3.66 \(\int \frac{1}{(a \csc ^4(x))^{3/2}} \, dx\)

Optimal. Leaf size=86 \[ \frac{5 x \csc ^2(x)}{16 a \sqrt{a \csc ^4(x)}}-\frac{5 \cot (x)}{16 a \sqrt{a \csc ^4(x)}}-\frac{\sin ^3(x) \cos (x)}{6 a \sqrt{a \csc ^4(x)}}-\frac{5 \sin (x) \cos (x)}{24 a \sqrt{a \csc ^4(x)}} \]

[Out]

(-5*Cot[x])/(16*a*Sqrt[a*Csc[x]^4]) + (5*x*Csc[x]^2)/(16*a*Sqrt[a*Csc[x]^4]) - (5*Cos[x]*Sin[x])/(24*a*Sqrt[a*
Csc[x]^4]) - (Cos[x]*Sin[x]^3)/(6*a*Sqrt[a*Csc[x]^4])

________________________________________________________________________________________

Rubi [A]  time = 0.0294559, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4123, 2635, 8} \[ \frac{5 x \csc ^2(x)}{16 a \sqrt{a \csc ^4(x)}}-\frac{5 \cot (x)}{16 a \sqrt{a \csc ^4(x)}}-\frac{\sin ^3(x) \cos (x)}{6 a \sqrt{a \csc ^4(x)}}-\frac{5 \sin (x) \cos (x)}{24 a \sqrt{a \csc ^4(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Cot[x])/(16*a*Sqrt[a*Csc[x]^4]) + (5*x*Csc[x]^2)/(16*a*Sqrt[a*Csc[x]^4]) - (5*Cos[x]*Sin[x])/(24*a*Sqrt[a*
Csc[x]^4]) - (Cos[x]*Sin[x]^3)/(6*a*Sqrt[a*Csc[x]^4])

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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a \csc ^4(x)\right )^{3/2}} \, dx &=\frac{\csc ^2(x) \int \sin ^6(x) \, dx}{a \sqrt{a \csc ^4(x)}}\\ &=-\frac{\cos (x) \sin ^3(x)}{6 a \sqrt{a \csc ^4(x)}}+\frac{\left (5 \csc ^2(x)\right ) \int \sin ^4(x) \, dx}{6 a \sqrt{a \csc ^4(x)}}\\ &=-\frac{5 \cos (x) \sin (x)}{24 a \sqrt{a \csc ^4(x)}}-\frac{\cos (x) \sin ^3(x)}{6 a \sqrt{a \csc ^4(x)}}+\frac{\left (5 \csc ^2(x)\right ) \int \sin ^2(x) \, dx}{8 a \sqrt{a \csc ^4(x)}}\\ &=-\frac{5 \cot (x)}{16 a \sqrt{a \csc ^4(x)}}-\frac{5 \cos (x) \sin (x)}{24 a \sqrt{a \csc ^4(x)}}-\frac{\cos (x) \sin ^3(x)}{6 a \sqrt{a \csc ^4(x)}}+\frac{\left (5 \csc ^2(x)\right ) \int 1 \, dx}{16 a \sqrt{a \csc ^4(x)}}\\ &=-\frac{5 \cot (x)}{16 a \sqrt{a \csc ^4(x)}}+\frac{5 x \csc ^2(x)}{16 a \sqrt{a \csc ^4(x)}}-\frac{5 \cos (x) \sin (x)}{24 a \sqrt{a \csc ^4(x)}}-\frac{\cos (x) \sin ^3(x)}{6 a \sqrt{a \csc ^4(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0429218, size = 38, normalized size = 0.44 \[ -\frac{(-60 x+45 \sin (2 x)-9 \sin (4 x)+\sin (6 x)) \csc ^6(x)}{192 \left (a \csc ^4(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Csc[x]^6*(-60*x + 45*Sin[2*x] - 9*Sin[4*x] + Sin[6*x]))/(192*(a*Csc[x]^4)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.205, size = 41, normalized size = 0.5 \begin{align*} -{\frac{8\,\sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{5}-26\,\sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{3}+33\,\cos \left ( x \right ) \sin \left ( x \right ) -15\,x}{48\, \left ( \sin \left ( x \right ) \right ) ^{6}} \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(1/(a*csc(x)^4)^(3/2),x)

[Out]

-1/48*(8*sin(x)*cos(x)^5-26*sin(x)*cos(x)^3+33*cos(x)*sin(x)-15*x)/(a/sin(x)^4)^(3/2)/sin(x)^6

________________________________________________________________________________________

Maxima [A]  time = 1.5244, size = 78, normalized size = 0.91 \begin{align*} -\frac{33 \, \tan \left (x\right )^{5} + 40 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )}{48 \,{\left (a^{\frac{3}{2}} \tan \left (x\right )^{6} + 3 \, a^{\frac{3}{2}} \tan \left (x\right )^{4} + 3 \, a^{\frac{3}{2}} \tan \left (x\right )^{2} + a^{\frac{3}{2}}\right )}} + \frac{5 \, x}{16 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^4)^(3/2),x, algorithm="maxima")

[Out]

-1/48*(33*tan(x)^5 + 40*tan(x)^3 + 15*tan(x))/(a^(3/2)*tan(x)^6 + 3*a^(3/2)*tan(x)^4 + 3*a^(3/2)*tan(x)^2 + a^
(3/2)) + 5/16*x/a^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 0.495052, size = 178, normalized size = 2.07 \begin{align*} -\frac{{\left (15 \, x \cos \left (x\right )^{2} -{\left (8 \, \cos \left (x\right )^{7} - 34 \, \cos \left (x\right )^{5} + 59 \, \cos \left (x\right )^{3} - 33 \, \cos \left (x\right )\right )} \sin \left (x\right ) - 15 \, x\right )} \sqrt{\frac{a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{48 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^4)^(3/2),x, algorithm="fricas")

[Out]

-1/48*(15*x*cos(x)^2 - (8*cos(x)^7 - 34*cos(x)^5 + 59*cos(x)^3 - 33*cos(x))*sin(x) - 15*x)*sqrt(a/(cos(x)^4 -
2*cos(x)^2 + 1))/a^2

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(a*csc(x)**4)**(3/2),x)

[Out]

Integral((a*csc(x)**4)**(-3/2), x)

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^4)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError