\(\int \csc ^6(a+b x) \, dx\) [6]

Optimal result

Integrand size = 8, antiderivative size = 42 \[ \int \csc ^6(a+b x) \, dx=-\frac {\cot (a+b x)}{b}-\frac {2 \cot ^3(a+b x)}{3 b}-\frac {\cot ^5(a+b x)}{5 b} \]

[Out]

-cot(b*x+a)/b-2/3*cot(b*x+a)^3/b-1/5*cot(b*x+a)^5/b

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, 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} \[ \int \csc ^6(a+b x) \, dx=-\frac {\cot ^5(a+b x)}{5 b}-\frac {2 \cot ^3(a+b x)}{3 b}-\frac {\cot (a+b x)}{b} \]

[In]

Int[Csc[a + b*x]^6,x]

[Out]

-(Cot[a + b*x]/b) - (2*Cot[a + b*x]^3)/(3*b) - Cot[a + b*x]^5/(5*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+2 x^2+x^4\right ) \, dx,x,\cot (a+b x)\right )}{b} \\ & = -\frac {\cot (a+b x)}{b}-\frac {2 \cot ^3(a+b x)}{3 b}-\frac {\cot ^5(a+b x)}{5 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.33 \[ \int \csc ^6(a+b x) \, dx=-\frac {8 \cot (a+b x)}{15 b}-\frac {4 \cot (a+b x) \csc ^2(a+b x)}{15 b}-\frac {\cot (a+b x) \csc ^4(a+b x)}{5 b} \]

[In]

Integrate[Csc[a + b*x]^6,x]

[Out]

(-8*Cot[a + b*x])/(15*b) - (4*Cot[a + b*x]*Csc[a + b*x]^2)/(15*b) - (Cot[a + b*x]*Csc[a + b*x]^4)/(5*b)

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {\left (-\frac {8}{15}-\frac {\csc \left (x b +a \right )^{4}}{5}-\frac {4 \csc \left (x b +a \right )^{2}}{15}\right ) \cot \left (x b +a \right )}{b}\)	\(33\)
default	\(\frac {\left (-\frac {8}{15}-\frac {\csc \left (x b +a \right )^{4}}{5}-\frac {4 \csc \left (x b +a \right )^{2}}{15}\right ) \cot \left (x b +a \right )}{b}\)	\(33\)
risch	\(-\frac {16 i \left (10 \,{\mathrm e}^{4 i \left (x b +a \right )}-5 \,{\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{15 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{5}}\)	\(44\)
parallelrisch	\(\frac {-3 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}+3 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}-25 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+25 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}-150 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )+150 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{480 b}\)	\(81\)
norman	\(\frac {-\frac {1}{160 b}-\frac {5 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{96 b}-\frac {5 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{16 b}+\frac {5 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{16 b}+\frac {5 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{96 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{10}}{160 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}\)	\(99\)

[In]

int(csc(b*x+a)^6,x,method=_RETURNVERBOSE)

[Out]

1/b*(-8/15-1/5*csc(b*x+a)^4-4/15*csc(b*x+a)^2)*cot(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.52 \[ \int \csc ^6(a+b x) \, dx=-\frac {8 \, \cos \left (b x + a\right )^{5} - 20 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )}{15 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )} \sin \left (b x + a\right )} \]

[In]

integrate(csc(b*x+a)^6,x, algorithm="fricas")

[Out]

-1/15*(8*cos(b*x + a)^5 - 20*cos(b*x + a)^3 + 15*cos(b*x + a))/((b*cos(b*x + a)^4 - 2*b*cos(b*x + a)^2 + b)*si
n(b*x + a))

Sympy [F]

\[ \int \csc ^6(a+b x) \, dx=\int \csc ^{6}{\left (a + b x \right )}\, dx \]

[In]

integrate(csc(b*x+a)**6,x)

[Out]

Integral(csc(a + b*x)**6, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83 \[ \int \csc ^6(a+b x) \, dx=-\frac {15 \, \tan \left (b x + a\right )^{4} + 10 \, \tan \left (b x + a\right )^{2} + 3}{15 \, b \tan \left (b x + a\right )^{5}} \]

[In]

integrate(csc(b*x+a)^6,x, algorithm="maxima")

[Out]

-1/15*(15*tan(b*x + a)^4 + 10*tan(b*x + a)^2 + 3)/(b*tan(b*x + a)^5)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83 \[ \int \csc ^6(a+b x) \, dx=-\frac {15 \, \tan \left (b x + a\right )^{4} + 10 \, \tan \left (b x + a\right )^{2} + 3}{15 \, b \tan \left (b x + a\right )^{5}} \]

[In]

integrate(csc(b*x+a)^6,x, algorithm="giac")

[Out]

-1/15*(15*tan(b*x + a)^4 + 10*tan(b*x + a)^2 + 3)/(b*tan(b*x + a)^5)

Mupad [B] (verification not implemented)

Time = 20.95 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90 \[ \int \csc ^6(a+b x) \, dx=-\frac {\mathrm {cot}\left (a+b\,x\right )}{b}-\frac {2\,{\mathrm {cot}\left (a+b\,x\right )}^3}{3\,b}-\frac {{\mathrm {cot}\left (a+b\,x\right )}^5}{5\,b} \]

[In]

int(1/sin(a + b*x)^6,x)

[Out]

- cot(a + b*x)/b - (2*cot(a + b*x)^3)/(3*b) - cot(a + b*x)^5/(5*b)