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

Optimal. Leaf size=55 \[ -\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{b}-\frac {3 \cot ^5(a+b x)}{5 b}-\frac {\cot ^7(a+b x)}{7 b} \]

[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

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Rubi [A]
time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3852} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \csc ^8(a+b x) \, dx &=-\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*}

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Mathematica [A]
time = 0.02, size = 77, normalized size = 1.40 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [A]
time = 0.05, size = 43, normalized size = 0.78

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]
time = 0.33, size = 45, normalized size = 0.82 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 3.18, size = 87, normalized size = 1.58 \begin {gather*} -\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \csc ^{8}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(a + b*x)**8, x)

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Giac [A]
time = 0.46, size = 45, normalized size = 0.82 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [B]
time = 0.13, size = 41, normalized size = 0.75 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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