\(\int (c+d x)^m \cos ^3(a+b x) \sin ^3(a+b x) \, dx\) [154]

Optimal result

Integrand size = 24, antiderivative size = 285 \[ \int (c+d x)^m \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {3\ 2^{-7-m} e^{2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 i b (c+d x)}{d}\right )}{b}-\frac {3\ 2^{-7-m} e^{-2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 i b (c+d x)}{d}\right )}{b}+\frac {2^{-7-m} 3^{-1-m} e^{6 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {6 i b (c+d x)}{d}\right )}{b}+\frac {2^{-7-m} 3^{-1-m} e^{-6 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {6 i b (c+d x)}{d}\right )}{b} \]

[Out]

-3*2^(-7-m)*exp(2*I*(a-b*c/d))*(d*x+c)^m*GAMMA(1+m,-2*I*b*(d*x+c)/d)/b/((-I*b*(d*x+c)/d)^m)-3*2^(-7-m)*(d*x+c)
^m*GAMMA(1+m,2*I*b*(d*x+c)/d)/b/exp(2*I*(a-b*c/d))/((I*b*(d*x+c)/d)^m)+2^(-7-m)*3^(-1-m)*exp(6*I*(a-b*c/d))*(d
*x+c)^m*GAMMA(1+m,-6*I*b*(d*x+c)/d)/b/((-I*b*(d*x+c)/d)^m)+2^(-7-m)*3^(-1-m)*(d*x+c)^m*GAMMA(1+m,6*I*b*(d*x+c)
/d)/b/exp(6*I*(a-b*c/d))/((I*b*(d*x+c)/d)^m)

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3389, 2212} \[ \int (c+d x)^m \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {3\ 2^{-m-7} e^{2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 i b (c+d x)}{d}\right )}{b}+\frac {2^{-m-7} 3^{-m-1} e^{6 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {6 i b (c+d x)}{d}\right )}{b}-\frac {3\ 2^{-m-7} e^{-2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 i b (c+d x)}{d}\right )}{b}+\frac {2^{-m-7} 3^{-m-1} e^{-6 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {6 i b (c+d x)}{d}\right )}{b} \]

[In]

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

[Out]

(-3*2^(-7 - m)*E^((2*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-2*I)*b*(c + d*x))/d])/(b*(((-I)*b*(c + d*x)
)/d)^m) - (3*2^(-7 - m)*(c + d*x)^m*Gamma[1 + m, ((2*I)*b*(c + d*x))/d])/(b*E^((2*I)*(a - (b*c)/d))*((I*b*(c +
d*x))/d)^m) + (2^(-7 - m)*3^(-1 - m)*E^((6*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-6*I)*b*(c + d*x))/d]
)/(b*(((-I)*b*(c + d*x))/d)^m) + (2^(-7 - m)*3^(-1 - m)*(c + d*x)^m*Gamma[1 + m, ((6*I)*b*(c + d*x))/d])/(b*E^
((6*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m)

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{32} (c+d x)^m \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^m \sin (6 a+6 b x)\right ) \, dx \\ & = -\left (\frac {1}{32} \int (c+d x)^m \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int (c+d x)^m \sin (2 a+2 b x) \, dx \\ & = -\left (\frac {1}{64} i \int e^{-i (6 a+6 b x)} (c+d x)^m \, dx\right )+\frac {1}{64} i \int e^{i (6 a+6 b x)} (c+d x)^m \, dx+\frac {3}{64} i \int e^{-i (2 a+2 b x)} (c+d x)^m \, dx-\frac {3}{64} i \int e^{i (2 a+2 b x)} (c+d x)^m \, dx \\ & = -\frac {3\ 2^{-7-m} e^{2 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 {2 i b (c+d x)}{d}\right )}{b}-\frac {3\ 2^{-7-m} e^{-2 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 {2 i b (c+d x)}{d}\right )}{b}+\frac {2^{-7-m} 3^{-1-m} e^{6 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 {6 i b (c+d x)}{d}\right )}{b}+\frac {2^{-7-m} 3^{-1-m} e^{-6 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 {6 i b (c+d x)}{d}\right )}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.27 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.89 \[ \int (c+d x)^m \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {2^{-7-m} 3^{-1-m} e^{-\frac {6 i (b c+a d)}{d}} (c+d x)^m \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (-3^{2+m} e^{4 i \left (2 a+\frac {b c}{d}\right )} \left (\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {2 i b (c+d x)}{d}\right )-3^{2+m} e^{4 i a+\frac {8 i b c}{d}} \left (-\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {2 i b (c+d x)}{d}\right )+e^{12 i a} \left (\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {6 i b (c+d x)}{d}\right )+e^{\frac {12 i b c}{d}} \left (-\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {6 i b (c+d x)}{d}\right )\right )}{b} \]

[In]

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

[Out]

(2^(-7 - m)*3^(-1 - m)*(c + d*x)^m*(-(3^(2 + m)*E^((4*I)*(2*a + (b*c)/d))*((I*b*(c + d*x))/d)^m*Gamma[1 + m, (
(-2*I)*b*(c + d*x))/d]) - 3^(2 + m)*E^((4*I)*a + ((8*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((2*I)*b
*(c + d*x))/d] + E^((12*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-6*I)*b*(c + d*x))/d] + E^(((12*I)*b*c)/d)*
(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((6*I)*b*(c + d*x))/d]))/(b*E^(((6*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)
/d^2)^m)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.67 \[ \int (c+d x)^m \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {9 \, e^{\left (-\frac {d m \log \left (-\frac {2 i \, b}{d}\right ) + 2 i \, b c - 2 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (i \, b d x + i \, b c\right )}}{d}\right ) - e^{\left (-\frac {d m \log \left (-\frac {6 i \, b}{d}\right ) + 6 i \, b c - 6 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {6 \, {\left (i \, b d x + i \, b c\right )}}{d}\right ) + 9 \, e^{\left (-\frac {d m \log \left (\frac {2 i \, b}{d}\right ) - 2 i \, b c + 2 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right ) - e^{\left (-\frac {d m \log \left (\frac {6 i \, b}{d}\right ) - 6 i \, b c + 6 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {6 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )}{384 \, b} \]

[In]

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

[Out]

-1/384*(9*e^(-(d*m*log(-2*I*b/d) + 2*I*b*c - 2*I*a*d)/d)*gamma(m + 1, -2*(I*b*d*x + I*b*c)/d) - e^(-(d*m*log(-
6*I*b/d) + 6*I*b*c - 6*I*a*d)/d)*gamma(m + 1, -6*(I*b*d*x + I*b*c)/d) + 9*e^(-(d*m*log(2*I*b/d) - 2*I*b*c + 2*
I*a*d)/d)*gamma(m + 1, -2*(-I*b*d*x - I*b*c)/d) - e^(-(d*m*log(6*I*b/d) - 6*I*b*c + 6*I*a*d)/d)*gamma(m + 1, -
6*(-I*b*d*x - I*b*c)/d))/b

Sympy [F(-2)]

Exception generated. \[ \int (c+d x)^m \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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