3.134 \(\int d^x \sin (x) \, dx\)

Optimal. Leaf size=32 \[ \frac{d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac{d^x \cos (x)}{\log ^2(d)+1} \]

[Out]

-((d^x*Cos[x])/(1 + Log[d]^2)) + (d^x*Log[d]*Sin[x])/(1 + Log[d]^2)

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Rubi [A]  time = 0.0109728, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4432} \[ \frac{d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac{d^x \cos (x)}{\log ^2(d)+1} \]

Antiderivative was successfully verified.

[In]

Int[d^x*Sin[x],x]

[Out]

-((d^x*Cos[x])/(1 + Log[d]^2)) + (d^x*Log[d]*Sin[x])/(1 + Log[d]^2)

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 d^x \sin (x) \, dx &=-\frac{d^x \cos (x)}{1+\log ^2(d)}+\frac{d^x \log (d) \sin (x)}{1+\log ^2(d)}\\ \end{align*}

Mathematica [A]  time = 0.0183267, size = 22, normalized size = 0.69 \[ \frac{d^x (\log (d) \sin (x)-\cos (x))}{\log ^2(d)+1} \]

Antiderivative was successfully verified.

[In]

Integrate[d^x*Sin[x],x]

[Out]

(d^x*(-Cos[x] + Log[d]*Sin[x]))/(1 + Log[d]^2)

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Maple [B]  time = 0.018, size = 69, normalized size = 2.2 \begin{align*}{ \left ({\frac{{{\rm e}^{x\ln \left ( d \right ) }}}{1+ \left ( \ln \left ( d \right ) \right ) ^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{{{\rm e}^{x\ln \left ( d \right ) }}}{1+ \left ( \ln \left ( d \right ) \right ) ^{2}}}+2\,{\frac{\ln \left ( d \right ){{\rm e}^{x\ln \left ( d \right ) }}\tan \left ( x/2 \right ) }{1+ \left ( \ln \left ( d \right ) \right ) ^{2}}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(d^x*sin(x),x)

[Out]

(1/(1+ln(d)^2)*exp(x*ln(d))*tan(1/2*x)^2-1/(1+ln(d)^2)*exp(x*ln(d))+2/(1+ln(d)^2)*ln(d)*exp(x*ln(d))*tan(1/2*x
))/(tan(1/2*x)^2+1)

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Maxima [A]  time = 0.976485, size = 34, normalized size = 1.06 \begin{align*} \frac{d^{x} \log \left (d\right ) \sin \left (x\right ) - d^{x} \cos \left (x\right )}{\log \left (d\right )^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d^x*sin(x),x, algorithm="maxima")

[Out]

(d^x*log(d)*sin(x) - d^x*cos(x))/(log(d)^2 + 1)

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Fricas [A]  time = 1.82861, size = 61, normalized size = 1.91 \begin{align*} \frac{{\left (\log \left (d\right ) \sin \left (x\right ) - \cos \left (x\right )\right )} d^{x}}{\log \left (d\right )^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d^x*sin(x),x, algorithm="fricas")

[Out]

(log(d)*sin(x) - cos(x))*d^x/(log(d)^2 + 1)

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Sympy [A]  time = 1.09542, size = 104, normalized size = 3.25 \begin{align*} \begin{cases} \frac{x e^{- i x} \sin{\left (x \right )}}{2} - \frac{i x e^{- i x} \cos{\left (x \right )}}{2} - \frac{e^{- i x} \cos{\left (x \right )}}{2} & \text{for}\: d = e^{- i} \\\frac{x e^{i x} \sin{\left (x \right )}}{2} + \frac{i x e^{i x} \cos{\left (x \right )}}{2} - \frac{e^{i x} \cos{\left (x \right )}}{2} & \text{for}\: d = e^{i} \\\frac{d^{x} \log{\left (d \right )} \sin{\left (x \right )}}{\log{\left (d \right )}^{2} + 1} - \frac{d^{x} \cos{\left (x \right )}}{\log{\left (d \right )}^{2} + 1} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d**x*sin(x),x)

[Out]

Piecewise((x*exp(-I*x)*sin(x)/2 - I*x*exp(-I*x)*cos(x)/2 - exp(-I*x)*cos(x)/2, Eq(d, exp(-I))), (x*exp(I*x)*si
n(x)/2 + I*x*exp(I*x)*cos(x)/2 - exp(I*x)*cos(x)/2, Eq(d, exp(I))), (d**x*log(d)*sin(x)/(log(d)**2 + 1) - d**x
*cos(x)/(log(d)**2 + 1), True))

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Giac [C]  time = 1.13138, size = 443, normalized size = 13.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d^x*sin(x),x, algorithm="giac")

[Out]

abs(d)^x*((pi - pi*sgn(d) - 2)*cos(1/2*pi*x*sgn(d) - 1/2*pi*x + x)/((pi - pi*sgn(d) - 2)^2 + 4*log(abs(d))^2)
+ 2*log(abs(d))*sin(1/2*pi*x*sgn(d) - 1/2*pi*x + x)/((pi - pi*sgn(d) - 2)^2 + 4*log(abs(d))^2)) - abs(d)^x*((p
i - pi*sgn(d) + 2)*cos(1/2*pi*x*sgn(d) - 1/2*pi*x - x)/((pi - pi*sgn(d) + 2)^2 + 4*log(abs(d))^2) + 2*log(abs(
d))*sin(1/2*pi*x*sgn(d) - 1/2*pi*x - x)/((pi - pi*sgn(d) + 2)^2 + 4*log(abs(d))^2)) + 1/2*abs(d)^x*(2*I*e^(1/2
*I*pi*x*sgn(d) - 1/2*I*pi*x + I*x)/(-2*I*pi + 2*I*pi*sgn(d) + 4*log(abs(d)) + 4*I) + 2*I*e^(-1/2*I*pi*x*sgn(d)
+ 1/2*I*pi*x - I*x)/(2*I*pi - 2*I*pi*sgn(d) + 4*log(abs(d)) - 4*I)) + 1/2*abs(d)^x*(-2*I*e^(1/2*I*pi*x*sgn(d)
- 1/2*I*pi*x - I*x)/(-2*I*pi + 2*I*pi*sgn(d) + 4*log(abs(d)) - 4*I) - 2*I*e^(-1/2*I*pi*x*sgn(d) + 1/2*I*pi*x
+ I*x)/(2*I*pi - 2*I*pi*sgn(d) + 4*log(abs(d)) + 4*I))