\(\int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx\) [65]

Optimal result

Integrand size = 10, antiderivative size = 36 \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=-\frac {\cot (x)}{2 \sqrt {a \csc ^4(x)}}+\frac {x \csc ^2(x)}{2 \sqrt {a \csc ^4(x)}} \]

[Out]

-1/2*cot(x)/(a*csc(x)^4)^(1/2)+1/2*x*csc(x)^2/(a*csc(x)^4)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4208, 2715, 8} \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=\frac {x \csc ^2(x)}{2 \sqrt {a \csc ^4(x)}}-\frac {\cot (x)}{2 \sqrt {a \csc ^4(x)}} \]

[In]

Int[1/Sqrt[a*Csc[x]^4],x]

[Out]

-1/2*Cot[x]/Sqrt[a*Csc[x]^4] + (x*Csc[x]^2)/(2*Sqrt[a*Csc[x]^4])

Rule 8

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

Rule 2715

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] && Integ
erQ[2*n]

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 ^2(x) \int \sin ^2(x) \, dx}{\sqrt {a \csc ^4(x)}} \\ & = -\frac {\cot (x)}{2 \sqrt {a \csc ^4(x)}}+\frac {\csc ^2(x) \int 1 \, dx}{2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {\cot (x)}{2 \sqrt {a \csc ^4(x)}}+\frac {x \csc ^2(x)}{2 \sqrt {a \csc ^4(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=\frac {-\cot (x)+x \csc ^2(x)}{2 \sqrt {a \csc ^4(x)}} \]

[In]

Integrate[1/Sqrt[a*Csc[x]^4],x]

[Out]

(-Cot[x] + x*Csc[x]^2)/(2*Sqrt[a*Csc[x]^4])

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67

method	result	size
default	\(-\frac {\left (\cot \left (x \right )-\csc \left (x \right )^{2} x \right ) \sqrt {16}}{8 \sqrt {a \csc \left (x \right )^{4}}}\)	\(24\)
risch	\(-\frac {{\mathrm e}^{2 i x} x}{2 \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {i {\mathrm e}^{4 i x}}{8 \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}-1\right )^{2}}+\frac {i}{8 \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}-1\right )^{2}}\)	\(102\)

[In]

int(1/(a*csc(x)^4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/8/(a*csc(x)^4)^(1/2)*(cot(x)-csc(x)^2*x)*16^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=-\frac {{\left (x \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sin \left (x\right ) - x\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{2 \, a} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=\int \frac {1}{\sqrt {a \csc ^{4}{\left (x \right )}}}\, dx \]

[In]

integrate(1/(a*csc(x)**4)**(1/2),x)

[Out]

Integral(1/sqrt(a*csc(x)**4), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=\frac {x}{2 \, \sqrt {a}} - \frac {\tan \left (x\right )}{2 \, {\left (\sqrt {a} \tan \left (x\right )^{2} + \sqrt {a}\right )}} \]

[In]

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

[Out]

1/2*x/sqrt(a) - 1/2*tan(x)/(sqrt(a)*tan(x)^2 + sqrt(a))

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=\int \frac {1}{\sqrt {\frac {a}{{\sin \left (x\right )}^4}}} \,d x \]

[In]

int(1/(a/sin(x)^4)^(1/2),x)

[Out]

int(1/(a/sin(x)^4)^(1/2), x)