\(\int \csc ^4(a+b x) \, dx\) [4]

Optimal result

Integrand size = 8, antiderivative size = 27 \[ \int \csc ^4(a+b x) \, dx=-\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b} \]

[Out]

-cot(b*x+a)/b-1/3*cot(b*x+a)^3/b

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3852} \[ \int \csc ^4(a+b x) \, dx=-\frac {\cot ^3(a+b x)}{3 b}-\frac {\cot (a+b x)}{b} \]

[In]

Int[Csc[a + b*x]^4,x]

[Out]

-(Cot[a + b*x]/b) - Cot[a + b*x]^3/(3*b)

Rule 3852

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (a+b x)\right )}{b} \\ & = -\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \csc ^4(a+b x) \, dx=-\frac {2 \cot (a+b x)}{3 b}-\frac {\cot (a+b x) \csc ^2(a+b x)}{3 b} \]

[In]

Integrate[Csc[a + b*x]^4,x]

[Out]

(-2*Cot[a + b*x])/(3*b) - (Cot[a + b*x]*Csc[a + b*x]^2)/(3*b)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(\frac {\left (-\frac {2}{3}-\frac {\csc \left (x b +a \right )^{2}}{3}\right ) \cot \left (x b +a \right )}{b}\)	\(23\)
default	\(\frac {\left (-\frac {2}{3}-\frac {\csc \left (x b +a \right )^{2}}{3}\right ) \cot \left (x b +a \right )}{b}\)	\(23\)
risch	\(\frac {4 i \left (3 \,{\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{3 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{3}}\)	\(33\)
parallelrisch	\(\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}-\cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+9 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )-9 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )}{24 b}\)	\(53\)
norman	\(\frac {-\frac {1}{24 b}-\frac {3 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{8 b}+\frac {3 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{8 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{24 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}\)	\(67\)

[In]

int(csc(b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

1/b*(-2/3-1/3*csc(b*x+a)^2)*cot(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \csc ^4(a+b x) \, dx=-\frac {2 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]

[In]

integrate(csc(b*x+a)^4,x, algorithm="fricas")

[Out]

-1/3*(2*cos(b*x + a)^3 - 3*cos(b*x + a))/((b*cos(b*x + a)^2 - b)*sin(b*x + a))

Sympy [F]

\[ \int \csc ^4(a+b x) \, dx=\int \csc ^{4}{\left (a + b x \right )}\, dx \]

[In]

integrate(csc(b*x+a)**4,x)

[Out]

Integral(csc(a + b*x)**4, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \csc ^4(a+b x) \, dx=-\frac {3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \]

[In]

integrate(csc(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/3*(3*tan(b*x + a)^2 + 1)/(b*tan(b*x + a)^3)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \csc ^4(a+b x) \, dx=-\frac {3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \]

[In]

integrate(csc(b*x+a)^4,x, algorithm="giac")

[Out]

-1/3*(3*tan(b*x + a)^2 + 1)/(b*tan(b*x + a)^3)

Mupad [B] (verification not implemented)

Time = 21.59 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \csc ^4(a+b x) \, dx=-\frac {\mathrm {cot}\left (a+b\,x\right )\,\left ({\mathrm {cot}\left (a+b\,x\right )}^2+3\right )}{3\,b} \]

[In]

int(1/sin(a + b*x)^4,x)

[Out]

-(cot(a + b*x)*(cot(a + b*x)^2 + 3))/(3*b)