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

Optimal. Leaf size=27 \[ -\frac {\cot ^3(a+b x)}{3 b}-\frac {\cot (a+b x)}{b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 27, 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 = {3767} \[ -\frac {\cot ^3(a+b x)}{3 b}-\frac {\cot (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^4,x]

[Out]

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

Rule 3767

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 ^4(a+b x) \, dx &=-\frac {\operatorname {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]  time = 0.01, size = 35, normalized size = 1.30 \[ -\frac {2 \cot (a+b x)}{3 b}-\frac {\cot (a+b x) \csc ^2(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 45, normalized size = 1.67 \[ -\frac {2 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

giac [A]  time = 0.15, size = 25, normalized size = 0.93 \[ -\frac {3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.87, size = 23, normalized size = 0.85 \[ \frac {\left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (b x +a \right )\right )}{3}\right ) \cot \left (b x +a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 25, normalized size = 0.93 \[ -\frac {3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 21, normalized size = 0.78 \[ -\frac {\mathrm {cot}\left (a+b\,x\right )\,\left ({\mathrm {cot}\left (a+b\,x\right )}^2+3\right )}{3\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(a + b*x)^4,x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^{4}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(a + b*x)**4, x)

________________________________________________________________________________________