\(\int (d \sin (e+f x))^m (a+b \sin (e+f x))^3 \, dx\) [213]

Optimal result

Integrand size = 23, antiderivative size = 270 \[ \int (d \sin (e+f x))^m (a+b \sin (e+f x))^3 \, dx=-\frac {a b^2 (7+2 m) \cos (e+f x) (d \sin (e+f x))^{1+m}}{d f (2+m) (3+m)}+\frac {a \left (3 b^2 (1+m)+a^2 (2+m)\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+m}}{d f (1+m) (2+m) \sqrt {\cos ^2(e+f x)}}+\frac {b \left (b^2 (2+m)+3 a^2 (3+m)\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+m}}{d^2 f (2+m) (3+m) \sqrt {\cos ^2(e+f x)}}-\frac {b^2 \cos (e+f x) (d \sin (e+f x))^{1+m} (a+b \sin (e+f x))}{d f (3+m)} \] Output:

-a*b^2*(7+2*m)*cos(f*x+e)*(d*sin(f*x+e))^(1+m)/d/f/(2+m)/(3+m)+a*(3*b^2*(1 
+m)+a^2*(2+m))*cos(f*x+e)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],sin(f*x+e 
)^2)*(d*sin(f*x+e))^(1+m)/d/f/(1+m)/(2+m)/(cos(f*x+e)^2)^(1/2)+b*(b^2*(2+m 
)+3*a^2*(3+m))*cos(f*x+e)*hypergeom([1/2, 1+1/2*m],[2+1/2*m],sin(f*x+e)^2) 
*(d*sin(f*x+e))^(2+m)/d^2/f/(2+m)/(3+m)/(cos(f*x+e)^2)^(1/2)-b^2*cos(f*x+e 
)*(d*sin(f*x+e))^(1+m)*(a+b*sin(f*x+e))/d/f/(3+m)
 

Mathematica [A] (verified)

Time = 1.33 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.81 \[ \int (d \sin (e+f x))^m (a+b \sin (e+f x))^3 \, dx=\frac {(d \sin (e+f x))^m \left (-\frac {a b^2 (7+2 m) \sin (2 (e+f x))}{2 (2+m)}-\frac {1}{2} b^2 (a+b \sin (e+f x)) \sin (2 (e+f x))+\frac {a (3+m) \left (3 b^2 (1+m)+a^2 (2+m)\right ) \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(e+f x)\right ) \tan (e+f x)}{(1+m) (2+m)}+\frac {b \left (b^2 (2+m)+3 a^2 (3+m)\right ) \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\sin ^2(e+f x)\right ) \sin (e+f x) \tan (e+f x)}{2+m}\right )}{f (3+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {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.

\(\displaystyle \int (a+b \sin (e+f x))^3 (d \sin (e+f x))^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+b \sin (e+f x))^3 (d \sin (e+f x))^mdx\)

\(\Big \downarrow \) 3272

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3502

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3227

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3122

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

Input:

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

Output:

-((b^2*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + m)*(a + b*Sin[e + f*x]))/(d*f*(3 
 + m))) + (-((a*b^2*(7 + 2*m)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + m))/(f*(2 
 + m))) + ((a*d*(3 + m)*(3*b^2*(1 + m) + a^2*(2 + m))*Cos[e + f*x]*Hyperge 
ometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 
+ m))/(f*(1 + m)*Sqrt[Cos[e + f*x]^2]) + (b*(b^2*(2 + m) + 3*a^2*(3 + m))* 
Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sin[e + f*x]^2]* 
(d*Sin[e + f*x])^(2 + m))/(f*Sqrt[Cos[e + f*x]^2]))/(d*(2 + m)))/(d*(3 + m 
))
 

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 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]

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

Input:

int((d*sin(f*x+e))^m*(a+b*sin(f*x+e))^3,x)
 

Output:

int((d*sin(f*x+e))^m*(a+b*sin(f*x+e))^3,x)
 

Fricas [F]

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

integrate((d*sin(f*x+e))^m*(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))^m, x)
 

Sympy [F(-1)]

Timed out. \[ \int (d \sin (e+f x))^m (a+b \sin (e+f x))^3 \, dx=\text {Timed out} \] Input:

integrate((d*sin(f*x+e))**m*(a+b*sin(f*x+e))**3,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((b*sin(f*x + e) + a)^3*(d*sin(f*x + e))^m, x)
 

Giac [F]

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

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

Output:

integrate((b*sin(f*x + e) + a)^3*(d*sin(f*x + e))^m, x)
 

Mupad [F(-1)]

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

int((d*sin(e + f*x))^m*(a + b*sin(e + f*x))^3,x)
 

Output:

int((d*sin(e + f*x))^m*(a + b*sin(e + f*x))^3, x)
 

Reduce [F]

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

int((d*sin(f*x+e))^m*(a+b*sin(f*x+e))^3,x)
 

Output:

d**m*(int(sin(e + f*x)**m,x)*a**3 + int(sin(e + f*x)**m*sin(e + f*x)**3,x) 
*b**3 + 3*int(sin(e + f*x)**m*sin(e + f*x)**2,x)*a*b**2 + 3*int(sin(e + f* 
x)**m*sin(e + f*x),x)*a**2*b)