\(\int (c+d x)^m \sin ^3(a+b x) \, dx\) [72]

Optimal result

Integrand size = 16, antiderivative size = 267 \[ \int (c+d x)^m \sin ^3(a+b x) \, dx=-\frac {3 e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )}{8 b}-\frac {3 e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )}{8 b}+\frac {3^{-1-m} e^{3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 i b (c+d x)}{d}\right )}{8 b}+\frac {3^{-1-m} e^{-3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 i b (c+d x)}{d}\right )}{8 b} \] Output:

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

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.94 \[ \int (c+d x)^m \sin ^3(a+b x) \, dx=\frac {3^{-1-m} e^{-\frac {3 i (b c+a d)}{d}} (c+d x)^m \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (-3^{2+m} e^{2 i \left (2 a+\frac {b c}{d}\right )} \left (\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )-3^{2+m} e^{2 i a+\frac {4 i b c}{d}} \left (-\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )+e^{6 i a} \left (\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {3 i b (c+d x)}{d}\right )+e^{\frac {6 i b c}{d}} \left (-\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {3 i b (c+d x)}{d}\right )\right )}{8 b} \] Input:

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

Output:

(3^(-1 - m)*(c + d*x)^m*(-(3^(2 + m)*E^((2*I)*(2*a + (b*c)/d))*((I*b*(c + 
d*x))/d)^m*Gamma[1 + m, ((-I)*b*(c + d*x))/d]) - 3^(2 + m)*E^((2*I)*a + (( 
4*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, (I*b*(c + d*x))/d] + E^ 
((6*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d] + E^( 
((6*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((3*I)*b*(c + d*x))/d 
]))/(8*b*E^(((3*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^m)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin (a+b x)^3 (c+d x)^mdx\)

\(\Big \downarrow \) 3793

\(\displaystyle \int \left (\frac {3}{4} \sin (a+b x) (c+d x)^m-\frac {1}{4} \sin (3 a+3 b x) (c+d x)^m\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.70 \[ \int (c+d x)^m \sin ^3(a+b x) \, dx=-\frac {9 \, e^{\left (-\frac {d m \log \left (\frac {i \, b}{d}\right ) - i \, b c + i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {i \, b d x + i \, b c}{d}\right ) - e^{\left (-\frac {d m \log \left (-\frac {3 i \, b}{d}\right ) + 3 i \, b c - 3 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (i \, b d x + i \, b c\right )}}{d}\right ) + 9 \, e^{\left (-\frac {d m \log \left (-\frac {i \, b}{d}\right ) + i \, b c - i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {-i \, b d x - i \, b c}{d}\right ) - e^{\left (-\frac {d m \log \left (\frac {3 i \, b}{d}\right ) - 3 i \, b c + 3 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )}{24 \, b} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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