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

Optimal. Leaf size=79 \[ -\frac{d e^{a+b x} \sin (4 c+4 d x)}{2 \left (b^2+16 d^2\right )}-\frac{b e^{a+b x} \cos (4 c+4 d x)}{8 \left (b^2+16 d^2\right )}+\frac{e^{a+b x}}{8 b} \]

[Out]

E^(a + b*x)/(8*b) - (b*E^(a + b*x)*Cos[4*c + 4*d*x])/(8*(b^2 + 16*d^2)) - (d*E^(a + b*x)*Sin[4*c + 4*d*x])/(2*
(b^2 + 16*d^2))

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Rubi [A]  time = 0.0756521, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4469, 2194, 4433} \[ -\frac{d e^{a+b x} \sin (4 c+4 d x)}{2 \left (b^2+16 d^2\right )}-\frac{b e^{a+b x} \cos (4 c+4 d x)}{8 \left (b^2+16 d^2\right )}+\frac{e^{a+b x}}{8 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(a + b*x)/(8*b) - (b*E^(a + b*x)*Cos[4*c + 4*d*x])/(8*(b^2 + 16*d^2)) - (d*E^(a + b*x)*Sin[4*c + 4*d*x])/(2*
(b^2 + 16*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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 4433

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*C
os[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sin[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 ^2(c+d x) \, dx &=\int \left (\frac{1}{8} e^{a+b x}-\frac{1}{8} e^{a+b x} \cos (4 c+4 d x)\right ) \, dx\\ &=\frac{1}{8} \int e^{a+b x} \, dx-\frac{1}{8} \int e^{a+b x} \cos (4 c+4 d x) \, dx\\ &=\frac{e^{a+b x}}{8 b}-\frac{b e^{a+b x} \cos (4 c+4 d x)}{8 \left (b^2+16 d^2\right )}-\frac{d e^{a+b x} \sin (4 c+4 d x)}{2 \left (b^2+16 d^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.385389, size = 57, normalized size = 0.72 \[ \frac{e^{a+b x} \left (b^2 (-\cos (4 (c+d x)))+b^2-4 b d \sin (4 (c+d x))+16 d^2\right )}{8 \left (b^3+16 b d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(a + b*x)*(b^2 + 16*d^2 - b^2*Cos[4*(c + d*x)] - 4*b*d*Sin[4*(c + d*x)]))/(8*(b^3 + 16*b*d^2))

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Maple [A]  time = 0.018, size = 71, normalized size = 0.9 \begin{align*}{\frac{{{\rm e}^{bx+a}}}{8\,b}}-{\frac{b{{\rm e}^{bx+a}}\cos \left ( 4\,dx+4\,c \right ) }{8\,{b}^{2}+128\,{d}^{2}}}-{\frac{d{{\rm e}^{bx+a}}\sin \left ( 4\,dx+4\,c \right ) }{2\,{b}^{2}+32\,{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)^2,x)

[Out]

1/8*exp(b*x+a)/b-1/8*b*exp(b*x+a)*cos(4*d*x+4*c)/(b^2+16*d^2)-1/2*d*exp(b*x+a)*sin(4*d*x+4*c)/(b^2+16*d^2)

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Maxima [B]  time = 1.04606, size = 319, normalized size = 4.04 \begin{align*} -\frac{{\left (b^{2} \cos \left (4 \, c\right ) e^{a} + 4 \, b d e^{a} \sin \left (4 \, c\right )\right )} \cos \left (4 \, d x\right ) e^{\left (b x\right )} +{\left (b^{2} \cos \left (4 \, c\right ) e^{a} - 4 \, b d e^{a} \sin \left (4 \, c\right )\right )} \cos \left (4 \, d x + 8 \, c\right ) e^{\left (b x\right )} +{\left (4 \, b d \cos \left (4 \, c\right ) e^{a} - b^{2} e^{a} \sin \left (4 \, c\right )\right )} e^{\left (b x\right )} \sin \left (4 \, d x\right ) +{\left (4 \, b d \cos \left (4 \, c\right ) e^{a} + b^{2} e^{a} \sin \left (4 \, c\right )\right )} e^{\left (b x\right )} \sin \left (4 \, d x + 8 \, c\right ) - 2 \,{\left (b^{2} \cos \left (4 \, c\right )^{2} e^{a} + b^{2} e^{a} \sin \left (4 \, c\right )^{2} + 16 \,{\left (\cos \left (4 \, c\right )^{2} e^{a} + e^{a} \sin \left (4 \, c\right )^{2}\right )} d^{2}\right )} e^{\left (b x\right )}}{16 \,{\left (b^{3} \cos \left (4 \, c\right )^{2} + b^{3} \sin \left (4 \, c\right )^{2} + 16 \,{\left (b \cos \left (4 \, c\right )^{2} + b \sin \left (4 \, c\right )^{2}\right )} d^{2}\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)^2,x, algorithm="maxima")

[Out]

-1/16*((b^2*cos(4*c)*e^a + 4*b*d*e^a*sin(4*c))*cos(4*d*x)*e^(b*x) + (b^2*cos(4*c)*e^a - 4*b*d*e^a*sin(4*c))*co
s(4*d*x + 8*c)*e^(b*x) + (4*b*d*cos(4*c)*e^a - b^2*e^a*sin(4*c))*e^(b*x)*sin(4*d*x) + (4*b*d*cos(4*c)*e^a + b^
2*e^a*sin(4*c))*e^(b*x)*sin(4*d*x + 8*c) - 2*(b^2*cos(4*c)^2*e^a + b^2*e^a*sin(4*c)^2 + 16*(cos(4*c)^2*e^a + e
^a*sin(4*c)^2)*d^2)*e^(b*x))/(b^3*cos(4*c)^2 + b^3*sin(4*c)^2 + 16*(b*cos(4*c)^2 + b*sin(4*c)^2)*d^2)

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Fricas [A]  time = 0.481435, size = 208, normalized size = 2.63 \begin{align*} -\frac{2 \,{\left (2 \, b d \cos \left (d x + c\right )^{3} - b d \cos \left (d x + c\right )\right )} e^{\left (b x + a\right )} \sin \left (d x + c\right ) +{\left (b^{2} \cos \left (d x + c\right )^{4} - b^{2} \cos \left (d x + c\right )^{2} - 2 \, d^{2}\right )} e^{\left (b x + a\right )}}{b^{3} + 16 \, b d^{2}} \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)^2,x, algorithm="fricas")

[Out]

-(2*(2*b*d*cos(d*x + c)^3 - b*d*cos(d*x + c))*e^(b*x + a)*sin(d*x + c) + (b^2*cos(d*x + c)^4 - b^2*cos(d*x + c
)^2 - 2*d^2)*e^(b*x + a))/(b^3 + 16*b*d^2)

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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)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.15453, size = 89, normalized size = 1.13 \begin{align*} -\frac{1}{8} \,{\left (\frac{b \cos \left (4 \, d x + 4 \, c\right )}{b^{2} + 16 \, d^{2}} + \frac{4 \, d \sin \left (4 \, d x + 4 \, c\right )}{b^{2} + 16 \, d^{2}}\right )} e^{\left (b x + a\right )} + \frac{e^{\left (b x + a\right )}}{8 \, b} \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)^2,x, algorithm="giac")

[Out]

-1/8*(b*cos(4*d*x + 4*c)/(b^2 + 16*d^2) + 4*d*sin(4*d*x + 4*c)/(b^2 + 16*d^2))*e^(b*x + a) + 1/8*e^(b*x + a)/b