\(\int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx\) [347]

Optimal result

Integrand size = 21, antiderivative size = 78 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {2^{\frac {5}{2}+m} \cos ^5(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-\frac {3}{2}-m,\frac {7}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {5}{2}-m} (a+a \sin (c+d x))^m}{5 d} \] Output:

-1/5*2^(5/2+m)*cos(d*x+c)^5*hypergeom([5/2, -3/2-m],[7/2],1/2-1/2*sin(d*x+ 
c))*(1+sin(d*x+c))^(-5/2-m)*(a+a*sin(d*x+c))^m/d
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {2^{\frac {5}{2}+m} \cos ^5(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-\frac {3}{2}-m,\frac {7}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {5}{2}-m} (a (1+\sin (c+d x)))^m}{5 d} \] Input:

Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^m,x]
 

Output:

-1/5*(2^(5/2 + m)*Cos[c + d*x]^5*Hypergeometric2F1[5/2, -3/2 - m, 7/2, (1 
- Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-5/2 - m)*(a*(1 + Sin[c + d*x]))^m) 
/d
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3168, 80, 79}

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.

\(\displaystyle \int \cos ^4(c+d x) (a \sin (c+d x)+a)^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^4 (a \sin (c+d x)+a)^mdx\)

\(\Big \downarrow \) 3168

\(\displaystyle \frac {a^2 \cos ^5(c+d x) \int (a-a \sin (c+d x))^{3/2} (\sin (c+d x) a+a)^{m+\frac {3}{2}}d\sin (c+d x)}{d (a-a \sin (c+d x))^{5/2} (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {a^3 2^{m+\frac {3}{2}} \cos ^5(c+d x) (\sin (c+d x)+1)^{-m-\frac {1}{2}} (a \sin (c+d x)+a)^{m-2} \int \left (\frac {1}{2} \sin (c+d x)+\frac {1}{2}\right )^{m+\frac {3}{2}} (a-a \sin (c+d x))^{3/2}d\sin (c+d x)}{d (a-a \sin (c+d x))^{5/2}}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {a^2 2^{m+\frac {5}{2}} \cos ^5(c+d x) (\sin (c+d x)+1)^{-m-\frac {1}{2}} (a \sin (c+d x)+a)^{m-2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-m-\frac {3}{2},\frac {7}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{5 d}\)

Input:

Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^m,x]
 

Output:

-1/5*(2^(5/2 + m)*a^2*Cos[c + d*x]^5*Hypergeometric2F1[5/2, -3/2 - m, 7/2, 
 (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/2 - m)*(a + a*Sin[c + d*x])^ 
(-2 + m))/d
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 3042

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

rule 3168

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.), x_Symbol] :> Simp[a^2*((g*Cos[e + f*x])^(p + 1)/(f*g*(a + b*Sin 
[e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2)))   Subst[Int[(a + 
b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; Fre 
eQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m]
 

Maple [F]

\[\int \cos \left (d x +c \right )^{4} \left (a +a \sin \left (d x +c \right )\right )^{m}d x\]

Input:

int(cos(d*x+c)^4*(a+a*sin(d*x+c))^m,x)
 

Output:

int(cos(d*x+c)^4*(a+a*sin(d*x+c))^m,x)
 

Fricas [F]

\[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )^{4} \,d x } \] Input:

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^m,x, algorithm="fricas")
 

Output:

integral((a*sin(d*x + c) + a)^m*cos(d*x + c)^4, x)
 

Sympy [F]

\[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \cos ^{4}{\left (c + d x \right )}\, dx \] Input:

integrate(cos(d*x+c)**4*(a+a*sin(d*x+c))**m,x)
 

Output:

Integral((a*(sin(c + d*x) + 1))**m*cos(c + d*x)**4, x)
 

Maxima [F]

\[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )^{4} \,d x } \] Input:

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^m,x, algorithm="maxima")
 

Output:

integrate((a*sin(d*x + c) + a)^m*cos(d*x + c)^4, x)
 

Giac [F]

\[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )^{4} \,d x } \] Input:

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^m,x, algorithm="giac")
 

Output:

integrate((a*sin(d*x + c) + a)^m*cos(d*x + c)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m \,d x \] Input:

int(cos(c + d*x)^4*(a + a*sin(c + d*x))^m,x)
 

Output:

int(cos(c + d*x)^4*(a + a*sin(c + d*x))^m, x)
 

Reduce [F]

\[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\int \left (\sin \left (d x +c \right ) a +a \right )^{m} \cos \left (d x +c \right )^{4}d x \] Input:

int(cos(d*x+c)^4*(a+a*sin(d*x+c))^m,x)
 

Output:

int((sin(c + d*x)*a + a)**m*cos(c + d*x)**4,x)