\(\int \csc (x) \sin (4 x) \, dx\) [65]

Optimal result

Integrand size = 7, antiderivative size = 13 \[ \int \csc (x) \sin (4 x) \, dx=4 \sin (x)-\frac {8 \sin ^3(x)}{3} \]

[Out]

4*sin(x)-8/3*sin(x)^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \csc (x) \sin (4 x) \, dx=4 \sin (x)-\frac {8 \sin ^3(x)}{3} \]

[In]

Int[Csc[x]*Sin[4*x],x]

[Out]

4*Sin[x] - (8*Sin[x]^3)/3

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (4-8 x^2\right ) \, dx,x,\sin (x)\right ) \\ & = 4 \sin (x)-\frac {8 \sin ^3(x)}{3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \csc (x) \sin (4 x) \, dx=4 \sin (x)-\frac {8 \sin ^3(x)}{3} \]

[In]

Integrate[Csc[x]*Sin[4*x],x]

[Out]

4*Sin[x] - (8*Sin[x]^3)/3

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(4 \sin \left (x \right )-\frac {8 \sin \left (x \right )^{3}}{3}\)	\(12\)
default	\(4 \sin \left (x \right )-\frac {8 \sin \left (x \right )^{3}}{3}\)	\(12\)
risch	\(2 \sin \left (x \right )+\frac {2 \sin \left (3 x \right )}{3}\)	\(12\)

[In]

int(sin(4*x)/sin(x),x,method=_RETURNVERBOSE)

[Out]

4*sin(x)-8/3*sin(x)^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \csc (x) \sin (4 x) \, dx=\frac {4}{3} \, {\left (2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) \]

[In]

integrate(sin(4*x)/sin(x),x, algorithm="fricas")

[Out]

4/3*(2*cos(x)^2 + 1)*sin(x)

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \csc (x) \sin (4 x) \, dx=- \frac {8 \sin ^{3}{\left (x \right )}}{3} + 4 \sin {\left (x \right )} \]

[In]

integrate(sin(4*x)/sin(x),x)

[Out]

-8*sin(x)**3/3 + 4*sin(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \csc (x) \sin (4 x) \, dx=-\frac {8}{3} \, \sin \left (x\right )^{3} + 4 \, \sin \left (x\right ) \]

[In]

integrate(sin(4*x)/sin(x),x, algorithm="maxima")

[Out]

-8/3*sin(x)^3 + 4*sin(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \csc (x) \sin (4 x) \, dx=\frac {2}{3} \, \sin \left (3 \, x\right ) + 2 \, \sin \left (x\right ) \]

[In]

integrate(sin(4*x)/sin(x),x, algorithm="giac")

[Out]

2/3*sin(3*x) + 2*sin(x)

Mupad [B] (verification not implemented)

Time = 14.36 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \csc (x) \sin (4 x) \, dx=4\,\sin \left (x\right )-\frac {8\,{\sin \left (x\right )}^3}{3} \]

[In]

int(sin(4*x)/sin(x),x)

[Out]

4*sin(x) - (8*sin(x)^3)/3