Integrand size = 18, antiderivative size = 148 \[ \int (c+d x)^m (a+a \sin (e+f x)) \, dx=\frac {a (c+d x)^{1+m}}{d (1+m)}-\frac {a e^{i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i f (c+d x)}{d}\right )}{2 f}-\frac {a e^{-i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i f (c+d x)}{d}\right )}{2 f} \] Output:
a*(d*x+c)^(1+m)/d/(1+m)-1/2*a*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*a*(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)
Time = 0.46 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.93 \[ \int (c+d x)^m (a+a \sin (e+f x)) \, dx=\frac {1}{2} a (c+d x)^m \left (\frac {2 c+2 d x}{d+d m}-\frac {e^{i \left (e-\frac {c f}{d}\right )} \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i f (c+d x)}{d}\right )}{f}-\frac {e^{-i \left (e-\frac {c f}{d}\right )} \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i f (c+d x)}{d}\right )}{f}\right ) \] Input:
Integrate[(c + d*x)^m*(a + a*Sin[e + f*x]),x]
Output:
(a*(c + d*x)^m*((2*c + 2*d*x)/(d + d*m) - (E^(I*(e - (c*f)/d))*Gamma[1 + m , ((-I)*f*(c + d*x))/d])/(f*(((-I)*f*(c + d*x))/d)^m) - Gamma[1 + m, (I*f* (c + d*x))/d]/(E^(I*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m)))/2
Time = 0.37 (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.
\(\displaystyle \int (c+d x)^m (a \sin (e+f x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d x)^m (a \sin (e+f x)+a)dx\) |
\(\Big \downarrow \) 3798 |
\(\displaystyle \int \left (a (c+d x)^m \sin (e+f x)+a (c+d x)^m\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a e^{i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {i f (c+d x)}{d}\right )}{2 f}-\frac {a e^{-i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {i f (c+d x)}{d}\right )}{2 f}+\frac {a (c+d x)^{m+1}}{d (m+1)}\) |
Input:
Int[(c + d*x)^m*(a + a*Sin[e + f*x]),x]
Output:
(a*(c + d*x)^(1 + m))/(d*(1 + m)) - (a*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) - (a*(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)
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])
\[\int \left (d x +c \right )^{m} \left (a +a \sin \left (f x +e \right )\right )d x\]
Input:
int((d*x+c)^m*(a+a*sin(f*x+e)),x)
Output:
int((d*x+c)^m*(a+a*sin(f*x+e)),x)
Time = 0.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.92 \[ \int (c+d x)^m (a+a \sin (e+f x)) \, dx=-\frac {{\left (a d m + a d\right )} e^{\left (-\frac {d m \log \left (\frac {i \, f}{d}\right ) + i \, d e - i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {i \, d f x + i \, c f}{d}\right ) + {\left (a d m + a d\right )} e^{\left (-\frac {d m \log \left (-\frac {i \, f}{d}\right ) - i \, d e + i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {-i \, d f x - i \, c f}{d}\right ) - 2 \, {\left (a d f x + a c f\right )} {\left (d x + c\right )}^{m}}{2 \, {\left (d f m + d f\right )}} \] Input:
integrate((d*x+c)^m*(a+a*sin(f*x+e)),x, algorithm="fricas")
Output:
-1/2*((a*d*m + a*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) + (a*d*m + a*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)
\[ \int (c+d x)^m (a+a \sin (e+f x)) \, dx=a \left (\int \left (c + d x\right )^{m} \sin {\left (e + f x \right )}\, dx + \int \left (c + d x\right )^{m}\, dx\right ) \] Input:
integrate((d*x+c)**m*(a+a*sin(f*x+e)),x)
Output:
a*(Integral((c + d*x)**m*sin(e + f*x), x) + Integral((c + d*x)**m, x))
\[ \int (c+d x)^m (a+a \sin (e+f x)) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} {\left (d x + c\right )}^{m} \,d x } \] Input:
integrate((d*x+c)^m*(a+a*sin(f*x+e)),x, algorithm="maxima")
Output:
a*integrate((d*x + c)^m*sin(f*x + e), x) + (d*x + c)^(m + 1)*a/(d*(m + 1))
\[ \int (c+d x)^m (a+a \sin (e+f x)) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} {\left (d x + c\right )}^{m} \,d x } \] Input:
integrate((d*x+c)^m*(a+a*sin(f*x+e)),x, algorithm="giac")
Output:
integrate((a*sin(f*x + e) + a)*(d*x + c)^m, x)
Timed out. \[ \int (c+d x)^m (a+a \sin (e+f x)) \, dx=\int \left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^m \,d x \] Input:
int((a + a*sin(e + f*x))*(c + d*x)^m,x)
Output:
int((a + a*sin(e + f*x))*(c + d*x)^m, x)
\[ \int (c+d x)^m (a+a \sin (e+f x)) \, dx=\frac {a \left (-\left (d x +c \right )^{m} \cos \left (f x +e \right ) d m -\left (d x +c \right )^{m} \cos \left (f x +e \right ) d +\left (d x +c \right )^{m} c f +\left (d x +c \right )^{m} d f x -\left (d x +c \right )^{m} d m -\left (d x +c \right )^{m} d +2 \left (\int \frac {\left (d x +c \right )^{m}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c +\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d x +c +d x}d x \right ) d^{2} m^{2}+2 \left (\int \frac {\left (d x +c \right )^{m}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c +\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d x +c +d x}d x \right ) d^{2} m \right )}{d f \left (m +1\right )} \] Input:
int((d*x+c)^m*(a+a*sin(f*x+e)),x)
Output:
(a*( - (c + d*x)**m*cos(e + f*x)*d*m - (c + d*x)**m*cos(e + f*x)*d + (c + d*x)**m*c*f + (c + d*x)**m*d*f*x - (c + d*x)**m*d*m - (c + d*x)**m*d + 2*i nt((c + d*x)**m/(tan((e + f*x)/2)**2*c + tan((e + f*x)/2)**2*d*x + c + d*x ),x)*d**2*m**2 + 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)*d**2*m))/(d*f*(m + 1))