3.43 \(\int e^{a+b x} \cos ^2(c+d x) \sin ^3(c+d x) \, dx\)

Optimal. Leaf size=183 \[ \frac{b e^{a+b x} \sin (c+d x)}{8 \left (b^2+d^2\right )}+\frac{b e^{a+b x} \sin (3 c+3 d x)}{16 \left (b^2+9 d^2\right )}-\frac{b e^{a+b x} \sin (5 c+5 d x)}{16 \left (b^2+25 d^2\right )}-\frac{d e^{a+b x} \cos (c+d x)}{8 \left (b^2+d^2\right )}-\frac{3 d e^{a+b x} \cos (3 c+3 d x)}{16 \left (b^2+9 d^2\right )}+\frac{5 d e^{a+b x} \cos (5 c+5 d x)}{16 \left (b^2+25 d^2\right )} \]

[Out]

-(d*E^(a + b*x)*Cos[c + d*x])/(8*(b^2 + d^2)) - (3*d*E^(a + b*x)*Cos[3*c + 3*d*x])/(16*(b^2 + 9*d^2)) + (5*d*E
^(a + b*x)*Cos[5*c + 5*d*x])/(16*(b^2 + 25*d^2)) + (b*E^(a + b*x)*Sin[c + d*x])/(8*(b^2 + d^2)) + (b*E^(a + b*
x)*Sin[3*c + 3*d*x])/(16*(b^2 + 9*d^2)) - (b*E^(a + b*x)*Sin[5*c + 5*d*x])/(16*(b^2 + 25*d^2))

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Rubi [A]  time = 0.125889, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4469, 4432} \[ \frac{b e^{a+b x} \sin (c+d x)}{8 \left (b^2+d^2\right )}+\frac{b e^{a+b x} \sin (3 c+3 d x)}{16 \left (b^2+9 d^2\right )}-\frac{b e^{a+b x} \sin (5 c+5 d x)}{16 \left (b^2+25 d^2\right )}-\frac{d e^{a+b x} \cos (c+d x)}{8 \left (b^2+d^2\right )}-\frac{3 d e^{a+b x} \cos (3 c+3 d x)}{16 \left (b^2+9 d^2\right )}+\frac{5 d e^{a+b x} \cos (5 c+5 d x)}{16 \left (b^2+25 d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*E^(a + b*x)*Cos[c + d*x])/(8*(b^2 + d^2)) - (3*d*E^(a + b*x)*Cos[3*c + 3*d*x])/(16*(b^2 + 9*d^2)) + (5*d*E
^(a + b*x)*Cos[5*c + 5*d*x])/(16*(b^2 + 25*d^2)) + (b*E^(a + b*x)*Sin[c + d*x])/(8*(b^2 + d^2)) + (b*E^(a + b*
x)*Sin[3*c + 3*d*x])/(16*(b^2 + 9*d^2)) - (b*E^(a + b*x)*Sin[5*c + 5*d*x])/(16*(b^2 + 25*d^2))

Rule 4469

Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :
> Int[ExpandTrigReduce[F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g
}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 4432

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*S
in[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] - Simp[(e*F^(c*(a + b*x))*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rubi steps

\begin{align*} \int e^{a+b x} \cos ^2(c+d x) \sin ^3(c+d x) \, dx &=\int \left (\frac{1}{8} e^{a+b x} \sin (c+d x)+\frac{1}{16} e^{a+b x} \sin (3 c+3 d x)-\frac{1}{16} e^{a+b x} \sin (5 c+5 d x)\right ) \, dx\\ &=\frac{1}{16} \int e^{a+b x} \sin (3 c+3 d x) \, dx-\frac{1}{16} \int e^{a+b x} \sin (5 c+5 d x) \, dx+\frac{1}{8} \int e^{a+b x} \sin (c+d x) \, dx\\ &=-\frac{d e^{a+b x} \cos (c+d x)}{8 \left (b^2+d^2\right )}-\frac{3 d e^{a+b x} \cos (3 c+3 d x)}{16 \left (b^2+9 d^2\right )}+\frac{5 d e^{a+b x} \cos (5 c+5 d x)}{16 \left (b^2+25 d^2\right )}+\frac{b e^{a+b x} \sin (c+d x)}{8 \left (b^2+d^2\right )}+\frac{b e^{a+b x} \sin (3 c+3 d x)}{16 \left (b^2+9 d^2\right )}-\frac{b e^{a+b x} \sin (5 c+5 d x)}{16 \left (b^2+25 d^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.913075, size = 110, normalized size = 0.6 \[ \frac{1}{16} e^{a+b x} \left (\frac{2 (b \sin (c+d x)-d \cos (c+d x))}{b^2+d^2}+\frac{b \sin (3 (c+d x))-3 d \cos (3 (c+d x))}{b^2+9 d^2}+\frac{5 d \cos (5 (c+d x))-b \sin (5 (c+d x))}{b^2+25 d^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(a + b*x)*((2*(-(d*Cos[c + d*x]) + b*Sin[c + d*x]))/(b^2 + d^2) + (-3*d*Cos[3*(c + d*x)] + b*Sin[3*(c + d*x
)])/(b^2 + 9*d^2) + (5*d*Cos[5*(c + d*x)] - b*Sin[5*(c + d*x)])/(b^2 + 25*d^2)))/16

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Maple [A]  time = 0.021, size = 166, normalized size = 0.9 \begin{align*} -{\frac{d{{\rm e}^{bx+a}}\cos \left ( dx+c \right ) }{8\,{b}^{2}+8\,{d}^{2}}}-{\frac{3\,d{{\rm e}^{bx+a}}\cos \left ( 3\,dx+3\,c \right ) }{16\,{b}^{2}+144\,{d}^{2}}}+{\frac{5\,d{{\rm e}^{bx+a}}\cos \left ( 5\,dx+5\,c \right ) }{16\,{b}^{2}+400\,{d}^{2}}}+{\frac{b{{\rm e}^{bx+a}}\sin \left ( dx+c \right ) }{8\,{b}^{2}+8\,{d}^{2}}}+{\frac{b{{\rm e}^{bx+a}}\sin \left ( 3\,dx+3\,c \right ) }{16\,{b}^{2}+144\,{d}^{2}}}-{\frac{b{{\rm e}^{bx+a}}\sin \left ( 5\,dx+5\,c \right ) }{16\,{b}^{2}+400\,{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*d*exp(b*x+a)*cos(d*x+c)/(b^2+d^2)-3/16*d*exp(b*x+a)*cos(3*d*x+3*c)/(b^2+9*d^2)+5/16*d*exp(b*x+a)*cos(5*d*
x+5*c)/(b^2+25*d^2)+1/8*b*exp(b*x+a)*sin(d*x+c)/(b^2+d^2)+1/16*b*exp(b*x+a)*sin(3*d*x+3*c)/(b^2+9*d^2)-1/16*b*
exp(b*x+a)*sin(5*d*x+5*c)/(b^2+25*d^2)

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Maxima [B]  time = 1.29158, size = 1550, normalized size = 8.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*((5*b^4*d*cos(5*c)*e^a + 50*b^2*d^3*cos(5*c)*e^a + 45*d^5*cos(5*c)*e^a - b^5*e^a*sin(5*c) - 10*b^3*d^2*e^
a*sin(5*c) - 9*b*d^4*e^a*sin(5*c))*cos(5*d*x)*e^(b*x) + (5*b^4*d*cos(5*c)*e^a + 50*b^2*d^3*cos(5*c)*e^a + 45*d
^5*cos(5*c)*e^a + b^5*e^a*sin(5*c) + 10*b^3*d^2*e^a*sin(5*c) + 9*b*d^4*e^a*sin(5*c))*cos(5*d*x + 10*c)*e^(b*x)
- (3*b^4*d*cos(5*c)*e^a + 78*b^2*d^3*cos(5*c)*e^a + 75*d^5*cos(5*c)*e^a + b^5*e^a*sin(5*c) + 26*b^3*d^2*e^a*s
in(5*c) + 25*b*d^4*e^a*sin(5*c))*cos(3*d*x + 8*c)*e^(b*x) - (3*b^4*d*cos(5*c)*e^a + 78*b^2*d^3*cos(5*c)*e^a +
75*d^5*cos(5*c)*e^a - b^5*e^a*sin(5*c) - 26*b^3*d^2*e^a*sin(5*c) - 25*b*d^4*e^a*sin(5*c))*cos(3*d*x - 2*c)*e^(
b*x) - 2*(b^4*d*cos(5*c)*e^a + 34*b^2*d^3*cos(5*c)*e^a + 225*d^5*cos(5*c)*e^a + b^5*e^a*sin(5*c) + 34*b^3*d^2*
e^a*sin(5*c) + 225*b*d^4*e^a*sin(5*c))*cos(d*x + 6*c)*e^(b*x) - 2*(b^4*d*cos(5*c)*e^a + 34*b^2*d^3*cos(5*c)*e^
a + 225*d^5*cos(5*c)*e^a - b^5*e^a*sin(5*c) - 34*b^3*d^2*e^a*sin(5*c) - 225*b*d^4*e^a*sin(5*c))*cos(d*x - 4*c)
*e^(b*x) - (b^5*cos(5*c)*e^a + 10*b^3*d^2*cos(5*c)*e^a + 9*b*d^4*cos(5*c)*e^a + 5*b^4*d*e^a*sin(5*c) + 50*b^2*
d^3*e^a*sin(5*c) + 45*d^5*e^a*sin(5*c))*e^(b*x)*sin(5*d*x) - (b^5*cos(5*c)*e^a + 10*b^3*d^2*cos(5*c)*e^a + 9*b
*d^4*cos(5*c)*e^a - 5*b^4*d*e^a*sin(5*c) - 50*b^2*d^3*e^a*sin(5*c) - 45*d^5*e^a*sin(5*c))*e^(b*x)*sin(5*d*x +
10*c) + (b^5*cos(5*c)*e^a + 26*b^3*d^2*cos(5*c)*e^a + 25*b*d^4*cos(5*c)*e^a - 3*b^4*d*e^a*sin(5*c) - 78*b^2*d^
3*e^a*sin(5*c) - 75*d^5*e^a*sin(5*c))*e^(b*x)*sin(3*d*x + 8*c) + (b^5*cos(5*c)*e^a + 26*b^3*d^2*cos(5*c)*e^a +
25*b*d^4*cos(5*c)*e^a + 3*b^4*d*e^a*sin(5*c) + 78*b^2*d^3*e^a*sin(5*c) + 75*d^5*e^a*sin(5*c))*e^(b*x)*sin(3*d
*x - 2*c) + 2*(b^5*cos(5*c)*e^a + 34*b^3*d^2*cos(5*c)*e^a + 225*b*d^4*cos(5*c)*e^a - b^4*d*e^a*sin(5*c) - 34*b
^2*d^3*e^a*sin(5*c) - 225*d^5*e^a*sin(5*c))*e^(b*x)*sin(d*x + 6*c) + 2*(b^5*cos(5*c)*e^a + 34*b^3*d^2*cos(5*c)
*e^a + 225*b*d^4*cos(5*c)*e^a + b^4*d*e^a*sin(5*c) + 34*b^2*d^3*e^a*sin(5*c) + 225*d^5*e^a*sin(5*c))*e^(b*x)*s
in(d*x - 4*c))/(b^6*cos(5*c)^2 + b^6*sin(5*c)^2 + 225*(cos(5*c)^2 + sin(5*c)^2)*d^6 + 259*(b^2*cos(5*c)^2 + b^
2*sin(5*c)^2)*d^4 + 35*(b^4*cos(5*c)^2 + b^4*sin(5*c)^2)*d^2)

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Fricas [A]  time = 0.511893, size = 454, normalized size = 2.48 \begin{align*} \frac{{\left (2 \, b^{3} d^{2} + 26 \, b d^{4} -{\left (b^{5} + 10 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cos \left (d x + c\right )^{4} +{\left (b^{5} + 14 \, b^{3} d^{2} + 13 \, b d^{4}\right )} \cos \left (d x + c\right )^{2}\right )} e^{\left (b x + a\right )} \sin \left (d x + c\right ) +{\left (5 \,{\left (b^{4} d + 10 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cos \left (d x + c\right )^{5} -{\left (7 \, b^{4} d + 82 \, b^{2} d^{3} + 75 \, d^{5}\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (b^{4} d + 13 \, b^{2} d^{3}\right )} \cos \left (d x + c\right )\right )} e^{\left (b x + a\right )}}{b^{6} + 35 \, b^{4} d^{2} + 259 \, b^{2} d^{4} + 225 \, d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*b^3*d^2 + 26*b*d^4 - (b^5 + 10*b^3*d^2 + 9*b*d^4)*cos(d*x + c)^4 + (b^5 + 14*b^3*d^2 + 13*b*d^4)*cos(d*x +
c)^2)*e^(b*x + a)*sin(d*x + c) + (5*(b^4*d + 10*b^2*d^3 + 9*d^5)*cos(d*x + c)^5 - (7*b^4*d + 82*b^2*d^3 + 75*
d^5)*cos(d*x + c)^3 + 2*(b^4*d + 13*b^2*d^3)*cos(d*x + c))*e^(b*x + a))/(b^6 + 35*b^4*d^2 + 259*b^2*d^4 + 225*
d^6)

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

[Out]

Timed out

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Giac [A]  time = 1.15763, size = 209, normalized size = 1.14 \begin{align*} \frac{1}{16} \,{\left (\frac{5 \, d \cos \left (5 \, d x + 5 \, c\right )}{b^{2} + 25 \, d^{2}} - \frac{b \sin \left (5 \, d x + 5 \, c\right )}{b^{2} + 25 \, d^{2}}\right )} e^{\left (b x + a\right )} - \frac{1}{16} \,{\left (\frac{3 \, d \cos \left (3 \, d x + 3 \, c\right )}{b^{2} + 9 \, d^{2}} - \frac{b \sin \left (3 \, d x + 3 \, c\right )}{b^{2} + 9 \, d^{2}}\right )} e^{\left (b x + a\right )} - \frac{1}{8} \,{\left (\frac{d \cos \left (d x + c\right )}{b^{2} + d^{2}} - \frac{b \sin \left (d x + c\right )}{b^{2} + d^{2}}\right )} e^{\left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(5*d*cos(5*d*x + 5*c)/(b^2 + 25*d^2) - b*sin(5*d*x + 5*c)/(b^2 + 25*d^2))*e^(b*x + a) - 1/16*(3*d*cos(3*d
*x + 3*c)/(b^2 + 9*d^2) - b*sin(3*d*x + 3*c)/(b^2 + 9*d^2))*e^(b*x + a) - 1/8*(d*cos(d*x + c)/(b^2 + d^2) - b*
sin(d*x + c)/(b^2 + d^2))*e^(b*x + a)