3.736 \(\int 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx\)

Optimal. Leaf size=14 \[ -\frac{5^{\csc (3 x)}}{3 \log (5)} \]

[Out]

-5^Csc[3*x]/(3*Log[5])

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Rubi [A]  time = 0.0247173, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4338, 2209} \[ -\frac{5^{\csc (3 x)}}{3 \log (5)} \]

Antiderivative was successfully verified.

[In]

Int[5^Csc[3*x]*Cot[3*x]*Csc[3*x],x]

[Out]

-5^Csc[3*x]/(3*Log[5])

Rule 4338

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[1/(b
*c), Subst[Int[SubstFor[1/x, 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, Cot] || EqQ[F, cot])

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 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{5^{\frac{1}{x}}}{x^2} \, dx,x,\sin (3 x)\right )\\ &=-\frac{5^{\csc (3 x)}}{3 \log (5)}\\ \end{align*}

Mathematica [A]  time = 0.0313623, size = 14, normalized size = 1. \[ -\frac{5^{\csc (3 x)}}{3 \log (5)} \]

Antiderivative was successfully verified.

[In]

Integrate[5^Csc[3*x]*Cot[3*x]*Csc[3*x],x]

[Out]

-5^Csc[3*x]/(3*Log[5])

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Maple [A]  time = 0.01, size = 13, normalized size = 0.9 \begin{align*} -{\frac{{5}^{\csc \left ( 3\,x \right ) }}{3\,\ln \left ( 5 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5^csc(3*x)*cot(3*x)*csc(3*x),x)

[Out]

-1/3*5^csc(3*x)/ln(5)

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Maxima [A]  time = 0.963998, size = 16, normalized size = 1.14 \begin{align*} -\frac{5^{\csc \left (3 \, x\right )}}{3 \, \log \left (5\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5^csc(3*x)*cot(3*x)*csc(3*x),x, algorithm="maxima")

[Out]

-1/3*5^csc(3*x)/log(5)

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Fricas [A]  time = 2.34667, size = 38, normalized size = 2.71 \begin{align*} -\frac{5^{\left (\frac{1}{\sin \left (3 \, x\right )}\right )}}{3 \, \log \left (5\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5^csc(3*x)*cot(3*x)*csc(3*x),x, algorithm="fricas")

[Out]

-1/3*5^(1/sin(3*x))/log(5)

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Sympy [A]  time = 0.821964, size = 12, normalized size = 0.86 \begin{align*} - \frac{5^{\csc{\left (3 x \right )}}}{3 \log{\left (5 \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5**csc(3*x)*cot(3*x)*csc(3*x),x)

[Out]

-5**csc(3*x)/(3*log(5))

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Giac [A]  time = 1.11444, size = 19, normalized size = 1.36 \begin{align*} -\frac{5^{\left (\frac{1}{\sin \left (3 \, x\right )}\right )}}{3 \, \log \left (5\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5^csc(3*x)*cot(3*x)*csc(3*x),x, algorithm="giac")

[Out]

-1/3*5^(1/sin(3*x))/log(5)