\(\int \frac {1}{(a \csc ^3(x))^{3/2}} \, dx\) [59]

Optimal result

Integrand size = 10, antiderivative size = 79 \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=-\frac {14 \cos (x)}{45 a \sqrt {a \csc ^3(x)}}-\frac {14 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{15 a \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)}-\frac {2 \cos (x) \sin ^2(x)}{9 a \sqrt {a \csc ^3(x)}} \]

[Out]

-14/45*cos(x)/a/(a*csc(x)^3)^(1/2)-14/15*(sin(1/4*Pi+1/2*x)^2)^(1/2)/sin(1/4*Pi+1/2*x)*EllipticE(cos(1/4*Pi+1/
2*x),2^(1/2))/a/sin(x)^(3/2)/(a*csc(x)^3)^(1/2)-2/9*cos(x)*sin(x)^2/a/(a*csc(x)^3)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3854, 3856, 2719} \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=-\frac {14 \cos (x)}{45 a \sqrt {a \csc ^3(x)}}-\frac {2 \sin ^2(x) \cos (x)}{9 a \sqrt {a \csc ^3(x)}}-\frac {14 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{15 a \sin ^{\frac {3}{2}}(x) \sqrt {a \csc ^3(x)}} \]

[In]

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

[Out]

(-14*Cos[x])/(45*a*Sqrt[a*Csc[x]^3]) - (14*EllipticE[Pi/4 - x/2, 2])/(15*a*Sqrt[a*Csc[x]^3]*Sin[x]^(3/2)) - (2
*Cos[x]*Sin[x]^2)/(9*a*Sqrt[a*Csc[x]^3])

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 3854

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

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

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}& = -\frac {(-\csc (x))^{3/2} \int \frac {1}{(-\csc (x))^{9/2}} \, dx}{a \sqrt {a \csc ^3(x)}} \\ & = -\frac {2 \cos (x) \sin ^2(x)}{9 a \sqrt {a \csc ^3(x)}}-\frac {\left (7 (-\csc (x))^{3/2}\right ) \int \frac {1}{(-\csc (x))^{5/2}} \, dx}{9 a \sqrt {a \csc ^3(x)}} \\ & = -\frac {14 \cos (x)}{45 a \sqrt {a \csc ^3(x)}}-\frac {2 \cos (x) \sin ^2(x)}{9 a \sqrt {a \csc ^3(x)}}-\frac {\left (7 (-\csc (x))^{3/2}\right ) \int \frac {1}{\sqrt {-\csc (x)}} \, dx}{15 a \sqrt {a \csc ^3(x)}} \\ & = -\frac {14 \cos (x)}{45 a \sqrt {a \csc ^3(x)}}-\frac {2 \cos (x) \sin ^2(x)}{9 a \sqrt {a \csc ^3(x)}}+\frac {7 \int \sqrt {\sin (x)} \, dx}{15 a \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)} \\ & = -\frac {14 \cos (x)}{45 a \sqrt {a \csc ^3(x)}}-\frac {14 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{15 a \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)}-\frac {2 \cos (x) \sin ^2(x)}{9 a \sqrt {a \csc ^3(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=\frac {-84 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )+(-33 \cos (x)+5 \cos (3 x)) \sin ^{\frac {3}{2}}(x)}{90 \left (a \csc ^3(x)\right )^{3/2} \sin ^{\frac {9}{2}}(x)} \]

[In]

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

[Out]

(-84*EllipticE[(Pi - 2*x)/4, 2] + (-33*Cos[x] + 5*Cos[3*x])*Sin[x]^(3/2))/(90*(a*Csc[x]^3)^(3/2)*Sin[x]^(9/2))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.29 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.85

method	result	size
default	\(-\frac {\csc \left (x \right )^{2} \left (5 \cos \left (x \right )^{5} \sqrt {2}+42 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )-21 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \cos \left (x \right )-17 \cos \left (x \right )^{3} \sqrt {2}+42 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )-21 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}+33 \cos \left (x \right ) \sqrt {2}-21 \sqrt {2}\right ) \sqrt {8}}{90 \sqrt {a \csc \left (x \right )^{3}}\, a}\)	\(304\)

[In]

int(1/(a*csc(x)^3)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/90*csc(x)^2*(5*cos(x)^5*2^(1/2)+42*(-I*(I+cot(x)-csc(x)))^(1/2)*(-I*(-csc(x)+cot(x)))^(1/2)*(I*(-I+cot(x)-c
sc(x)))^(1/2)*EllipticE((I*(-I+cot(x)-csc(x)))^(1/2),1/2*2^(1/2))*cos(x)-21*(-I*(I+cot(x)-csc(x)))^(1/2)*(-I*(
-csc(x)+cot(x)))^(1/2)*EllipticF((I*(-I+cot(x)-csc(x)))^(1/2),1/2*2^(1/2))*(I*(-I+cot(x)-csc(x)))^(1/2)*cos(x)
-17*cos(x)^3*2^(1/2)+42*(-I*(I+cot(x)-csc(x)))^(1/2)*(-I*(-csc(x)+cot(x)))^(1/2)*(I*(-I+cot(x)-csc(x)))^(1/2)*
EllipticE((I*(-I+cot(x)-csc(x)))^(1/2),1/2*2^(1/2))-21*(-I*(I+cot(x)-csc(x)))^(1/2)*(-I*(-csc(x)+cot(x)))^(1/2
)*EllipticF((I*(-I+cot(x)-csc(x)))^(1/2),1/2*2^(1/2))*(I*(-I+cot(x)-csc(x)))^(1/2)+33*cos(x)*2^(1/2)-21*2^(1/2
))/(a*csc(x)^3)^(1/2)/a*8^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=-\frac {2 \, {\left (5 \, \cos \left (x\right )^{5} - 17 \, \cos \left (x\right )^{3} + 12 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}} \sin \left (x\right ) - 21 \, \sqrt {2 i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) - 21 \, \sqrt {-2 i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right )}{45 \, a^{2}} \]

[In]

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

[Out]

-1/45*(2*(5*cos(x)^5 - 17*cos(x)^3 + 12*cos(x))*sqrt(-a/((cos(x)^2 - 1)*sin(x)))*sin(x) - 21*sqrt(2*I*a)*weier
strassZeta(4, 0, weierstrassPInverse(4, 0, cos(x) + I*sin(x))) - 21*sqrt(-2*I*a)*weierstrassZeta(4, 0, weierst
rassPInverse(4, 0, cos(x) - I*sin(x))))/a^2

Sympy [F]

\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \csc ^{3}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(a*csc(x)**3)**(3/2),x)

[Out]

Integral((a*csc(x)**3)**(-3/2), x)

Maxima [F]

\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((a*csc(x)^3)^(-3/2), x)

Giac [F]

\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((a*csc(x)^3)^(-3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^3}\right )}^{3/2}} \,d x \]

[In]

int(1/(a/sin(x)^3)^(3/2),x)

[Out]

int(1/(a/sin(x)^3)^(3/2), x)