3.72 \(\int e^x \csc ^2(e^x) \, dx\)

Optimal. Leaf size=6 \[ -\cot \left (e^x\right ) \]

[Out]

-Cot[E^x]

________________________________________________________________________________________

Rubi [A]  time = 0.0174201, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2282, 3767, 8} \[ -\cot \left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Csc[E^x]^2,x]

[Out]

-Cot[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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int e^x \csc ^2\left (e^x\right ) \, dx &=\operatorname{Subst}\left (\int \csc ^2(x) \, dx,x,e^x\right )\\ &=-\operatorname{Subst}\left (\int 1 \, dx,x,\cot \left (e^x\right )\right )\\ &=-\cot \left (e^x\right )\\ \end{align*}

Mathematica [A]  time = 0.0167705, size = 6, normalized size = 1. \[ -\cot \left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Csc[E^x]^2,x]

[Out]

-Cot[E^x]

________________________________________________________________________________________

Maple [A]  time = 0.054, size = 6, normalized size = 1. \begin{align*} -\cot \left ({{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*csc(exp(x))^2,x)

[Out]

-cot(exp(x))

________________________________________________________________________________________

Maxima [A]  time = 1.03101, size = 9, normalized size = 1.5 \begin{align*} -\frac{1}{\tan \left (e^{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*csc(exp(x))^2,x, algorithm="maxima")

[Out]

-1/tan(e^x)

________________________________________________________________________________________

Fricas [A]  time = 0.439292, size = 27, normalized size = 4.5 \begin{align*} -\frac{\cos \left (e^{x}\right )}{\sin \left (e^{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*csc(exp(x))^2,x, algorithm="fricas")

[Out]

-cos(e^x)/sin(e^x)

________________________________________________________________________________________

Sympy [A]  time = 3.2511, size = 5, normalized size = 0.83 \begin{align*} - \cot{\left (e^{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*csc(exp(x))**2,x)

[Out]

-cot(exp(x))

________________________________________________________________________________________

Giac [A]  time = 1.27695, size = 9, normalized size = 1.5 \begin{align*} -\frac{1}{\tan \left (e^{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*csc(exp(x))^2,x, algorithm="giac")

[Out]

-1/tan(e^x)