3.63 \(\int e^x \csc (e^x) \sec (e^x) \, dx\)

Optimal. Leaf size=5 \[ \log \left (\tan \left (e^x\right )\right ) \]

[Out]

Log[Tan[E^x]]

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Rubi [A]  time = 0.0215424, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 2620, 29} \[ \log \left (\tan \left (e^x\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Csc[E^x]*Sec[E^x],x]

[Out]

Log[Tan[E^x]]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2620

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int e^x \csc \left (e^x\right ) \sec \left (e^x\right ) \, dx &=\operatorname{Subst}\left (\int \csc (x) \sec (x) \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\tan \left (e^x\right )\right )\\ &=\log \left (\tan \left (e^x\right )\right )\\ \end{align*}

Mathematica [B]  time = 0.0177249, size = 21, normalized size = 4.2 \[ 2 \left (\frac{1}{2} \log \left (\sin \left (e^x\right )\right )-\frac{1}{2} \log \left (\cos \left (e^x\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Csc[E^x]*Sec[E^x],x]

[Out]

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

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Maple [A]  time = 0.022, size = 5, normalized size = 1. \begin{align*} \ln \left ( \tan \left ({{\rm e}^{x}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*csc(exp(x))*sec(exp(x)),x)

[Out]

ln(tan(exp(x)))

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Maxima [B]  time = 1.07253, size = 26, normalized size = 5.2 \begin{align*} -\frac{1}{2} \, \log \left (\sin \left (e^{x}\right )^{2} - 1\right ) + \frac{1}{2} \, \log \left (\sin \left (e^{x}\right )^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*csc(exp(x))*sec(exp(x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.475521, size = 74, normalized size = 14.8 \begin{align*} -\frac{1}{2} \, \log \left (\cos \left (e^{x}\right )^{2}\right ) + \frac{1}{2} \, \log \left (-\frac{1}{4} \, \cos \left (e^{x}\right )^{2} + \frac{1}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*csc(exp(x))*sec(exp(x)),x, algorithm="fricas")

[Out]

-1/2*log(cos(e^x)^2) + 1/2*log(-1/4*cos(e^x)^2 + 1/4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{x} \csc{\left (e^{x} \right )} \sec{\left (e^{x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*csc(exp(x))*sec(exp(x)),x)

[Out]

Integral(exp(x)*csc(exp(x))*sec(exp(x)), x)

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Giac [B]  time = 1.11425, size = 27, normalized size = 5.4 \begin{align*} \frac{1}{2} \, \log \left (\sin \left (e^{x}\right )^{2}\right ) - \frac{1}{2} \, \log \left ({\left | \sin \left (e^{x}\right )^{2} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*csc(exp(x))*sec(exp(x)),x, algorithm="giac")

[Out]

1/2*log(sin(e^x)^2) - 1/2*log(abs(sin(e^x)^2 - 1))