3.840 \(\int 2^{\sin (x)} \cos (x) \, dx\)

Optimal. Leaf size=9 \[ \frac{2^{\sin (x)}}{\log (2)} \]

[Out]

2^Sin[x]/Log[2]

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Rubi [A]  time = 0.0089618, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4334, 2194} \[ \frac{2^{\sin (x)}}{\log (2)} \]

Antiderivative was successfully verified.

[In]

Int[2^Sin[x]*Cos[x],x]

[Out]

2^Sin[x]/Log[2]

Rule 4334

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

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]

Rubi steps

\begin{align*} \int 2^{\sin (x)} \cos (x) \, dx &=\operatorname{Subst}\left (\int 2^x \, dx,x,\sin (x)\right )\\ &=\frac{2^{\sin (x)}}{\log (2)}\\ \end{align*}

Mathematica [A]  time = 0.0063061, size = 9, normalized size = 1. \[ \frac{2^{\sin (x)}}{\log (2)} \]

Antiderivative was successfully verified.

[In]

Integrate[2^Sin[x]*Cos[x],x]

[Out]

2^Sin[x]/Log[2]

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Maple [A]  time = 0.006, size = 10, normalized size = 1.1 \begin{align*}{\frac{{2}^{\sin \left ( x \right ) }}{\ln \left ( 2 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2^sin(x)*cos(x),x)

[Out]

2^sin(x)/ln(2)

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Maxima [A]  time = 0.959556, size = 12, normalized size = 1.33 \begin{align*} \frac{2^{\sin \left (x\right )}}{\log \left (2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^sin(x)*cos(x),x, algorithm="maxima")

[Out]

2^sin(x)/log(2)

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Fricas [A]  time = 1.94472, size = 23, normalized size = 2.56 \begin{align*} \frac{2^{\sin \left (x\right )}}{\log \left (2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^sin(x)*cos(x),x, algorithm="fricas")

[Out]

2^sin(x)/log(2)

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Sympy [A]  time = 0.289783, size = 7, normalized size = 0.78 \begin{align*} \frac{2^{\sin{\left (x \right )}}}{\log{\left (2 \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2**sin(x)*cos(x),x)

[Out]

2**sin(x)/log(2)

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Giac [A]  time = 1.06874, size = 12, normalized size = 1.33 \begin{align*} \frac{2^{\sin \left (x\right )}}{\log \left (2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^sin(x)*cos(x),x, algorithm="giac")

[Out]

2^sin(x)/log(2)