\(\int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx\) [287]

Optimal result

Integrand size = 28, antiderivative size = 449 \[ \int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {(e+f x)^{1+m}}{2 a f (1+m)}+\frac {e^{i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {i d (e+f x)}{f}\right )}{8 a d}+\frac {e^{-i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {i d (e+f x)}{f}\right )}{8 a d}-\frac {i 2^{-3-m} e^{2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {2 i d (e+f x)}{f}\right )}{a d}+\frac {i 2^{-3-m} e^{-2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {2 i d (e+f x)}{f}\right )}{a d}+\frac {3^{-1-m} e^{3 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {3 i d (e+f x)}{f}\right )}{8 a d}+\frac {3^{-1-m} e^{-3 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {3 i d (e+f x)}{f}\right )}{8 a d} \] Output:

1/2*(f*x+e)^(1+m)/a/f/(1+m)+1/8*exp(I*(c-d*e/f))*(f*x+e)^m*GAMMA(1+m,-I*d* 
(f*x+e)/f)/a/d/((-I*d*(f*x+e)/f)^m)+1/8*(f*x+e)^m*GAMMA(1+m,I*d*(f*x+e)/f) 
/a/d/exp(I*(c-d*e/f))/((I*d*(f*x+e)/f)^m)-I*2^(-3-m)*exp(2*I*(c-d*e/f))*(f 
*x+e)^m*GAMMA(1+m,-2*I*d*(f*x+e)/f)/a/d/((-I*d*(f*x+e)/f)^m)+I*2^(-3-m)*(f 
*x+e)^m*GAMMA(1+m,2*I*d*(f*x+e)/f)/a/d/exp(2*I*(c-d*e/f))/((I*d*(f*x+e)/f) 
^m)+1/8*3^(-1-m)*exp(3*I*(c-d*e/f))*(f*x+e)^m*GAMMA(1+m,-3*I*d*(f*x+e)/f)/ 
a/d/((-I*d*(f*x+e)/f)^m)+1/8*3^(-1-m)*(f*x+e)^m*GAMMA(1+m,3*I*d*(f*x+e)/f) 
/a/d/exp(3*I*(c-d*e/f))/((I*d*(f*x+e)/f)^m)
 

Mathematica [A] (verified)

Time = 12.20 (sec) , antiderivative size = 405, normalized size of antiderivative = 0.90 \[ \int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {i (e+f x)^m \left (-\frac {12 i d (e+f x)}{f (1+m)}-3 i e^{i \left (c-\frac {d e}{f}\right )} \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {i d (e+f x)}{f}\right )-3 i e^{-i \left (c-\frac {d e}{f}\right )} \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {i d (e+f x)}{f}\right )-3\ 2^{-m} e^{2 i \left (c-\frac {d e}{f}\right )} \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {2 i d (e+f x)}{f}\right )+3\ 2^{-m} e^{-2 i \left (c-\frac {d e}{f}\right )} \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {2 i d (e+f x)}{f}\right )-i 3^{-m} e^{3 i \left (c-\frac {d e}{f}\right )} \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {3 i d (e+f x)}{f}\right )-i 3^{-m} e^{-3 i \left (c-\frac {d e}{f}\right )} \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {3 i d (e+f x)}{f}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}{24 a d (1+\sin (c+d x))} \] Input:

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

Output:

((I/24)*(e + f*x)^m*(((-12*I)*d*(e + f*x))/(f*(1 + m)) - ((3*I)*E^(I*(c - 
(d*e)/f))*Gamma[1 + m, ((-I)*d*(e + f*x))/f])/(((-I)*d*(e + f*x))/f)^m - ( 
(3*I)*Gamma[1 + m, (I*d*(e + f*x))/f])/(E^(I*(c - (d*e)/f))*((I*d*(e + f*x 
))/f)^m) - (3*E^((2*I)*(c - (d*e)/f))*Gamma[1 + m, ((-2*I)*d*(e + f*x))/f] 
)/(2^m*(((-I)*d*(e + f*x))/f)^m) + (3*Gamma[1 + m, ((2*I)*d*(e + f*x))/f]) 
/(2^m*E^((2*I)*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m) - (I*E^((3*I)*(c - (d 
*e)/f))*Gamma[1 + m, ((-3*I)*d*(e + f*x))/f])/(3^m*(((-I)*d*(e + f*x))/f)^ 
m) - (I*Gamma[1 + m, ((3*I)*d*(e + f*x))/f])/(3^m*E^((3*I)*(c - (d*e)/f))* 
((I*d*(e + f*x))/f)^m))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(a*d*(1 + 
 Sin[c + d*x]))
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 439, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5034, 3042, 3793, 2009, 4906, 2009}

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.

\(\displaystyle \int \frac {\cos ^4(c+d x) (e+f x)^m}{a \sin (c+d x)+a} \, dx\)

\(\Big \downarrow \) 5034

\(\displaystyle \frac {\int (e+f x)^m \cos ^2(c+d x)dx}{a}-\frac {\int (e+f x)^m \cos ^2(c+d x) \sin (c+d x)dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (e+f x)^m \sin \left (c+d x+\frac {\pi }{2}\right )^2dx}{a}-\frac {\int (e+f x)^m \cos ^2(c+d x) \sin (c+d x)dx}{a}\)

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {\int \left (\frac {1}{4} \sin (c+d x) (e+f x)^m+\frac {1}{4} \sin (3 c+3 d x) (e+f x)^m\right )dx}{a}+\frac {-\frac {i 2^{-m-3} e^{2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,-\frac {2 i d (e+f x)}{f}\right )}{d}+\frac {i 2^{-m-3} e^{-2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,\frac {2 i d (e+f x)}{f}\right )}{d}+\frac {(e+f x)^{m+1}}{2 f (m+1)}}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {i 2^{-m-3} e^{2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,-\frac {2 i d (e+f x)}{f}\right )}{d}+\frac {i 2^{-m-3} e^{-2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,\frac {2 i d (e+f x)}{f}\right )}{d}+\frac {(e+f x)^{m+1}}{2 f (m+1)}}{a}-\frac {-\frac {e^{i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,-\frac {i d (e+f x)}{f}\right )}{8 d}-\frac {3^{-m-1} e^{3 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,-\frac {3 i d (e+f x)}{f}\right )}{8 d}-\frac {e^{-i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,\frac {i d (e+f x)}{f}\right )}{8 d}-\frac {3^{-m-1} e^{-3 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,\frac {3 i d (e+f x)}{f}\right )}{8 d}}{a}\)

Input:

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

Output:

((e + f*x)^(1 + m)/(2*f*(1 + m)) - (I*2^(-3 - m)*E^((2*I)*(c - (d*e)/f))*( 
e + f*x)^m*Gamma[1 + m, ((-2*I)*d*(e + f*x))/f])/(d*(((-I)*d*(e + f*x))/f) 
^m) + (I*2^(-3 - m)*(e + f*x)^m*Gamma[1 + m, ((2*I)*d*(e + f*x))/f])/(d*E^ 
((2*I)*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m))/a - (-1/8*(E^(I*(c - (d*e)/f 
))*(e + f*x)^m*Gamma[1 + m, ((-I)*d*(e + f*x))/f])/(d*(((-I)*d*(e + f*x))/ 
f)^m) - ((e + f*x)^m*Gamma[1 + m, (I*d*(e + f*x))/f])/(8*d*E^(I*(c - (d*e) 
/f))*((I*d*(e + f*x))/f)^m) - (3^(-1 - m)*E^((3*I)*(c - (d*e)/f))*(e + f*x 
)^m*Gamma[1 + m, ((-3*I)*d*(e + f*x))/f])/(8*d*(((-I)*d*(e + f*x))/f)^m) - 
 (3^(-1 - m)*(e + f*x)^m*Gamma[1 + m, ((3*I)*d*(e + f*x))/f])/(8*d*E^((3*I 
)*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m))/a
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 5034

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

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.76 \[ \int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \, {\left (f m + f\right )} e^{\left (-\frac {f m \log \left (\frac {i \, d}{f}\right ) - i \, d e + i \, c f}{f}\right )} \Gamma \left (m + 1, \frac {i \, d f x + i \, d e}{f}\right ) - 3 \, {\left (i \, f m + i \, f\right )} e^{\left (-\frac {f m \log \left (-\frac {2 i \, d}{f}\right ) + 2 i \, d e - 2 i \, c f}{f}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (i \, d f x + i \, d e\right )}}{f}\right ) + {\left (f m + f\right )} e^{\left (-\frac {f m \log \left (-\frac {3 i \, d}{f}\right ) + 3 i \, d e - 3 i \, c f}{f}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (i \, d f x + i \, d e\right )}}{f}\right ) + 3 \, {\left (f m + f\right )} e^{\left (-\frac {f m \log \left (-\frac {i \, d}{f}\right ) + i \, d e - i \, c f}{f}\right )} \Gamma \left (m + 1, \frac {-i \, d f x - i \, d e}{f}\right ) - 3 \, {\left (-i \, f m - i \, f\right )} e^{\left (-\frac {f m \log \left (\frac {2 i \, d}{f}\right ) - 2 i \, d e + 2 i \, c f}{f}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (-i \, d f x - i \, d e\right )}}{f}\right ) + {\left (f m + f\right )} e^{\left (-\frac {f m \log \left (\frac {3 i \, d}{f}\right ) - 3 i \, d e + 3 i \, c f}{f}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (-i \, d f x - i \, d e\right )}}{f}\right ) + 12 \, {\left (d f x + d e\right )} {\left (f x + e\right )}^{m}}{24 \, {\left (a d f m + a d f\right )}} \] Input:

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

Output:

1/24*(3*(f*m + f)*e^(-(f*m*log(I*d/f) - I*d*e + I*c*f)/f)*gamma(m + 1, (I* 
d*f*x + I*d*e)/f) - 3*(I*f*m + I*f)*e^(-(f*m*log(-2*I*d/f) + 2*I*d*e - 2*I 
*c*f)/f)*gamma(m + 1, -2*(I*d*f*x + I*d*e)/f) + (f*m + f)*e^(-(f*m*log(-3* 
I*d/f) + 3*I*d*e - 3*I*c*f)/f)*gamma(m + 1, -3*(I*d*f*x + I*d*e)/f) + 3*(f 
*m + f)*e^(-(f*m*log(-I*d/f) + I*d*e - I*c*f)/f)*gamma(m + 1, (-I*d*f*x - 
I*d*e)/f) - 3*(-I*f*m - I*f)*e^(-(f*m*log(2*I*d/f) - 2*I*d*e + 2*I*c*f)/f) 
*gamma(m + 1, -2*(-I*d*f*x - I*d*e)/f) + (f*m + f)*e^(-(f*m*log(3*I*d/f) - 
 3*I*d*e + 3*I*c*f)/f)*gamma(m + 1, -3*(-I*d*f*x - I*d*e)/f) + 12*(d*f*x + 
 d*e)*(f*x + e)^m)/(a*d*f*m + a*d*f)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(((e + f*x)**m*cos(c + d*x)**4)/(sin(c + d*x) + 1),x)/a