\(\int (c+d x)^m \cos ^3(a+b x) \sin ^2(a+b x) \, dx\) [145]

Optimal result

Integrand size = 24, antiderivative size = 419 \[ \int (c+d x)^m \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {i e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )}{16 b}+\frac {i e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )}{16 b}+\frac {i 3^{-1-m} e^{3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 i b (c+d x)}{d}\right )}{32 b}-\frac {i 3^{-1-m} e^{-3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 i b (c+d x)}{d}\right )}{32 b}+\frac {i 5^{-1-m} e^{5 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {5 i b (c+d x)}{d}\right )}{32 b}-\frac {i 5^{-1-m} e^{-5 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {5 i b (c+d x)}{d}\right )}{32 b} \]

[Out]

-1/16*I*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)+1/16*I*(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)+1/32*I*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)-1/32*I*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)+1/32*I*5^(-1-m)*exp(5*I*(a-b*c/d))*(d*x+c)^m*GAMMA(1+m,-5*I*b*(d*x+c)/d)/b/((-I*b*(d*
x+c)/d)^m)-1/32*I*5^(-1-m)*(d*x+c)^m*GAMMA(1+m,5*I*b*(d*x+c)/d)/b/exp(5*I*(a-b*c/d))/((I*b*(d*x+c)/d)^m)

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3388, 2212} \[ \int (c+d x)^m \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {i e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {i b (c+d x)}{d}\right )}{16 b}+\frac {i 3^{-m-1} e^{3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {3 i b (c+d x)}{d}\right )}{32 b}+\frac {i 5^{-m-1} e^{5 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {5 i b (c+d x)}{d}\right )}{32 b}+\frac {i e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {i b (c+d x)}{d}\right )}{16 b}-\frac {i 3^{-m-1} e^{-3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {3 i b (c+d x)}{d}\right )}{32 b}-\frac {i 5^{-m-1} e^{-5 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {5 i b (c+d x)}{d}\right )}{32 b} \]

[In]

Int[(c + d*x)^m*Cos[a + b*x]^3*Sin[a + b*x]^2,x]

[Out]

((-1/16*I)*E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(b*(((-I)*b*(c + d*x))/d)^m) +
((I/16)*(c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(b*E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) + ((I/32)*3
^(-1 - m)*E^((3*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d])/(b*(((-I)*b*(c + d*x))/d)^
m) - ((I/32)*3^(-1 - m)*(c + d*x)^m*Gamma[1 + m, ((3*I)*b*(c + d*x))/d])/(b*E^((3*I)*(a - (b*c)/d))*((I*b*(c +
d*x))/d)^m) + ((I/32)*5^(-1 - m)*E^((5*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-5*I)*b*(c + d*x))/d])/(b
*(((-I)*b*(c + d*x))/d)^m) - ((I/32)*5^(-1 - m)*(c + d*x)^m*Gamma[1 + m, ((5*I)*b*(c + d*x))/d])/(b*E^((5*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 3388

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

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 {1}{8} (c+d x)^m \cos (a+b x)-\frac {1}{16} (c+d x)^m \cos (3 a+3 b x)-\frac {1}{16} (c+d x)^m \cos (5 a+5 b x)\right ) \, dx \\ & = -\left (\frac {1}{16} \int (c+d x)^m \cos (3 a+3 b x) \, dx\right )-\frac {1}{16} \int (c+d x)^m \cos (5 a+5 b x) \, dx+\frac {1}{8} \int (c+d x)^m \cos (a+b x) \, dx \\ & = -\left (\frac {1}{32} \int e^{-i (3 a+3 b x)} (c+d x)^m \, dx\right )-\frac {1}{32} \int e^{i (3 a+3 b x)} (c+d x)^m \, dx-\frac {1}{32} \int e^{-i (5 a+5 b x)} (c+d x)^m \, dx-\frac {1}{32} \int e^{i (5 a+5 b x)} (c+d x)^m \, dx+\frac {1}{16} \int e^{-i (a+b x)} (c+d x)^m \, dx+\frac {1}{16} \int e^{i (a+b x)} (c+d x)^m \, dx \\ & = -\frac {i 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 )}{16 b}+\frac {i 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 )}{16 b}+\frac {i 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 )}{32 b}-\frac {i 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 )}{32 b}+\frac {i 5^{-1-m} e^{5 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 {5 i b (c+d x)}{d}\right )}{32 b}-\frac {i 5^{-1-m} e^{-5 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 {5 i b (c+d x)}{d}\right )}{32 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.98 \[ \int (c+d x)^m \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {i e^{-\frac {i (b c+a d)}{d}} (c+d x)^m \left (e^{2 i a} \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )-e^{\frac {2 i b c}{d}} \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )\right )}{16 b}-\frac {i 3^{-1-m} e^{-\frac {3 i (b c+a d)}{d}} (c+d x)^m \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (-e^{6 i a} \left (\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {3 i b (c+d x)}{d}\right )+e^{\frac {6 i b c}{d}} \left (-\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {3 i b (c+d x)}{d}\right )\right )}{32 b}-\frac {i 5^{-1-m} e^{-\frac {5 i (b c+a d)}{d}} (c+d x)^m \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (-e^{10 i a} \left (\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {5 i b (c+d x)}{d}\right )+e^{\frac {10 i b c}{d}} \left (-\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {5 i b (c+d x)}{d}\right )\right )}{32 b} \]

[In]

Integrate[(c + d*x)^m*Cos[a + b*x]^3*Sin[a + b*x]^2,x]

[Out]

((-1/16*I)*(c + d*x)^m*((E^((2*I)*a)*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^m - (E^(((2*I)
*b*c)/d)*Gamma[1 + m, (I*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^m))/(b*E^((I*(b*c + a*d))/d)) - ((I/32)*3^(-1 -
m)*(c + d*x)^m*(-(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]))/(b*E^(((3*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)
/d^2)^m) - ((I/32)*5^(-1 - m)*(c + d*x)^m*(-(E^((10*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-5*I)*b*(c + d*
x))/d]) + E^(((10*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((5*I)*b*(c + d*x))/d]))/(b*E^(((5*I)*(b*c
+ a*d))/d)*((b^2*(c + d*x)^2)/d^2)^m)

Maple [F]

\[\int \left (d x +c \right )^{m} \cos \left (x b +a \right )^{3} \sin \left (x b +a \right )^{2}d x\]

[In]

int((d*x+c)^m*cos(b*x+a)^3*sin(b*x+a)^2,x)

[Out]

int((d*x+c)^m*cos(b*x+a)^3*sin(b*x+a)^2,x)

Fricas [A] (verification not implemented)

none

Time = 0.10 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.67 \[ \int (c+d x)^m \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\frac {30 i \, e^{\left (-\frac {d m \log \left (\frac {i \, b}{d}\right ) - i \, b c + i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {i \, b d x + i \, b c}{d}\right ) + 5 i \, e^{\left (-\frac {d m \log \left (-\frac {3 i \, b}{d}\right ) + 3 i \, b c - 3 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (i \, b d x + i \, b c\right )}}{d}\right ) + 3 i \, e^{\left (-\frac {d m \log \left (-\frac {5 i \, b}{d}\right ) + 5 i \, b c - 5 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {5 \, {\left (i \, b d x + i \, b c\right )}}{d}\right ) - 30 i \, e^{\left (-\frac {d m \log \left (-\frac {i \, b}{d}\right ) + i \, b c - i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {-i \, b d x - i \, b c}{d}\right ) - 5 i \, e^{\left (-\frac {d m \log \left (\frac {3 i \, b}{d}\right ) - 3 i \, b c + 3 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right ) - 3 i \, e^{\left (-\frac {d m \log \left (\frac {5 i \, b}{d}\right ) - 5 i \, b c + 5 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {5 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )}{480 \, b} \]

[In]

integrate((d*x+c)^m*cos(b*x+a)^3*sin(b*x+a)^2,x, algorithm="fricas")

[Out]

1/480*(30*I*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) + 5*I*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) + 3*I*e^(-(d*m*log(-5*I*b/d) + 5*I*b*c - 5*I
*a*d)/d)*gamma(m + 1, -5*(I*b*d*x + I*b*c)/d) - 30*I*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) - 5*I*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) -
3*I*e^(-(d*m*log(5*I*b/d) - 5*I*b*c + 5*I*a*d)/d)*gamma(m + 1, -5*(-I*b*d*x - I*b*c)/d))/b

Sympy [F(-2)]

Exception generated. \[ \int (c+d x)^m \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate((d*x+c)**m*cos(b*x+a)**3*sin(b*x+a)**2,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

integrate((d*x+c)^m*cos(b*x+a)^3*sin(b*x+a)^2,x, algorithm="maxima")

[Out]

integrate((d*x + c)^m*cos(b*x + a)^3*sin(b*x + a)^2, x)

Giac [F]

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

[In]

integrate((d*x+c)^m*cos(b*x+a)^3*sin(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^m*cos(b*x + a)^3*sin(b*x + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^m \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^m \,d x \]

[In]

int(cos(a + b*x)^3*sin(a + b*x)^2*(c + d*x)^m,x)

[Out]

int(cos(a + b*x)^3*sin(a + b*x)^2*(c + d*x)^m, x)