\(\int (a \csc ^4(x))^{3/2} \, dx\) [63]

Optimal result

Integrand size = 10, antiderivative size = 62 \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=-\frac {2}{3} a \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-\frac {1}{5} a \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-a \cos (x) \sqrt {a \csc ^4(x)} \sin (x) \]

[Out]

-2/3*a*cos(x)^2*cot(x)*(a*csc(x)^4)^(1/2)-1/5*a*cos(x)^2*cot(x)^3*(a*csc(x)^4)^(1/2)-a*cos(x)*sin(x)*(a*csc(x)
^4)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4208, 3852} \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=-\frac {1}{5} a \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-\frac {2}{3} a \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-a \sin (x) \cos (x) \sqrt {a \csc ^4(x)} \]

[In]

Int[(a*Csc[x]^4)^(3/2),x]

[Out]

(-2*a*Cos[x]^2*Cot[x]*Sqrt[a*Csc[x]^4])/3 - (a*Cos[x]^2*Cot[x]^3*Sqrt[a*Csc[x]^4])/5 - a*Cos[x]*Sqrt[a*Csc[x]^
4]*Sin[x]

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]

Rule 4208

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Sec[e + f*x])^n)^
FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \left (a \sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \int \csc ^6(x) \, dx \\ & = -\left (\left (a \sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (x)\right )\right ) \\ & = -\frac {2}{3} a \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-\frac {1}{5} a \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-a \cos (x) \sqrt {a \csc ^4(x)} \sin (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.53 \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=-\frac {1}{15} a \cos (x) \sqrt {a \csc ^4(x)} \left (8+4 \csc ^2(x)+3 \csc ^4(x)\right ) \sin (x) \]

[In]

Integrate[(a*Csc[x]^4)^(3/2),x]

[Out]

-1/15*(a*Cos[x]*Sqrt[a*Csc[x]^4]*(8 + 4*Csc[x]^2 + 3*Csc[x]^4)*Sin[x])

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.56

method	result	size
default	\(-\frac {\csc \left (x \right )^{2} \cot \left (x \right ) a \sqrt {a \csc \left (x \right )^{4}}\, \left (8 \cos \left (x \right )^{4}-20 \cos \left (x \right )^{2}+15\right ) \sqrt {16}}{60}\)	\(35\)
risch	\(\frac {16 i a \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left (-5+11 \cos \left (2 x \right )+9 i \sin \left (2 x \right )\right )}{15 \left ({\mathrm e}^{2 i x}-1\right )^{3}}\)	\(47\)

[In]

int((a*csc(x)^4)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/60*csc(x)^2*cot(x)*a*(a*csc(x)^4)^(1/2)*(8*cos(x)^4-20*cos(x)^2+15)*16^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.84 \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=\frac {{\left (8 \, a \cos \left (x\right )^{5} - 20 \, a \cos \left (x\right )^{3} + 15 \, a \cos \left (x\right )\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{15 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]

[In]

integrate((a*csc(x)^4)^(3/2),x, algorithm="fricas")

[Out]

1/15*(8*a*cos(x)^5 - 20*a*cos(x)^3 + 15*a*cos(x))*sqrt(a/(cos(x)^4 - 2*cos(x)^2 + 1))/((cos(x)^2 - 1)*sin(x))

Sympy [F]

\[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=\int \left (a \csc ^{4}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((a*csc(x)**4)**(3/2),x)

[Out]

Integral((a*csc(x)**4)**(3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.48 \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=-\frac {15 \, a^{\frac {3}{2}} \tan \left (x\right )^{4} + 10 \, a^{\frac {3}{2}} \tan \left (x\right )^{2} + 3 \, a^{\frac {3}{2}}}{15 \, \tan \left (x\right )^{5}} \]

[In]

integrate((a*csc(x)^4)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.37 \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=-\frac {{\left (15 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 3\right )} a^{\frac {3}{2}}}{15 \, \tan \left (x\right )^{5}} \]

[In]

integrate((a*csc(x)^4)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 17.62 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.71 \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=\frac {\frac {a^{3/2}\,8{}\mathrm {i}}{15}-\frac {4\,a^{3/2}\,\left (2\,{\sin \left (2\,x\right )}^3-9\,\sin \left (2\,x\right )+3\,\sin \left (4\,x\right )+2{}\mathrm {i}\right )}{15}}{{\left (\cos \left (2\,x\right )-1\right )}^3} \]

[In]

int((a/sin(x)^4)^(3/2),x)

[Out]

((a^(3/2)*8i)/15 - (4*a^(3/2)*(3*sin(4*x) - 9*sin(2*x) + 2*sin(2*x)^3 + 2i))/15)/(cos(2*x) - 1)^3