Integrand size = 33, antiderivative size = 120 \[ \int \frac {\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\frac {4 \sqrt {2} \operatorname {AppellF1}\left (\frac {3}{2},-\frac {3}{2},-n,\frac {5}{2},\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (1+\sin (e+f x)) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{3 a f \sqrt {1-\sin (e+f x)}} \] Output:
4/3*2^(1/2)*AppellF1(3/2,-n,-3/2,5/2,-d*(1+sin(f*x+e))/(c-d),1/2+1/2*sin(f *x+e))*cos(f*x+e)*(1+sin(f*x+e))*(c+d*sin(f*x+e))^n/a/f/(1-sin(f*x+e))^(1/ 2)/(((c+d*sin(f*x+e))/(c-d))^n)
\[ \int \frac {\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\int \frac {\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx \] Input:
Integrate[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]),x]
Output:
Integrate[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]), x]
Time = 0.39 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3042, 3396, 156, 155}
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.
\(\displaystyle \int \frac {\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a \sin (e+f x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (e+f x)^4 (c+d \sin (e+f x))^n}{a \sin (e+f x)+a}dx\) |
\(\Big \downarrow \) 3396 |
\(\displaystyle \frac {\cos (e+f x) \int (1-\sin (e+f x))^{3/2} \sqrt {\sin (e+f x)+1} (c+d \sin (e+f x))^nd\sin (e+f x)}{a f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}\) |
\(\Big \downarrow \) 156 |
\(\displaystyle \frac {\cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \int (1-\sin (e+f x))^{3/2} \sqrt {\sin (e+f x)+1} \left (\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}\right )^nd\sin (e+f x)}{a f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}\) |
\(\Big \downarrow \) 155 |
\(\displaystyle -\frac {2 \sqrt {2} (1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {AppellF1}\left (\frac {5}{2},-\frac {1}{2},-n,\frac {7}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{5 a f \sqrt {\sin (e+f x)+1}}\) |
Input:
Int[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]),x]
Output:
(-2*Sqrt[2]*AppellF1[5/2, -1/2, -n, 7/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin [e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x] )^n)/(5*a*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ (b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && GtQ[Sim plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] && !(GtQ[Simpl ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d *x, a + b*x]) && !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c - e*d)], 0] && SimplerQ[e + f*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p ]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]) Int[(a + b*x)^m*(c + d*x)^n*Si mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & GtQ[Simplify[b/(b*c - a*d)], 0] && !GtQ[Simplify[b/(b*e - a*f)], 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ )*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a^m*(Cos[e + f*x]/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]])) Subst[Int[(1 + (b/a)*x)^(m + (p - 1)/2)*(1 - (b/a)*x)^((p - 1)/2)*(c + d*x)^n, x], x, Sin [e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && I ntegerQ[p/2] && IntegerQ[m]
\[\int \frac {\cos \left (f x +e \right )^{4} \left (c +d \sin \left (f x +e \right )\right )^{n}}{a +a \sin \left (f x +e \right )}d x\]
Input:
int(cos(f*x+e)^4*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x)
Output:
int(cos(f*x+e)^4*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x)
\[ \int \frac {\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{4}}{a \sin \left (f x + e\right ) + a} \,d x } \] Input:
integrate(cos(f*x+e)^4*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x, algorithm="f ricas")
Output:
integral((d*sin(f*x + e) + c)^n*cos(f*x + e)^4/(a*sin(f*x + e) + a), x)
Timed out. \[ \int \frac {\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\text {Timed out} \] Input:
integrate(cos(f*x+e)**4*(c+d*sin(f*x+e))**n/(a+a*sin(f*x+e)),x)
Output:
Timed out
\[ \int \frac {\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{4}}{a \sin \left (f x + e\right ) + a} \,d x } \] Input:
integrate(cos(f*x+e)^4*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x, algorithm="m axima")
Output:
integrate((d*sin(f*x + e) + c)^n*cos(f*x + e)^4/(a*sin(f*x + e) + a), x)
\[ \int \frac {\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{4}}{a \sin \left (f x + e\right ) + a} \,d x } \] Input:
integrate(cos(f*x+e)^4*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x, algorithm="g iac")
Output:
integrate((d*sin(f*x + e) + c)^n*cos(f*x + e)^4/(a*sin(f*x + e) + a), x)
Timed out. \[ \int \frac {\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^4\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{a+a\,\sin \left (e+f\,x\right )} \,d x \] Input:
int((cos(e + f*x)^4*(c + d*sin(e + f*x))^n)/(a + a*sin(e + f*x)),x)
Output:
int((cos(e + f*x)^4*(c + d*sin(e + f*x))^n)/(a + a*sin(e + f*x)), x)
\[ \int \frac {\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\frac {\int \frac {\left (\sin \left (f x +e \right ) d +c \right )^{n} \cos \left (f x +e \right )^{4}}{\sin \left (f x +e \right )+1}d x}{a} \] Input:
int(cos(f*x+e)^4*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x)
Output:
int(((sin(e + f*x)*d + c)**n*cos(e + f*x)**4)/(sin(e + f*x) + 1),x)/a