3.12 \(\int (e^x+\sin (x)) \, dx\)

Optimal. Leaf size=8 \[ e^x-\cos (x) \]

[Out]

E^x - Cos[x]

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Rubi [A]  time = 0.00333, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2194, 2638} \[ e^x-\cos (x) \]

Antiderivative was successfully verified.

[In]

Int[E^x + Sin[x],x]

[Out]

E^x - Cos[x]

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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \left (e^x+\sin (x)\right ) \, dx &=\int e^x \, dx+\int \sin (x) \, dx\\ &=e^x-\cos (x)\\ \end{align*}

Mathematica [A]  time = 0.0028654, size = 8, normalized size = 1. \[ e^x-\cos (x) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x + Sin[x],x]

[Out]

E^x - Cos[x]

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Maple [A]  time = 0.002, size = 8, normalized size = 1. \begin{align*}{{\rm e}^{x}}-\cos \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)+sin(x),x)

[Out]

exp(x)-cos(x)

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Maxima [A]  time = 0.945783, size = 9, normalized size = 1.12 \begin{align*} -\cos \left (x\right ) + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+sin(x),x, algorithm="maxima")

[Out]

-cos(x) + e^x

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Fricas [A]  time = 1.82012, size = 20, normalized size = 2.5 \begin{align*} -\cos \left (x\right ) + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+sin(x),x, algorithm="fricas")

[Out]

-cos(x) + e^x

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Sympy [A]  time = 0.071743, size = 5, normalized size = 0.62 \begin{align*} e^{x} - \cos{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+sin(x),x)

[Out]

exp(x) - cos(x)

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Giac [A]  time = 1.06968, size = 9, normalized size = 1.12 \begin{align*} -\cos \left (x\right ) + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+sin(x),x, algorithm="giac")

[Out]

-cos(x) + e^x