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

Optimal. Leaf size=271 \[ -\frac{2^{-m-4} e^{2 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i b (c+d x)}{d}\right )}{b}+\frac{2^{-2 (m+3)} e^{4 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{4 i b (c+d x)}{d}\right )}{b}-\frac{2^{-m-4} e^{-2 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i b (c+d x)}{d}\right )}{b}+\frac{2^{-2 (m+3)} e^{-4 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{4 i b (c+d x)}{d}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.33105, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4406, 3308, 2181} \[ -\frac{2^{-m-4} e^{2 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i b (c+d x)}{d}\right )}{b}+\frac{2^{-2 (m+3)} e^{4 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{4 i b (c+d x)}{d}\right )}{b}-\frac{2^{-m-4} e^{-2 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i b (c+d x)}{d}\right )}{b}+\frac{2^{-2 (m+3)} e^{-4 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{4 i b (c+d x)}{d}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2^(-4 - 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)) - (2^(-4 - 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) + (E^((4*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-4*I)*b*(c + d*x))/d])/(2^(2*(3 + m))*b*(((-I
)*b*(c + d*x))/d)^m) + ((c + d*x)^m*Gamma[1 + m, ((4*I)*b*(c + d*x))/d])/(2^(2*(3 + m))*b*E^((4*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 (a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac{1}{4} (c+d x)^m \sin (2 a+2 b x)-\frac{1}{8} (c+d x)^m \sin (4 a+4 b x)\right ) \, dx\\ &=-\left (\frac{1}{8} \int (c+d x)^m \sin (4 a+4 b x) \, dx\right )+\frac{1}{4} \int (c+d x)^m \sin (2 a+2 b x) \, dx\\ &=-\left (\frac{1}{16} i \int e^{-i (4 a+4 b x)} (c+d x)^m \, dx\right )+\frac{1}{16} i \int e^{i (4 a+4 b x)} (c+d x)^m \, dx+\frac{1}{8} i \int e^{-i (2 a+2 b x)} (c+d x)^m \, dx-\frac{1}{8} i \int e^{i (2 a+2 b x)} (c+d x)^m \, dx\\ &=-\frac{2^{-4-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^{-4-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{4^{-3-m} e^{4 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{4 i b (c+d x)}{d}\right )}{b}+\frac{4^{-3-m} e^{-4 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{4 i b (c+d x)}{d}\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.299893, size = 246, normalized size = 0.91 \[ \frac{4^{-m-3} e^{-\frac{4 i (a d+b c)}{d}} (c+d x)^m \left (\frac{b^2 (c+d x)^2}{d^2}\right )^{-m} \left (-2^{m+2} e^{2 i \left (a+\frac{3 b c}{d}\right )} \left (-\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{2 i b (c+d x)}{d}\right )-2^{m+2} e^{2 i \left (3 a+\frac{b c}{d}\right )} \left (\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,-\frac{2 i b (c+d x)}{d}\right )+e^{8 i a} \left (\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,-\frac{4 i b (c+d x)}{d}\right )+e^{\frac{8 i b c}{d}} \left (-\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{4 i b (c+d x)}{d}\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.265, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m}\cos \left ( bx+a \right ) \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)*sin(b*x+a)^3,x)

[Out]

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

[Out]

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

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Fricas [A]  time = 0.551049, size = 486, normalized size = 1.79 \begin{align*} \frac{e^{\left (-\frac{d m \log \left (\frac{4 i \, b}{d}\right ) - 4 i \, b c + 4 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac{4 i \, b d x + 4 i \, b c}{d}\right ) - 4 \, 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 ) - 4 \, 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{4 i \, b}{d}\right ) + 4 i \, b c - 4 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac{-4 i \, b d x - 4 i \, b c}{d}\right )}{64 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(e^(-(d*m*log(4*I*b/d) - 4*I*b*c + 4*I*a*d)/d)*gamma(m + 1, (4*I*b*d*x + 4*I*b*c)/d) - 4*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) - 4*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(-4*I*b/d) + 4*I*b*c - 4*I*a*d)/d)*gamma(m + 1,
(-4*I*b*d*x - 4*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)*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 ) \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)*sin(b*x+a)^3,x, algorithm="giac")

[Out]

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