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]

ln(tan(exp(x)))

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Rubi [A]  time = 0.02, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 29

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

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*}

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Mathematica [B]  time = 0.02, size = 21, normalized size = 4.20 \[ 2 \left (\frac {1}{2} \log \left (\sin \left (e^x\right )\right )-\frac {1}{2} \log \left (\cos \left (e^x\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*(-1/2*Log[Cos[E^x]] + Log[Sin[E^x]]/2)

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fricas [B]  time = 0.57, size = 21, normalized size = 4.20 \[ -\frac {1}{2} \, \log \left (\cos \left (e^{x}\right )^{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{4} \, \cos \left (e^{x}\right )^{2} + \frac {1}{4}\right ) \]

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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giac [B]  time = 0.14, size = 17, normalized size = 3.40 \[ -\frac {1}{2} \, \log \left ({\left | \sin \left (e^{x}\right )^{2} - 1 \right |}\right ) + \log \left ({\left | \sin \left (e^{x}\right ) \right |}\right ) \]

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(abs(sin(e^x)^2 - 1)) + log(abs(sin(e^x)))

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maple [A]  time = 0.08, size = 5, normalized size = 1.00 \[ \ln \left (\tan \left ({\mathrm e}^{x}\right )\right ) \]

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 = 0.31, size = 19, normalized size = 3.80 \[ -\frac {1}{2} \, \log \left (\sin \left (e^{x}\right )^{2} - 1\right ) + \frac {1}{2} \, \log \left (\sin \left (e^{x}\right )^{2}\right ) \]

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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mupad [B]  time = 2.51, size = 43, normalized size = 8.60 \[ -\ln \left (-16\,{\mathrm {e}}^{2\,x}-16\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x\,2{}\mathrm {i}}\right )+\ln \left (16\,{\mathrm {e}}^{2\,x}-16\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x\,2{}\mathrm {i}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(cos(exp(x))*sin(exp(x))),x)

[Out]

log(16*exp(2*x) - 16*exp(2*x)*exp(exp(x)*2i)) - log(- 16*exp(2*x) - 16*exp(2*x)*exp(exp(x)*2i))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{x} \csc {\left (e^{x} \right )} \sec {\left (e^{x} \right )}\, dx \]

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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