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

Optimal. Leaf size=285 \[ -\frac{3\ 2^{-m-7} e^{2 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i b (c+d x)}{d}\right )}{b}+\frac{2^{-m-7} 3^{-m-1} e^{6 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{6 i b (c+d x)}{d}\right )}{b}-\frac{3\ 2^{-m-7} e^{-2 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i b (c+d x)}{d}\right )}{b}+\frac{2^{-m-7} 3^{-m-1} e^{-6 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{6 i b (c+d x)}{d}\right )}{b} \]

[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)

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Rubi [A]  time = 0.316992, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4406, 3308, 2181} \[ -\frac{3\ 2^{-m-7} e^{2 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i b (c+d x)}{d}\right )}{b}+\frac{2^{-m-7} 3^{-m-1} e^{6 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{6 i b (c+d x)}{d}\right )}{b}-\frac{3\ 2^{-m-7} e^{-2 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i b (c+d x)}{d}\right )}{b}+\frac{2^{-m-7} 3^{-m-1} e^{-6 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{6 i b (c+d x)}{d}\right )}{b} \]

Antiderivative was successfully verified.

[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 4406

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]

Rule 3308

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 2181

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

Rubi steps

\begin{align*} \int (c+d x)^m \cos ^3(a+b x) \sin ^3(a+b x) \, dx &=\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]  time = 3.41225, size = 255, normalized size = 0.89 \[ \frac{2^{-m-7} 3^{-m-1} e^{-\frac{6 i (a d+b c)}{d}} (c+d x)^m \left (\frac{b^2 (c+d x)^2}{d^2}\right )^{-m} \left (-3^{m+2} e^{4 i a+\frac{8 i b c}{d}} \left (-\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{2 i b (c+d x)}{d}\right )-3^{m+2} e^{4 i \left (2 a+\frac{b c}{d}\right )} \left (\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,-\frac{2 i b (c+d x)}{d}\right )+e^{12 i a} \left (\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,-\frac{6 i b (c+d x)}{d}\right )+e^{\frac{12 i b c}{d}} \left (-\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{6 i b (c+d x)}{d}\right )\right )}{b} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.264, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( \cos \left ( bx+a \right ) \right ) ^{3} \left ( \sin \left ( bx+a \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} \cos \left (b x + a\right )^{3} \sin \left (b x + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0.556983, size = 487, normalized size = 1.71 \begin{align*} \frac{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 i \, b d x + 6 i \, b c}{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 i \, b d x + 2 i \, b c}{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 i \, b d x - 2 i \, b c}{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 i \, b d x - 6 i \, b c}{d}\right )}{384 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(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 + 6*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 + 2*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 - 2*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 - 6*I*b*c)/d))/b

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} \cos \left (b x + a\right )^{3} \sin \left (b x + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)