2 (2+n p)+3 a^2 (3+n p)\right ) \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+\frac {n p}{2},2+\frac {n p}{2},\sin ^2(e+f x)\right ) \sin (e+f x) \tan (e+f x)}{2+n p}\right )}{f (3+n p)} \] Input:

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

Output:

((c*(d*Sin[e + f*x])^p)^n*(-((a*b^2*(7 + 2*n*p)*Sin[2*(e + f*x)])/(4 + 2*n 
*p)) - (b^2*(a + b*Sin[e + f*x])*Sin[2*(e + f*x)])/2 + (a*(3 + n*p)*(3*b^2 
*(1 + n*p) + a^2*(2 + n*p))*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/2, (1 
 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Tan[e + f*x])/((1 + n*p)*(2 + n*p) 
) + (b*(b^2*(2 + n*p) + 3*a^2*(3 + n*p))*Sqrt[Cos[e + f*x]^2]*Hypergeometr 
ic2F1[1/2, 1 + (n*p)/2, 2 + (n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*Tan[e + 
f*x])/(2 + n*p)))/(f*(3 + n*p))
 

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 3305, 3042, 3272, 3042, 3502, 3042, 3227, 3042, 3122}

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.

\(\displaystyle \int (a+b \sin (e+f x))^3 \left (c (d \sin (e+f x))^p\right )^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+b \sin (e+f x))^3 \left (c (d \sin (e+f x))^p\right )^ndx\)

\(\Big \downarrow \) 3305

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \int (d \sin (e+f x))^{n p} (a+b \sin (e+f x))^3dx\)

\(\Big \downarrow \) 3042

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \int (d \sin (e+f x))^{n p} (a+b \sin (e+f x))^3dx\)

\(\Big \downarrow \) 3272

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\int (d \sin (e+f x))^{n p} \left (a b^2 d (2 n p+7) \sin ^2(e+f x)+b d \left (3 (n p+3) a^2+b^2 (n p+2)\right ) \sin (e+f x)+a d \left ((n p+3) a^2+b^2 (n p+1)\right )\right )dx}{d (n p+3)}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (d \sin (e+f x))^{n p+1}}{d f (n p+3)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\int (d \sin (e+f x))^{n p} \left (a b^2 d (2 n p+7) \sin (e+f x)^2+b d \left (3 (n p+3) a^2+b^2 (n p+2)\right ) \sin (e+f x)+a d \left ((n p+3) a^2+b^2 (n p+1)\right )\right )dx}{d (n p+3)}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (d \sin (e+f x))^{n p+1}}{d f (n p+3)}\right )\)

\(\Big \downarrow \) 3502

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\frac {\int (d \sin (e+f x))^{n p} \left (a (n p+3) \left ((n p+2) a^2+3 b^2 (n p+1)\right ) d^2+b (n p+2) \left (3 (n p+3) a^2+b^2 (n p+2)\right ) \sin (e+f x) d^2\right )dx}{d (n p+2)}-\frac {a b^2 (2 n p+7) \cos (e+f x) (d \sin (e+f x))^{n p+1}}{f (n p+2)}}{d (n p+3)}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (d \sin (e+f x))^{n p+1}}{d f (n p+3)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\frac {\int (d \sin (e+f x))^{n p} \left (a (n p+3) \left ((n p+2) a^2+3 b^2 (n p+1)\right ) d^2+b (n p+2) \left (3 (n p+3) a^2+b^2 (n p+2)\right ) \sin (e+f x) d^2\right )dx}{d (n p+2)}-\frac {a b^2 (2 n p+7) \cos (e+f x) (d \sin (e+f x))^{n p+1}}{f (n p+2)}}{d (n p+3)}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (d \sin (e+f x))^{n p+1}}{d f (n p+3)}\right )\)

\(\Big \downarrow \) 3227

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\frac {a d^2 (n p+3) \left (a^2 (n p+2)+3 b^2 (n p+1)\right ) \int (d \sin (e+f x))^{n p}dx+b d (n p+2) \left (3 a^2 (n p+3)+b^2 (n p+2)\right ) \int (d \sin (e+f x))^{n p+1}dx}{d (n p+2)}-\frac {a b^2 (2 n p+7) \cos (e+f x) (d \sin (e+f x))^{n p+1}}{f (n p+2)}}{d (n p+3)}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (d \sin (e+f x))^{n p+1}}{d f (n p+3)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\frac {a d^2 (n p+3) \left (a^2 (n p+2)+3 b^2 (n p+1)\right ) \int (d \sin (e+f x))^{n p}dx+b d (n p+2) \left (3 a^2 (n p+3)+b^2 (n p+2)\right ) \int (d \sin (e+f x))^{n p+1}dx}{d (n p+2)}-\frac {a b^2 (2 n p+7) \cos (e+f x) (d \sin (e+f x))^{n p+1}}{f (n p+2)}}{d (n p+3)}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (d \sin (e+f x))^{n p+1}}{d f (n p+3)}\right )\)

\(\Big \downarrow \) 3122

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\frac {\frac {a d (n p+3) \left (a^2 (n p+2)+3 b^2 (n p+1)\right ) \cos (e+f x) (d \sin (e+f x))^{n p+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+1),\frac {1}{2} (n p+3),\sin ^2(e+f x)\right )}{f (n p+1) \sqrt {\cos ^2(e+f x)}}+\frac {b \left (3 a^2 (n p+3)+b^2 (n p+2)\right ) \cos (e+f x) (d \sin (e+f x))^{n p+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+2),\frac {1}{2} (n p+4),\sin ^2(e+f x)\right )}{f \sqrt {\cos ^2(e+f x)}}}{d (n p+2)}-\frac {a b^2 (2 n p+7) \cos (e+f x) (d \sin (e+f x))^{n p+1}}{f (n p+2)}}{d (n p+3)}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (d \sin (e+f x))^{n p+1}}{d f (n p+3)}\right )\)

Input:

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

Output:

((c*(d*Sin[e + f*x])^p)^n*(-((b^2*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n*p)* 
(a + b*Sin[e + f*x]))/(d*f*(3 + n*p))) + (-((a*b^2*(7 + 2*n*p)*Cos[e + f*x 
]*(d*Sin[e + f*x])^(1 + n*p))/(f*(2 + n*p))) + ((a*d*(3 + n*p)*(3*b^2*(1 + 
 n*p) + a^2*(2 + n*p))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 
 + n*p)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n*p))/(f*(1 + n*p)*Sqrt[C 
os[e + f*x]^2]) + (b*(b^2*(2 + n*p) + 3*a^2*(3 + n*p))*Cos[e + f*x]*Hyperg 
eometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x] 
)^(2 + n*p))/(f*Sqrt[Cos[e + f*x]^2]))/(d*(2 + n*p)))/(d*(3 + n*p))))/(d*S 
in[e + f*x])^(n*p)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3122

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]))*Hypergeometric2 
F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] 
 &&  !IntegerQ[2*n]
 

rule 3227

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

rule 3272

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* 
x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m 
 + n))   Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d 
*(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 
 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si 
n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* 
d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m 
] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] 
&& NeQ[c, 0])))
 

rule 3305

Int[((c_.)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(p_))^(n_)*((a_.) + (b_.)*sin[(e 
_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Simp[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 3502

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

Maple [F]

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

Input:

int((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^3,x)
 

Output:

int((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^3,x)
 

Fricas [F]

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

integrate((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^3,x, algorithm="fricas")
 

Output:

integral(-(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 
3*a^2*b - b^3)*sin(f*x + e))*((d*sin(f*x + e))^p*c)^n, x)
 

Sympy [F]

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

integrate((c*(d*sin(f*x+e))**p)**n*(a+b*sin(f*x+e))**3,x)
 

Output:

Integral((c*(d*sin(e + f*x))**p)**n*(a + b*sin(e + f*x))**3, x)
 

Maxima [F]

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

integrate((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^3,x, algorithm="maxima")
 

Output:

integrate((b*sin(f*x + e) + a)^3*((d*sin(f*x + e))^p*c)^n, x)
 

Giac [F]

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

integrate((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^3,x, algorithm="giac")
 

Output:

integrate((b*sin(f*x + e) + a)^3*((d*sin(f*x + e))^p*c)^n, x)
 

Mupad [F(-1)]

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

int((c*(d*sin(e + f*x))^p)^n*(a + b*sin(e + f*x))^3,x)
 

Output:

int((c*(d*sin(e + f*x))^p)^n*(a + b*sin(e + f*x))^3, x)
 

Reduce [F]

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

int((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^3,x)
                                                                                    
                                                                                    
 

Output:

d**(n*p)*c**n*(int(sin(e + f*x)**(n*p),x)*a**3 + int(sin(e + f*x)**(n*p)*s 
in(e + f*x)**3,x)*b**3 + 3*int(sin(e + f*x)**(n*p)*sin(e + f*x)**2,x)*a*b* 
*2 + 3*int(sin(e + f*x)**(n*p)*sin(e + f*x),x)*a**2*b)