\(\int (c+d x)^m (a+b \sin (e+f x)) \, dx\) [176]

Optimal result

Integrand size = 18, antiderivative size = 148 \[ \int (c+d x)^m (a+b \sin (e+f x)) \, dx=\frac {a (c+d x)^{1+m}}{d (1+m)}-\frac {b e^{i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i f (c+d x)}{d}\right )}{2 f}-\frac {b e^{-i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i f (c+d x)}{d}\right )}{2 f} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.93 \[ \int (c+d x)^m (a+b \sin (e+f x)) \, dx=\frac {1}{2} (c+d x)^m \left (\frac {2 a (c+d x)}{d (1+m)}-\frac {b e^{i \left (e-\frac {c f}{d}\right )} \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i f (c+d x)}{d}\right )}{f}-\frac {b e^{-i \left (e-\frac {c f}{d}\right )} \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i f (c+d x)}{d}\right )}{f}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3798, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3798

\(\displaystyle \int \left (a (c+d x)^m+b (c+d x)^m \sin (e+f x)\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 3798

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

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.92 \[ \int (c+d x)^m (a+b \sin (e+f x)) \, dx=-\frac {{\left (b d m + b d\right )} e^{\left (-\frac {d m \log \left (\frac {i \, f}{d}\right ) + i \, d e - i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {i \, d f x + i \, c f}{d}\right ) + {\left (b d m + b d\right )} e^{\left (-\frac {d m \log \left (-\frac {i \, f}{d}\right ) - i \, d e + i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {-i \, d f x - i \, c f}{d}\right ) - 2 \, {\left (a d f x + a c f\right )} {\left (d x + c\right )}^{m}}{2 \, {\left (d f m + d f\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*sin(e + f*x))*(c + d*x)**m, x)
 

Maxima [F]

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

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

Output:

b*integrate((d*x + c)^m*sin(f*x + e), x) + (d*x + c)^(m + 1)*a/(d*(m + 1))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^m (a+b \sin (e+f x)) \, dx=\frac {-\left (d x +c \right )^{m} \cos \left (f x +e \right ) b d m -\left (d x +c \right )^{m} \cos \left (f x +e \right ) b d +\left (d x +c \right )^{m} a c f +\left (d x +c \right )^{m} a d f x -\left (d x +c \right )^{m} b d m -\left (d x +c \right )^{m} b d +2 \left (\int \frac {\left (d x +c \right )^{m}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c +\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d x +c +d x}d x \right ) b \,d^{2} m^{2}+2 \left (\int \frac {\left (d x +c \right )^{m}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c +\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d x +c +d x}d x \right ) b \,d^{2} m}{d f \left (m +1\right )} \] Input:

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

Output:

( - (c + d*x)**m*cos(e + f*x)*b*d*m - (c + d*x)**m*cos(e + f*x)*b*d + (c + 
 d*x)**m*a*c*f + (c + d*x)**m*a*d*f*x - (c + d*x)**m*b*d*m - (c + d*x)**m* 
b*d + 2*int((c + d*x)**m/(tan((e + f*x)/2)**2*c + tan((e + f*x)/2)**2*d*x 
+ c + d*x),x)*b*d**2*m**2 + 2*int((c + d*x)**m/(tan((e + f*x)/2)**2*c + ta 
n((e + f*x)/2)**2*d*x + c + d*x),x)*b*d**2*m)/(d*f*(m + 1))