3.92 \(\int (c+d x) \cos ^2(a+b x) \sin ^3(a+b x) \, dx\)

Optimal. Leaf size=109 \[ \frac{d \sin (a+b x)}{8 b^2}+\frac{d \sin (3 a+3 b x)}{144 b^2}-\frac{d \sin (5 a+5 b x)}{400 b^2}-\frac{(c+d x) \cos (a+b x)}{8 b}-\frac{(c+d x) \cos (3 a+3 b x)}{48 b}+\frac{(c+d x) \cos (5 a+5 b x)}{80 b} \]

[Out]

-((c + d*x)*Cos[a + b*x])/(8*b) - ((c + d*x)*Cos[3*a + 3*b*x])/(48*b) + ((c + d*x)*Cos[5*a + 5*b*x])/(80*b) +
(d*Sin[a + b*x])/(8*b^2) + (d*Sin[3*a + 3*b*x])/(144*b^2) - (d*Sin[5*a + 5*b*x])/(400*b^2)

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Rubi [A]  time = 0.0969841, antiderivative size = 109, 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, 3296, 2637} \[ \frac{d \sin (a+b x)}{8 b^2}+\frac{d \sin (3 a+3 b x)}{144 b^2}-\frac{d \sin (5 a+5 b x)}{400 b^2}-\frac{(c+d x) \cos (a+b x)}{8 b}-\frac{(c+d x) \cos (3 a+3 b x)}{48 b}+\frac{(c+d x) \cos (5 a+5 b x)}{80 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c + d*x)*Cos[a + b*x])/(8*b) - ((c + d*x)*Cos[3*a + 3*b*x])/(48*b) + ((c + d*x)*Cos[5*a + 5*b*x])/(80*b) +
(d*Sin[a + b*x])/(8*b^2) + (d*Sin[3*a + 3*b*x])/(144*b^2) - (d*Sin[5*a + 5*b*x])/(400*b^2)

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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (c+d x) \cos ^2(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac{1}{8} (c+d x) \sin (a+b x)+\frac{1}{16} (c+d x) \sin (3 a+3 b x)-\frac{1}{16} (c+d x) \sin (5 a+5 b x)\right ) \, dx\\ &=\frac{1}{16} \int (c+d x) \sin (3 a+3 b x) \, dx-\frac{1}{16} \int (c+d x) \sin (5 a+5 b x) \, dx+\frac{1}{8} \int (c+d x) \sin (a+b x) \, dx\\ &=-\frac{(c+d x) \cos (a+b x)}{8 b}-\frac{(c+d x) \cos (3 a+3 b x)}{48 b}+\frac{(c+d x) \cos (5 a+5 b x)}{80 b}-\frac{d \int \cos (5 a+5 b x) \, dx}{80 b}+\frac{d \int \cos (3 a+3 b x) \, dx}{48 b}+\frac{d \int \cos (a+b x) \, dx}{8 b}\\ &=-\frac{(c+d x) \cos (a+b x)}{8 b}-\frac{(c+d x) \cos (3 a+3 b x)}{48 b}+\frac{(c+d x) \cos (5 a+5 b x)}{80 b}+\frac{d \sin (a+b x)}{8 b^2}+\frac{d \sin (3 a+3 b x)}{144 b^2}-\frac{d \sin (5 a+5 b x)}{400 b^2}\\ \end{align*}

Mathematica [A]  time = 0.324346, size = 94, normalized size = 0.86 \[ \frac{-450 b (c+d x) \cos (a+b x)-75 b (c+d x) \cos (3 (a+b x))+45 b c \cos (5 (a+b x))+450 d \sin (a+b x)+25 d \sin (3 (a+b x))-9 d \sin (5 (a+b x))+45 b d x \cos (5 (a+b x))}{3600 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-450*b*(c + d*x)*Cos[a + b*x] - 75*b*(c + d*x)*Cos[3*(a + b*x)] + 45*b*c*Cos[5*(a + b*x)] + 45*b*d*x*Cos[5*(a
+ b*x)] + 450*d*Sin[a + b*x] + 25*d*Sin[3*(a + b*x)] - 9*d*Sin[5*(a + b*x)])/(3600*b^2)

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Maple [A]  time = 0.023, size = 163, normalized size = 1.5 \begin{align*}{\frac{1}{b} \left ({\frac{d}{b} \left ( -{\frac{ \left ( bx+a \right ) \left ( 2+ \left ( \sin \left ( bx+a \right ) \right ) ^{2} \right ) \cos \left ( bx+a \right ) }{3}}+{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{45}}+{\frac{2\,\sin \left ( bx+a \right ) }{15}}+{\frac{ \left ( bx+a \right ) \cos \left ( bx+a \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( bx+a \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) }-{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{5}}{25}} \right ) }-{\frac{ad}{b} \left ( -{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{5}}-{\frac{2\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{15}} \right ) }+c \left ( -{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{5}}-{\frac{2\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{15}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(d/b*(-1/3*(b*x+a)*(2+sin(b*x+a)^2)*cos(b*x+a)+1/45*sin(b*x+a)^3+2/15*sin(b*x+a)+1/5*(b*x+a)*(8/3+sin(b*x+
a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a)-1/25*sin(b*x+a)^5)-1/b*d*a*(-1/5*sin(b*x+a)^2*cos(b*x+a)^3-2/15*cos(b*x+a)^3
)+c*(-1/5*sin(b*x+a)^2*cos(b*x+a)^3-2/15*cos(b*x+a)^3))

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Maxima [A]  time = 1.23504, size = 188, normalized size = 1.72 \begin{align*} \frac{240 \,{\left (3 \, \cos \left (b x + a\right )^{5} - 5 \, \cos \left (b x + a\right )^{3}\right )} c - \frac{240 \,{\left (3 \, \cos \left (b x + a\right )^{5} - 5 \, \cos \left (b x + a\right )^{3}\right )} a d}{b} + \frac{{\left (45 \,{\left (b x + a\right )} \cos \left (5 \, b x + 5 \, a\right ) - 75 \,{\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) - 450 \,{\left (b x + a\right )} \cos \left (b x + a\right ) - 9 \, \sin \left (5 \, b x + 5 \, a\right ) + 25 \, \sin \left (3 \, b x + 3 \, a\right ) + 450 \, \sin \left (b x + a\right )\right )} d}{b}}{3600 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3600*(240*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*c - 240*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*a*d/b + (45*(b
*x + a)*cos(5*b*x + 5*a) - 75*(b*x + a)*cos(3*b*x + 3*a) - 450*(b*x + a)*cos(b*x + a) - 9*sin(5*b*x + 5*a) + 2
5*sin(3*b*x + 3*a) + 450*sin(b*x + a))*d/b)/b

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Fricas [A]  time = 0.490192, size = 194, normalized size = 1.78 \begin{align*} \frac{45 \,{\left (b d x + b c\right )} \cos \left (b x + a\right )^{5} - 75 \,{\left (b d x + b c\right )} \cos \left (b x + a\right )^{3} -{\left (9 \, d \cos \left (b x + a\right )^{4} - 13 \, d \cos \left (b x + a\right )^{2} - 26 \, d\right )} \sin \left (b x + a\right )}{225 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/225*(45*(b*d*x + b*c)*cos(b*x + a)^5 - 75*(b*d*x + b*c)*cos(b*x + a)^3 - (9*d*cos(b*x + a)^4 - 13*d*cos(b*x
+ a)^2 - 26*d)*sin(b*x + a))/b^2

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Sympy [A]  time = 9.73825, size = 163, normalized size = 1.5 \begin{align*} \begin{cases} - \frac{c \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac{2 c \cos ^{5}{\left (a + b x \right )}}{15 b} - \frac{d x \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac{2 d x \cos ^{5}{\left (a + b x \right )}}{15 b} + \frac{26 d \sin ^{5}{\left (a + b x \right )}}{225 b^{2}} + \frac{13 d \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{45 b^{2}} + \frac{2 d \sin{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{15 b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \sin ^{3}{\left (a \right )} \cos ^{2}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-c*sin(a + b*x)**2*cos(a + b*x)**3/(3*b) - 2*c*cos(a + b*x)**5/(15*b) - d*x*sin(a + b*x)**2*cos(a +
b*x)**3/(3*b) - 2*d*x*cos(a + b*x)**5/(15*b) + 26*d*sin(a + b*x)**5/(225*b**2) + 13*d*sin(a + b*x)**3*cos(a +
b*x)**2/(45*b**2) + 2*d*sin(a + b*x)*cos(a + b*x)**4/(15*b**2), Ne(b, 0)), ((c*x + d*x**2/2)*sin(a)**3*cos(a)
**2, True))

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Giac [A]  time = 1.20442, size = 143, normalized size = 1.31 \begin{align*} \frac{{\left (b d x + b c\right )} \cos \left (5 \, b x + 5 \, a\right )}{80 \, b^{2}} - \frac{{\left (b d x + b c\right )} \cos \left (3 \, b x + 3 \, a\right )}{48 \, b^{2}} - \frac{{\left (b d x + b c\right )} \cos \left (b x + a\right )}{8 \, b^{2}} - \frac{d \sin \left (5 \, b x + 5 \, a\right )}{400 \, b^{2}} + \frac{d \sin \left (3 \, b x + 3 \, a\right )}{144 \, b^{2}} + \frac{d \sin \left (b x + a\right )}{8 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80*(b*d*x + b*c)*cos(5*b*x + 5*a)/b^2 - 1/48*(b*d*x + b*c)*cos(3*b*x + 3*a)/b^2 - 1/8*(b*d*x + b*c)*cos(b*x
+ a)/b^2 - 1/400*d*sin(5*b*x + 5*a)/b^2 + 1/144*d*sin(3*b*x + 3*a)/b^2 + 1/8*d*sin(b*x + a)/b^2