\(\int (c+d x)^m (a+b \sin (e+f x))^3 \, dx\) [174]

Optimal result

Integrand size = 20, antiderivative size = 607 \[ \int (c+d x)^m (a+b \sin (e+f x))^3 \, dx=\frac {a^3 (c+d x)^{1+m}}{d (1+m)}+\frac {3 a b^2 (c+d x)^{1+m}}{2 d (1+m)}-\frac {3 a^2 b e^{i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i f (c+d x)}{d}\right )}{2 f}-\frac {3 b^3 e^{i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i f (c+d x)}{d}\right )}{8 f}-\frac {3 a^2 b e^{-i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i f (c+d x)}{d}\right )}{2 f}-\frac {3 b^3 e^{-i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i f (c+d x)}{d}\right )}{8 f}+\frac {3 i 2^{-3-m} a b^2 e^{2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 i f (c+d x)}{d}\right )}{f}-\frac {3 i 2^{-3-m} a b^2 e^{-2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 i f (c+d x)}{d}\right )}{f}+\frac {3^{-1-m} b^3 e^{3 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 i f (c+d x)}{d}\right )}{8 f}+\frac {3^{-1-m} b^3 e^{-3 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 i f (c+d x)}{d}\right )}{8 f} \] Output:

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

Mathematica [A] (verified)

Time = 11.71 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.68 \[ \int (c+d x)^m (a+b \sin (e+f x))^3 \, dx=\frac {i (c+d x)^m \left (-\frac {12 i a \left (2 a^2+3 b^2\right ) f (c+d x)}{d (1+m)}+9 i b \left (4 a^2+b^2\right ) e^{i \left (e-\frac {c f}{d}\right )} \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i f (c+d x)}{d}\right )+9 i b \left (4 a^2+b^2\right ) e^{-i \left (e-\frac {c f}{d}\right )} \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i f (c+d x)}{d}\right )+9\ 2^{-m} a b^2 e^{2 i \left (e-\frac {c f}{d}\right )} \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 i f (c+d x)}{d}\right )-9\ 2^{-m} a b^2 e^{-2 i \left (e-\frac {c f}{d}\right )} \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 i f (c+d x)}{d}\right )-i 3^{-m} b^3 e^{3 i \left (e-\frac {c f}{d}\right )} \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 i f (c+d x)}{d}\right )-i 3^{-m} b^3 e^{-3 i \left (e-\frac {c f}{d}\right )} \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 i f (c+d x)}{d}\right )\right )}{24 f} \] Input:

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

Output:

((I/24)*(c + d*x)^m*(((-12*I)*a*(2*a^2 + 3*b^2)*f*(c + d*x))/(d*(1 + m)) + 
 ((9*I)*b*(4*a^2 + b^2)*E^(I*(e - (c*f)/d))*Gamma[1 + m, ((-I)*f*(c + d*x) 
)/d])/(((-I)*f*(c + d*x))/d)^m + ((9*I)*b*(4*a^2 + b^2)*Gamma[1 + m, (I*f* 
(c + d*x))/d])/(E^(I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m) + (9*a*b^2*E^(( 
2*I)*(e - (c*f)/d))*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d])/(2^m*(((-I)*f*(c 
 + d*x))/d)^m) - (9*a*b^2*Gamma[1 + m, ((2*I)*f*(c + d*x))/d])/(2^m*E^((2* 
I)*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m) - (I*b^3*E^((3*I)*(e - (c*f)/d))* 
Gamma[1 + m, ((-3*I)*f*(c + d*x))/d])/(3^m*(((-I)*f*(c + d*x))/d)^m) - (I* 
b^3*Gamma[1 + m, ((3*I)*f*(c + d*x))/d])/(3^m*E^((3*I)*(e - (c*f)/d))*((I* 
f*(c + d*x))/d)^m)))/f
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 607, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3798

\(\displaystyle \int \left (a^3 (c+d x)^m+3 a^2 b (c+d x)^m \sin (e+f x)+3 a b^2 (c+d x)^m \sin ^2(e+f x)+b^3 (c+d x)^m \sin ^3(e+f x)\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.72 \[ \int (c+d x)^m (a+b \sin (e+f x))^3 \, dx=-\frac {9 \, {\left ({\left (4 \, a^{2} b + b^{3}\right )} d m + {\left (4 \, a^{2} b + b^{3}\right )} d\right )} e^{\left (-\frac {d m \log \left (\frac {i \, f}{d}\right ) + i \, d e - i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {i \, d f x + i \, c f}{d}\right ) + 9 \, {\left (-i \, a b^{2} d m - i \, a b^{2} d\right )} e^{\left (-\frac {d m \log \left (-\frac {2 i \, f}{d}\right ) - 2 i \, d e + 2 i \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) - {\left (b^{3} d m + b^{3} d\right )} e^{\left (-\frac {d m \log \left (-\frac {3 i \, f}{d}\right ) - 3 i \, d e + 3 i \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) + 9 \, {\left ({\left (4 \, a^{2} b + b^{3}\right )} d m + {\left (4 \, a^{2} b + b^{3}\right )} d\right )} e^{\left (-\frac {d m \log \left (-\frac {i \, f}{d}\right ) - i \, d e + i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {-i \, d f x - i \, c f}{d}\right ) + 9 \, {\left (i \, a b^{2} d m + i \, a b^{2} d\right )} e^{\left (-\frac {d m \log \left (\frac {2 i \, f}{d}\right ) + 2 i \, d e - 2 i \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) - {\left (b^{3} d m + b^{3} d\right )} e^{\left (-\frac {d m \log \left (\frac {3 i \, f}{d}\right ) + 3 i \, d e - 3 i \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) - 12 \, {\left ({\left (2 \, a^{3} + 3 \, a b^{2}\right )} d f x + {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c f\right )} {\left (d x + c\right )}^{m}}{24 \, {\left (d f m + d f\right )}} \] Input:

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

Output:

-1/24*(9*((4*a^2*b + b^3)*d*m + (4*a^2*b + b^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*b^2*d*m - I*a 
*b^2*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) - (b^3*d*m + b^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) + 9*((4*a^2*b + b^3)*d*m + 
 (4*a^2*b + b^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*b^2*d*m + I*a*b^2*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) - (b^3*d*m + 
 b^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) - 12*((2*a^3 + 3*a*b^2)*d*f*x + (2*a^3 + 3*a*b^2)*c*f)*( 
d*x + c)^m)/(d*f*m + d*f)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^m (a+b \sin (e+f x))^3 \, dx=\text {too large to display} \] Input:

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

Output:

( - 15*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)**2*tan((e + f*x)/2)**6*b**3* 
c*d*f*m - 15*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)**2*tan((e + f*x)/2)**6 
*b**3*c*d*f - 45*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)**2*tan((e + f*x)/2 
)**4*b**3*c*d*f*m - 45*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)**2*tan((e + 
f*x)/2)**4*b**3*c*d*f - 45*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)**2*tan(( 
e + f*x)/2)**2*b**3*c*d*f*m - 45*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)**2 
*tan((e + f*x)/2)**2*b**3*c*d*f - 15*(c + d*x)**m*cos(e + f*x)*sin(e + f*x 
)**2*b**3*c*d*f*m - 15*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)**2*b**3*c*d* 
f - 108*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)*tan((e + f*x)/2)**6*a*b**2* 
c*d*f*m - 108*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)*tan((e + f*x)/2)**6*a 
*b**2*c*d*f - 324*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)*tan((e + f*x)/2)* 
*4*a*b**2*c*d*f*m - 324*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)*tan((e + f* 
x)/2)**4*a*b**2*c*d*f - 324*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)*tan((e 
+ f*x)/2)**2*a*b**2*c*d*f*m - 324*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)*t 
an((e + f*x)/2)**2*a*b**2*c*d*f - 108*(c + d*x)**m*cos(e + f*x)*sin(e + f* 
x)*a*b**2*c*d*f*m - 108*(c + d*x)**m*cos(e + f*x)*sin(e + f*x)*a*b**2*c*d* 
f - 135*(c + d*x)**m*cos(e + f*x)*tan((e + f*x)/2)**6*a**2*b*c*d*f*m - 135 
*(c + d*x)**m*cos(e + f*x)*tan((e + f*x)/2)**6*a**2*b*c*d*f - 30*(c + d*x) 
**m*cos(e + f*x)*tan((e + f*x)/2)**6*b**3*c*d*f*m - 30*(c + d*x)**m*cos(e 
+ f*x)*tan((e + f*x)/2)**6*b**3*c*d*f - 405*(c + d*x)**m*cos(e + f*x)*t...