\(\int \csc ^8(a+b x) \, dx\) [8]

Optimal result

Integrand size = 8, antiderivative size = 55 \[ \int \csc ^8(a+b x) \, dx=-\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{b}-\frac {3 \cot ^5(a+b x)}{5 b}-\frac {\cot ^7(a+b x)}{7 b} \]

[Out]

-cot(b*x+a)/b-cot(b*x+a)^3/b-3/5*cot(b*x+a)^5/b-1/7*cot(b*x+a)^7/b

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, 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} \[ \int \csc ^8(a+b x) \, dx=-\frac {\cot ^7(a+b x)}{7 b}-\frac {3 \cot ^5(a+b x)}{5 b}-\frac {\cot ^3(a+b x)}{b}-\frac {\cot (a+b x)}{b} \]

[In]

Int[Csc[a + b*x]^8,x]

[Out]

-(Cot[a + b*x]/b) - Cot[a + b*x]^3/b - (3*Cot[a + b*x]^5)/(5*b) - Cot[a + b*x]^7/(7*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+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (a+b x)\right )}{b} \\ & = -\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{b}-\frac {3 \cot ^5(a+b x)}{5 b}-\frac {\cot ^7(a+b x)}{7 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.40 \[ \int \csc ^8(a+b x) \, dx=-\frac {16 \cot (a+b x)}{35 b}-\frac {8 \cot (a+b x) \csc ^2(a+b x)}{35 b}-\frac {6 \cot (a+b x) \csc ^4(a+b x)}{35 b}-\frac {\cot (a+b x) \csc ^6(a+b x)}{7 b} \]

[In]

Integrate[Csc[a + b*x]^8,x]

[Out]

(-16*Cot[a + b*x])/(35*b) - (8*Cot[a + b*x]*Csc[a + b*x]^2)/(35*b) - (6*Cot[a + b*x]*Csc[a + b*x]^4)/(35*b) -
(Cot[a + b*x]*Csc[a + b*x]^6)/(7*b)

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {\left (-\frac {16}{35}-\frac {\csc \left (x b +a \right )^{6}}{7}-\frac {6 \csc \left (x b +a \right )^{4}}{35}-\frac {8 \csc \left (x b +a \right )^{2}}{35}\right ) \cot \left (x b +a \right )}{b}\)	\(43\)
default	\(\frac {\left (-\frac {16}{35}-\frac {\csc \left (x b +a \right )^{6}}{7}-\frac {6 \csc \left (x b +a \right )^{4}}{35}-\frac {8 \csc \left (x b +a \right )^{2}}{35}\right ) \cot \left (x b +a \right )}{b}\)	\(43\)
risch	\(\frac {32 i \left (35 \,{\mathrm e}^{6 i \left (x b +a \right )}-21 \,{\mathrm e}^{4 i \left (x b +a \right )}+7 \,{\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{35 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{7}}\)	\(55\)
parallelrisch	\(\frac {5 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}-5 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}+49 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}-49 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}+245 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}-245 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+1225 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )-1225 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )}{4480 b}\)	\(107\)
norman	\(\frac {-\frac {1}{896 b}-\frac {7 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{640 b}-\frac {7 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{128 b}-\frac {35 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{128 b}+\frac {35 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{128 b}+\frac {7 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{10}}{128 b}+\frac {7 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{12}}{640 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{14}}{896 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}}\)	\(131\)

[In]

int(csc(b*x+a)^8,x,method=_RETURNVERBOSE)

[Out]

1/b*(-16/35-1/7*csc(b*x+a)^6-6/35*csc(b*x+a)^4-8/35*csc(b*x+a)^2)*cot(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.58 \[ \int \csc ^8(a+b x) \, dx=-\frac {16 \, \cos \left (b x + a\right )^{7} - 56 \, \cos \left (b x + a\right )^{5} + 70 \, \cos \left (b x + a\right )^{3} - 35 \, \cos \left (b x + a\right )}{35 \, {\left (b \cos \left (b x + a\right )^{6} - 3 \, b \cos \left (b x + a\right )^{4} + 3 \, b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]

[In]

integrate(csc(b*x+a)^8,x, algorithm="fricas")

[Out]

-1/35*(16*cos(b*x + a)^7 - 56*cos(b*x + a)^5 + 70*cos(b*x + a)^3 - 35*cos(b*x + a))/((b*cos(b*x + a)^6 - 3*b*c
os(b*x + a)^4 + 3*b*cos(b*x + a)^2 - b)*sin(b*x + a))

Sympy [F]

\[ \int \csc ^8(a+b x) \, dx=\int \csc ^{8}{\left (a + b x \right )}\, dx \]

[In]

integrate(csc(b*x+a)**8,x)

[Out]

Integral(csc(a + b*x)**8, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82 \[ \int \csc ^8(a+b x) \, dx=-\frac {35 \, \tan \left (b x + a\right )^{6} + 35 \, \tan \left (b x + a\right )^{4} + 21 \, \tan \left (b x + a\right )^{2} + 5}{35 \, b \tan \left (b x + a\right )^{7}} \]

[In]

integrate(csc(b*x+a)^8,x, algorithm="maxima")

[Out]

-1/35*(35*tan(b*x + a)^6 + 35*tan(b*x + a)^4 + 21*tan(b*x + a)^2 + 5)/(b*tan(b*x + a)^7)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82 \[ \int \csc ^8(a+b x) \, dx=-\frac {35 \, \tan \left (b x + a\right )^{6} + 35 \, \tan \left (b x + a\right )^{4} + 21 \, \tan \left (b x + a\right )^{2} + 5}{35 \, b \tan \left (b x + a\right )^{7}} \]

[In]

integrate(csc(b*x+a)^8,x, algorithm="giac")

[Out]

-1/35*(35*tan(b*x + a)^6 + 35*tan(b*x + a)^4 + 21*tan(b*x + a)^2 + 5)/(b*tan(b*x + a)^7)

Mupad [B] (verification not implemented)

Time = 20.41 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.75 \[ \int \csc ^8(a+b x) \, dx=-\frac {{\mathrm {tan}\left (a+b\,x\right )}^6+{\mathrm {tan}\left (a+b\,x\right )}^4+\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^2}{5}+\frac {1}{7}}{b\,{\mathrm {tan}\left (a+b\,x\right )}^7} \]

[In]

int(1/sin(a + b*x)^8,x)

[Out]

-((3*tan(a + b*x)^2)/5 + tan(a + b*x)^4 + tan(a + b*x)^6 + 1/7)/(b*tan(a + b*x)^7)