3.1.12 \(\int (e^x+\sin (x)) \, dx\) [12]

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

[Out]

exp(x)-cos(x)

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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 = {2225, 2718} \begin {gather*} e^x-\cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x + Sin[x],x]

[Out]

E^x - Cos[x]

Rule 2225

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 2718

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*}

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Mathematica [A]
time = 0.01, size = 8, normalized size = 1.00 \begin {gather*} e^x-\cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x + Sin[x],x]

[Out]

E^x - Cos[x]

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Mathics [A]
time = 1.63, size = 8, normalized size = 1.00 \begin {gather*} E^x-\text {Cos}\left [x\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Sin[x] + E^x,x]')

[Out]

E ^ x - Cos[x]

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Maple [A]
time = 0.02, size = 8, normalized size = 1.00

method	result	size
default	\({\mathrm e}^{x}-\cos \left (x \right )\)	\(8\)
risch	\({\mathrm e}^{x}-\cos \left (x \right )\)	\(8\)
meijerg	\(-1+{\mathrm e}^{x}+\sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (x \right )}{\sqrt {\pi }}\right )\)	\(20\)
norman	\(\frac {{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2+{\mathrm e}^{x}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)+sin(x),x,method=_RETURNVERBOSE)

[Out]

exp(x)-cos(x)

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Maxima [A]
time = 0.27, size = 7, normalized size = 0.88 \begin {gather*} -\cos \left (x\right ) + e^{x} \end {gather*}

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 = 0.32, size = 7, normalized size = 0.88 \begin {gather*} -\cos \left (x\right ) + e^{x} \end {gather*}

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 = 0.04, size = 5, normalized size = 0.62 \begin {gather*} e^{x} - \cos {\left (x \right )} \end {gather*}

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 = 0.00, size = 6, normalized size = 0.75 \begin {gather*} \mathrm {e}^{x}-\cos x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^x - cos(x)

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Mupad [B]
time = 0.04, size = 7, normalized size = 0.88 \begin {gather*} {\mathrm {e}}^x-\cos \left (x\right ) \end {gather*}

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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