3.2.46 \(\int (c+d x)^m (a+a \sin (e+f x))^3 \, dx\) [146]

3.2.46.1 Optimal result

Integrand size = 20, antiderivative size = 449 \[ \int (c+d x)^m (a+a \sin (e+f x))^3 \, dx=\frac {5 a^3 (c+d x)^{1+m}}{2 d (1+m)}-\frac {15 a^3 e^{i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i f (c+d x)}{d}\right )}{8 f}-\frac {15 a^3 e^{-i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i f (c+d x)}{d}\right )}{8 f}+\frac {3 i 2^{-3-m} a^3 e^{2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 i f (c+d x)}{d}\right )}{f}-\frac {3 i 2^{-3-m} a^3 e^{-2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 i f (c+d x)}{d}\right )}{f}+\frac {3^{-1-m} a^3 e^{3 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 i f (c+d x)}{d}\right )}{8 f}+\frac {3^{-1-m} a^3 e^{-3 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 i f (c+d x)}{d}\right )}{8 f} \]

output

5/2*a^3*(d*x+c)^(1+m)/d/(1+m)-15/8*a^3*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)-15/8*a^3*(d*x+c)^m*GAMMA(1+m,I*f* 
(d*x+c)/d)/exp(I*(e-c*f/d))/f/((I*f*(d*x+c)/d)^m)+3*I*2^(-3-m)*a^3*exp(2*I 
*(e-c*f/d))*(d*x+c)^m*GAMMA(1+m,-2*I*f*(d*x+c)/d)/f/((-I*f*(d*x+c)/d)^m)-3 
*I*2^(-3-m)*a^3*(d*x+c)^m*GAMMA(1+m,2*I*f*(d*x+c)/d)/exp(2*I*(e-c*f/d))/f/ 
((I*f*(d*x+c)/d)^m)+1/8*3^(-1-m)*a^3*exp(3*I*(e-c*f/d))*(d*x+c)^m*GAMMA(1+ 
m,-3*I*f*(d*x+c)/d)/f/((-I*f*(d*x+c)/d)^m)+1/8*3^(-1-m)*a^3*(d*x+c)^m*GAMM 
A(1+m,3*I*f*(d*x+c)/d)/exp(3*I*(e-c*f/d))/f/((I*f*(d*x+c)/d)^m)
 

3.2.46.2 Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.84 \[ \int (c+d x)^m (a+a \sin (e+f x))^3 \, dx=\frac {1}{24} a^3 (c+d x)^m \left (\frac {60 (c+d x)}{d (1+m)}-\frac {45 e^{i \left (e-\frac {c f}{d}\right )} \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i f (c+d x)}{d}\right )}{f}-\frac {45 e^{-i \left (e-\frac {c f}{d}\right )} \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i f (c+d x)}{d}\right )}{f}+\frac {9 i 2^{-m} e^{2 i \left (e-\frac {c f}{d}\right )} \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 i f (c+d x)}{d}\right )}{f}-\frac {9 i 2^{-m} e^{-2 i \left (e-\frac {c f}{d}\right )} \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 i f (c+d x)}{d}\right )}{f}+\frac {3^{-m} e^{3 i \left (e-\frac {c f}{d}\right )} \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 i f (c+d x)}{d}\right )}{f}+\frac {3^{-m} e^{-3 i \left (e-\frac {c f}{d}\right )} \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 i f (c+d x)}{d}\right )}{f}\right ) \]

input

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

output

(a^3*(c + d*x)^m*((60*(c + d*x))/(d*(1 + m)) - (45*E^(I*(e - (c*f)/d))*Gam 
ma[1 + m, ((-I)*f*(c + d*x))/d])/(f*(((-I)*f*(c + d*x))/d)^m) - (45*Gamma[ 
1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m) + 
 ((9*I)*E^((2*I)*(e - (c*f)/d))*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d])/(2^m 
*f*(((-I)*f*(c + d*x))/d)^m) - ((9*I)*Gamma[1 + m, ((2*I)*f*(c + d*x))/d]) 
/(2^m*E^((2*I)*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m) + (E^((3*I)*(e - (c 
*f)/d))*Gamma[1 + m, ((-3*I)*f*(c + d*x))/d])/(3^m*f*(((-I)*f*(c + d*x))/d 
)^m) + Gamma[1 + m, ((3*I)*f*(c + d*x))/d]/(3^m*E^((3*I)*(e - (c*f)/d))*f* 
((I*f*(c + d*x))/d)^m)))/24
 

3.2.46.3 Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3799, 3042, 3793, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^m (a \sin (e+f x)+a)^3dx\)

\(\Big \downarrow \) 3799

\(\displaystyle 8 a^3 \int (c+d x)^m \sin ^6\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 8 a^3 \int (c+d x)^m \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^6dx\)

\(\Big \downarrow \) 3793

\(\displaystyle 8 a^3 \int \left (-\frac {3}{16} \cos (2 e+2 f x) (c+d x)^m+\frac {15}{32} \sin (e+f x) (c+d x)^m-\frac {1}{32} \sin (3 e+3 f x) (c+d x)^m+\frac {5}{16} (c+d x)^m\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 8 a^3 \left (-\frac {15 e^{i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {i f (c+d x)}{d}\right )}{64 f}+\frac {3 i 2^{-m-6} e^{2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 i f (c+d x)}{d}\right )}{f}+\frac {3^{-m-1} e^{3 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {3 i f (c+d x)}{d}\right )}{64 f}-\frac {15 e^{-i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {i f (c+d x)}{d}\right )}{64 f}-\frac {3 i 2^{-m-6} e^{-2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 i f (c+d x)}{d}\right )}{f}+\frac {3^{-m-1} e^{-3 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {3 i f (c+d x)}{d}\right )}{64 f}+\frac {5 (c+d x)^{m+1}}{16 d (m+1)}\right )\)

input

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

output

8*a^3*((5*(c + d*x)^(1 + m))/(16*d*(1 + m)) - (15*E^(I*(e - (c*f)/d))*(c + 
 d*x)^m*Gamma[1 + m, ((-I)*f*(c + d*x))/d])/(64*f*(((-I)*f*(c + d*x))/d)^m 
) - (15*(c + d*x)^m*Gamma[1 + m, (I*f*(c + d*x))/d])/(64*E^(I*(e - (c*f)/d 
))*f*((I*f*(c + d*x))/d)^m) + ((3*I)*2^(-6 - m)*E^((2*I)*(e - (c*f)/d))*(c 
 + d*x)^m*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d])/(f*(((-I)*f*(c + d*x))/d)^ 
m) - ((3*I)*2^(-6 - m)*(c + d*x)^m*Gamma[1 + m, ((2*I)*f*(c + d*x))/d])/(E 
^((2*I)*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m) + (3^(-1 - m)*E^((3*I)*(e 
- (c*f)/d))*(c + d*x)^m*Gamma[1 + m, ((-3*I)*f*(c + d*x))/d])/(64*f*(((-I) 
*f*(c + d*x))/d)^m) + (3^(-1 - m)*(c + d*x)^m*Gamma[1 + m, ((3*I)*f*(c + d 
*x))/d])/(64*E^((3*I)*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m))
 

3.2.46.3.1 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 3799

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

3.2.46.4 Maple [F]

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

input

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

output

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

3.2.46.5 Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.86 \[ \int (c+d x)^m (a+a \sin (e+f x))^3 \, dx=-\frac {45 \, {\left (a^{3} d m + a^{3} d\right )} e^{\left (-\frac {d m \log \left (\frac {i \, f}{d}\right ) + i \, d e - i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {i \, d f x + i \, c f}{d}\right ) + 9 \, {\left (-i \, a^{3} d m - i \, a^{3} d\right )} e^{\left (-\frac {d m \log \left (-\frac {2 i \, f}{d}\right ) - 2 i \, d e + 2 i \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) - {\left (a^{3} d m + a^{3} d\right )} e^{\left (-\frac {d m \log \left (-\frac {3 i \, f}{d}\right ) - 3 i \, d e + 3 i \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) + 45 \, {\left (a^{3} d m + a^{3} d\right )} e^{\left (-\frac {d m \log \left (-\frac {i \, f}{d}\right ) - i \, d e + i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {-i \, d f x - i \, c f}{d}\right ) + 9 \, {\left (i \, a^{3} d m + i \, a^{3} d\right )} e^{\left (-\frac {d m \log \left (\frac {2 i \, f}{d}\right ) + 2 i \, d e - 2 i \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) - {\left (a^{3} d m + a^{3} d\right )} e^{\left (-\frac {d m \log \left (\frac {3 i \, f}{d}\right ) + 3 i \, d e - 3 i \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) - 60 \, {\left (a^{3} d f x + a^{3} c f\right )} {\left (d x + c\right )}^{m}}{24 \, {\left (d f m + d f\right )}} \]

input

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

output

-1/24*(45*(a^3*d*m + a^3*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) + 9*(-I*a^3*d*m - I*a^3*d)*e^(-(d*m*log(-2*I*f 
/d) - 2*I*d*e + 2*I*c*f)/d)*gamma(m + 1, -2*(I*d*f*x + I*c*f)/d) - (a^3*d* 
m + a^3*d)*e^(-(d*m*log(-3*I*f/d) - 3*I*d*e + 3*I*c*f)/d)*gamma(m + 1, -3* 
(I*d*f*x + I*c*f)/d) + 45*(a^3*d*m + a^3*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) + 9*(I*a^3*d*m + I*a^3*d)*e^ 
(-(d*m*log(2*I*f/d) + 2*I*d*e - 2*I*c*f)/d)*gamma(m + 1, -2*(-I*d*f*x - I* 
c*f)/d) - (a^3*d*m + a^3*d)*e^(-(d*m*log(3*I*f/d) + 3*I*d*e - 3*I*c*f)/d)* 
gamma(m + 1, -3*(-I*d*f*x - I*c*f)/d) - 60*(a^3*d*f*x + a^3*c*f)*(d*x + c) 
^m)/(d*f*m + d*f)
 

3.2.46.6 Sympy [F]

\[ \int (c+d x)^m (a+a \sin (e+f x))^3 \, dx=a^{3} \left (\int 3 \left (c + d x\right )^{m} \sin {\left (e + f x \right )}\, dx + \int 3 \left (c + d x\right )^{m} \sin ^{2}{\left (e + f x \right )}\, dx + \int \left (c + d x\right )^{m} \sin ^{3}{\left (e + f x \right )}\, dx + \int \left (c + d x\right )^{m}\, dx\right ) \]

input

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

output

a**3*(Integral(3*(c + d*x)**m*sin(e + f*x), x) + Integral(3*(c + d*x)**m*s 
in(e + f*x)**2, x) + Integral((c + d*x)**m*sin(e + f*x)**3, x) + Integral( 
(c + d*x)**m, x))
 

3.2.46.7 Maxima [F]

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

input

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

output

(d*x + c)^(m + 1)*a^3/(d*(m + 1)) + 1/4*(6*a^3*e^(m*log(d*x + c) + log(d*x 
 + c)) - 6*(a^3*d*m + a^3*d)*integrate((d*x + c)^m*cos(2*f*x + 2*e), x) - 
(a^3*d*m + a^3*d)*integrate((d*x + c)^m*sin(3*f*x + 3*e), x) + 15*(a^3*d*m 
 + a^3*d)*integrate((d*x + c)^m*sin(f*x + e), x))/(d*m + d)
 

3.2.46.8 Giac [F]

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

input

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

output

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

3.2.46.9 Mupad [F(-1)]

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

input

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

output

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