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

Optimal. Leaf size=63 \[ \frac{b e^{a+b x} \sin (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}-\frac{d e^{a+b x} \cos (2 c+2 d x)}{b^2+4 d^2} \]

[Out]

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

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Rubi [A]  time = 0.0471897, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4469, 12, 4432} \[ \frac{b e^{a+b x} \sin (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}-\frac{d e^{a+b x} \cos (2 c+2 d x)}{b^2+4 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d*E^(a + b*x)*Cos[2*c + 2*d*x])/(b^2 + 4*d^2)) + (b*E^(a + b*x)*Sin[2*c + 2*d*x])/(2*(b^2 + 4*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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 (c+d x) \sin (c+d x) \, dx &=\int \frac{1}{2} e^{a+b x} \sin (2 c+2 d x) \, dx\\ &=\frac{1}{2} \int e^{a+b x} \sin (2 c+2 d x) \, dx\\ &=-\frac{d e^{a+b x} \cos (2 c+2 d x)}{b^2+4 d^2}+\frac{b e^{a+b x} \sin (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.152665, size = 44, normalized size = 0.7 \[ \frac{e^{a+b x} (b \sin (2 (c+d x))-2 d \cos (2 (c+d x)))}{2 \left (b^2+4 d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.014, size = 60, normalized size = 1. \begin{align*} -{\frac{d{{\rm e}^{bx+a}}\cos \left ( 2\,dx+2\,c \right ) }{{b}^{2}+4\,{d}^{2}}}+{\frac{b{{\rm e}^{bx+a}}\sin \left ( 2\,dx+2\,c \right ) }{2\,{b}^{2}+8\,{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04499, size = 59, normalized size = 0.94 \begin{align*} -\frac{{\left (2 \, d \cos \left (2 \, d x + 2 \, c\right ) - b \sin \left (2 \, d x + 2 \, c\right )\right )} e^{\left (b x + a\right )}}{2 \,{\left (b^{2} + 4 \, 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)*sin(d*x+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.471155, size = 130, normalized size = 2.06 \begin{align*} \frac{b \cos \left (d x + c\right ) e^{\left (b x + a\right )} \sin \left (d x + c\right ) -{\left (2 \, d \cos \left (d x + c\right )^{2} - d\right )} e^{\left (b x + a\right )}}{b^{2} + 4 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 70.2396, size = 342, normalized size = 5.43 \begin{align*} \begin{cases} x e^{a} \sin{\left (c \right )} \cos{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\\frac{i x e^{a} e^{- 2 i d x} \sin ^{2}{\left (c + d x \right )}}{4} + \frac{x e^{a} e^{- 2 i d x} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2} - \frac{i x e^{a} e^{- 2 i d x} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{e^{a} e^{- 2 i d x} \sin ^{2}{\left (c + d x \right )}}{8 d} - \frac{e^{a} e^{- 2 i d x} \cos ^{2}{\left (c + d x \right )}}{8 d} & \text{for}\: b = - 2 i d \\- \frac{i x e^{a} e^{2 i d x} \sin ^{2}{\left (c + d x \right )}}{4} + \frac{x e^{a} e^{2 i d x} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2} + \frac{i x e^{a} e^{2 i d x} \cos ^{2}{\left (c + d x \right )}}{4} - \frac{i e^{a} e^{2 i d x} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{4 d} & \text{for}\: b = 2 i d \\\frac{b e^{a} e^{b x} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{b^{2} + 4 d^{2}} + \frac{d e^{a} e^{b x} \sin ^{2}{\left (c + d x \right )}}{b^{2} + 4 d^{2}} - \frac{d e^{a} e^{b x} \cos ^{2}{\left (c + d x \right )}}{b^{2} + 4 d^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*exp(a)*sin(c)*cos(c), Eq(b, 0) & Eq(d, 0)), (I*x*exp(a)*exp(-2*I*d*x)*sin(c + d*x)**2/4 + x*exp(a
)*exp(-2*I*d*x)*sin(c + d*x)*cos(c + d*x)/2 - I*x*exp(a)*exp(-2*I*d*x)*cos(c + d*x)**2/4 + exp(a)*exp(-2*I*d*x
)*sin(c + d*x)**2/(8*d) - exp(a)*exp(-2*I*d*x)*cos(c + d*x)**2/(8*d), Eq(b, -2*I*d)), (-I*x*exp(a)*exp(2*I*d*x
)*sin(c + d*x)**2/4 + x*exp(a)*exp(2*I*d*x)*sin(c + d*x)*cos(c + d*x)/2 + I*x*exp(a)*exp(2*I*d*x)*cos(c + d*x)
**2/4 - I*exp(a)*exp(2*I*d*x)*sin(c + d*x)*cos(c + d*x)/(4*d), Eq(b, 2*I*d)), (b*exp(a)*exp(b*x)*sin(c + d*x)*
cos(c + d*x)/(b**2 + 4*d**2) + d*exp(a)*exp(b*x)*sin(c + d*x)**2/(b**2 + 4*d**2) - d*exp(a)*exp(b*x)*cos(c + d
*x)**2/(b**2 + 4*d**2), True))

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Giac [A]  time = 1.11114, size = 74, normalized size = 1.17 \begin{align*} -\frac{1}{2} \,{\left (\frac{2 \, d \cos \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}} - \frac{b \sin \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, 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)*sin(d*x+c),x, algorithm="giac")

[Out]

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