3.680 \(\int \frac{e^{\sqrt{\sin (x)}} \cos (x)}{\sqrt{\sin (x)}} \, dx\)

Optimal. Leaf size=10 \[ 2 e^{\sqrt{\sin (x)}} \]

[Out]

2*E^Sqrt[Sin[x]]

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Rubi [A]  time = 0.026561, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4334, 2209} \[ 2 e^{\sqrt{\sin (x)}} \]

Antiderivative was successfully verified.

[In]

Int[(E^Sqrt[Sin[x]]*Cos[x])/Sqrt[Sin[x]],x]

[Out]

2*E^Sqrt[Sin[x]]

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 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

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

Mathematica [A]  time = 0.0130626, size = 10, normalized size = 1. \[ 2 e^{\sqrt{\sin (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^Sqrt[Sin[x]]*Cos[x])/Sqrt[Sin[x]],x]

[Out]

2*E^Sqrt[Sin[x]]

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Maple [A]  time = 0.007, size = 8, normalized size = 0.8 \begin{align*} 2\,{{\rm e}^{\sqrt{\sin \left ( x \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(sin(x)^(1/2))*cos(x)/sin(x)^(1/2),x)

[Out]

2*exp(sin(x)^(1/2))

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Maxima [A]  time = 0.964918, size = 9, normalized size = 0.9 \begin{align*} 2 \, e^{\sqrt{\sin \left (x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(sin(x)^(1/2))*cos(x)/sin(x)^(1/2),x, algorithm="maxima")

[Out]

2*e^sqrt(sin(x))

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Fricas [A]  time = 2.0343, size = 24, normalized size = 2.4 \begin{align*} 2 \, e^{\sqrt{\sin \left (x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(sin(x)^(1/2))*cos(x)/sin(x)^(1/2),x, algorithm="fricas")

[Out]

2*e^sqrt(sin(x))

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Sympy [A]  time = 0.595964, size = 8, normalized size = 0.8 \begin{align*} 2 e^{\sqrt{\sin{\left (x \right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(sin(x)**(1/2))*cos(x)/sin(x)**(1/2),x)

[Out]

2*exp(sqrt(sin(x)))

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Giac [A]  time = 1.10442, size = 9, normalized size = 0.9 \begin{align*} 2 \, e^{\sqrt{\sin \left (x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(sin(x)^(1/2))*cos(x)/sin(x)^(1/2),x, algorithm="giac")

[Out]

2*e^sqrt(sin(x))