\(\int 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx\) [736]

Optimal result

Integrand size = 15, antiderivative size = 14 \[ \int 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx=-\frac {5^{\csc (3 x)}}{3 \log (5)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4423, 2240} \[ \int 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx=-\frac {5^{\csc (3 x)}}{3 \log (5)} \]

[In]

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

[Out]

-1/3*5^Csc[3*x]/Log[5]

Rule 2240

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]

Rule 4423

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])

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx=-\frac {5^{\csc (3 x)}}{3 \log (5)} \]

[In]

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

[Out]

-1/3*5^Csc[3*x]/Log[5]

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {5^{\csc \left (3 x \right )}}{3 \ln \left (5\right )}\)	\(13\)
derivativedivides	\(-\frac {5^{\frac {1}{4 \cos \left (x \right )^{2} \sin \left (x \right )-\sin \left (x \right )}}}{3 \ln \left (5\right )}\)	\(24\)
default	\(-\frac {5^{\frac {1}{4 \cos \left (x \right )^{2} \sin \left (x \right )-\sin \left (x \right )}}}{3 \ln \left (5\right )}\)	\(24\)

[In]

int(5^csc(3*x)*cot(3*x)*csc(3*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx=-\frac {5^{\left (\frac {1}{\sin \left (3 \, x\right )}\right )}}{3 \, \log \left (5\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx=- \frac {5^{\csc {\left (3 x \right )}}}{3 \log {\left (5 \right )}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx=-\frac {5^{\csc \left (3 \, x\right )}}{3 \, \log \left (5\right )} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx=-\frac {5^{\left (\frac {1}{\sin \left (3 \, x\right )}\right )}}{3 \, \log \left (5\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 26.65 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx=-\frac {5^{\frac {1}{\sin \left (3\,x\right )}}}{3\,\ln \left (5\right )} \]

[In]

int((5^(1/sin(3*x))*cot(3*x))/sin(3*x),x)

[Out]

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