\(\int \frac {(e+f x)^m \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [288]

Optimal result

Integrand size = 28, antiderivative size = 277 \[ \int \frac {(e+f x)^m \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {i e^{i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {i d (e+f x)}{f}\right )}{2 a d}+\frac {i e^{-i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {i d (e+f x)}{f}\right )}{2 a d}+\frac {2^{-3-m} e^{2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {2 i d (e+f x)}{f}\right )}{a d}+\frac {2^{-3-m} e^{-2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {2 i d (e+f x)}{f}\right )}{a d} \] Output:

-1/2*I*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)+1/2*I*(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)+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)+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)
 

Mathematica [A] (verified)

Time = 9.89 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.91 \[ \int \frac {(e+f x)^m \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2^{-3-m} e^{-\frac {2 i (d e+c f)}{f}} (e+f x)^m \left (\frac {d^2 (e+f x)^2}{f^2}\right )^{-m} \left (-i 2^{2+m} e^{i \left (3 c+\frac {d e}{f}\right )} \left (\frac {i d (e+f x)}{f}\right )^m \Gamma \left (1+m,-\frac {i d (e+f x)}{f}\right )+i 2^{2+m} e^{i \left (c+\frac {3 d e}{f}\right )} \left (-\frac {i d (e+f x)}{f}\right )^m \Gamma \left (1+m,\frac {i d (e+f x)}{f}\right )+e^{4 i c} \left (\frac {i d (e+f x)}{f}\right )^m \Gamma \left (1+m,-\frac {2 i d (e+f x)}{f}\right )+e^{\frac {4 i d e}{f}} \left (-\frac {i d (e+f x)}{f}\right )^m \Gamma \left (1+m,\frac {2 i d (e+f x)}{f}\right )\right )}{a d} \] Input:

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

Output:

(2^(-3 - m)*(e + f*x)^m*((-I)*2^(2 + m)*E^(I*(3*c + (d*e)/f))*((I*d*(e + f 
*x))/f)^m*Gamma[1 + m, ((-I)*d*(e + f*x))/f] + I*2^(2 + m)*E^(I*(c + (3*d* 
e)/f))*(((-I)*d*(e + f*x))/f)^m*Gamma[1 + m, (I*d*(e + f*x))/f] + E^((4*I) 
*c)*((I*d*(e + f*x))/f)^m*Gamma[1 + m, ((-2*I)*d*(e + f*x))/f] + E^(((4*I) 
*d*e)/f)*(((-I)*d*(e + f*x))/f)^m*Gamma[1 + m, ((2*I)*d*(e + f*x))/f]))/(a 
*d*E^(((2*I)*(d*e + c*f))/f)*((d^2*(e + f*x)^2)/f^2)^m)
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5034, 3042, 3788, 26, 2612, 4906, 27, 3042, 3789, 2612}

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.

\(\displaystyle \int \frac {\cos ^3(c+d x) (e+f x)^m}{a \sin (c+d x)+a} \, dx\)

\(\Big \downarrow \) 5034

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2612

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3789

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

\(\Big \downarrow \) 2612

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2612

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] 
:> Simp[(-F^(g*(e - c*(f/d))))*((c + d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d) 
)^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m + 1, 
 ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] && 
!IntegerQ[m]
 

rule 3042

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

rule 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[I/2   Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e 
, f, m}, x] && IntegerQ[2*k]
 

rule 3789

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
/2   Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2   Int[(c + d*x)^m*E 
^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
 

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]

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

Input:

int((f*x+e)^m*cos(d*x+c)^3/(a+a*sin(d*x+c)),x)
 

Output:

int((f*x+e)^m*cos(d*x+c)^3/(a+a*sin(d*x+c)),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.68 \[ \int \frac {(e+f x)^m \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {4 i \, e^{\left (-\frac {f m \log \left (\frac {i \, d}{f}\right ) - i \, d e + i \, c f}{f}\right )} \Gamma \left (m + 1, \frac {i \, d f x + i \, d e}{f}\right ) + e^{\left (-\frac {f m \log \left (-\frac {2 i \, d}{f}\right ) + 2 i \, d e - 2 i \, c f}{f}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (i \, d f x + i \, d e\right )}}{f}\right ) - 4 i \, e^{\left (-\frac {f m \log \left (-\frac {i \, d}{f}\right ) + i \, d e - i \, c f}{f}\right )} \Gamma \left (m + 1, \frac {-i \, d f x - i \, d e}{f}\right ) + e^{\left (-\frac {f m \log \left (\frac {2 i \, d}{f}\right ) - 2 i \, d e + 2 i \, c f}{f}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (-i \, d f x - i \, d e\right )}}{f}\right )}{8 \, a d} \] Input:

integrate((f*x+e)^m*cos(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

1/8*(4*I*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) + 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) - 4*I*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) + 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))/(a*d)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^m \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((f*x+e)**m*cos(d*x+c)**3/(a+a*sin(d*x+c)),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

integrate((f*x + e)^m*cos(d*x + c)^3/(a*sin(d*x + c) + a), x)
 

Giac [F]

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

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

Output:

integrate((f*x + e)^m*cos(d*x + c)^3/(a*sin(d*x + c) + a), x)
 

Mupad [F(-1)]

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

int((cos(c + d*x)^3*(e + f*x)^m)/(a + a*sin(c + d*x)),x)
 

Output:

int((cos(c + d*x)^3*(e + f*x)^m)/(a + a*sin(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {(e+f x)^m \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\left (f x +e \right )^{m} \cos \left (d x +c \right )^{3}}{\sin \left (d x +c \right )+1}d x}{a} \] Input:

int((f*x+e)^m*cos(d*x+c)^3/(a+a*sin(d*x+c)),x)
 

Output:

int(((e + f*x)**m*cos(c + d*x)**3)/(sin(c + d*x) + 1),x)/a