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

Optimal. Leaf size=62 \[ -\frac{1}{5} a \cos ^2(x) \cot ^3(x) \sqrt{a \csc ^4(x)}-\frac{2}{3} a \cos ^2(x) \cot (x) \sqrt{a \csc ^4(x)}-a \sin (x) \cos (x) \sqrt{a \csc ^4(x)} \]

[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]

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Rubi [A]  time = 0.0209453, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4123, 3767} \[ -\frac{1}{5} a \cos ^2(x) \cot ^3(x) \sqrt{a \csc ^4(x)}-\frac{2}{3} a \cos ^2(x) \cot (x) \sqrt{a \csc ^4(x)}-a \sin (x) \cos (x) \sqrt{a \csc ^4(x)} \]

Antiderivative was successfully verified.

[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 4123

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]

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 \left (a \csc ^4(x)\right )^{3/2} \, dx &=\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 ) \operatorname{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]  time = 0.0199325, size = 33, normalized size = 0.53 \[ -\frac{1}{15} a \sin (x) \cos (x) \left (3 \csc ^4(x)+4 \csc ^2(x)+8\right ) \sqrt{a \csc ^4(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.096, size = 29, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 8\, \left ( \cos \left ( x \right ) \right ) ^{4}-20\, \left ( \cos \left ( x \right ) \right ) ^{2}+15 \right ) \cos \left ( x \right ) \sin \left ( x \right ) }{15} \left ({\frac{a}{ \left ( \sin \left ( x \right ) \right ) ^{4}}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.51077, size = 41, normalized size = 0.66 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 0.471147, size = 150, normalized size = 2.42 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \csc ^{4}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*csc(x)**4)**(3/2), x)

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Giac [A]  time = 1.2962, size = 31, normalized size = 0.5 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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