\(\int \csc ^2(x)^{5/2} \, dx\) [40]

Optimal result

Integrand size = 8, antiderivative size = 36 \[ \int \csc ^2(x)^{5/2} \, dx=-\frac {3}{8} \text {arcsinh}(\cot (x))-\frac {3}{8} \cot (x) \sqrt {\csc ^2(x)}-\frac {1}{4} \cot (x) \csc ^2(x)^{3/2} \]

[Out]

-3/8*arcsinh(cot(x))-1/4*cot(x)*(csc(x)^2)^(3/2)-3/8*cot(x)*(csc(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4207, 201, 221} \[ \int \csc ^2(x)^{5/2} \, dx=-\frac {3}{8} \text {arcsinh}(\cot (x))-\frac {1}{4} \cot (x) \csc ^2(x)^{3/2}-\frac {3}{8} \cot (x) \sqrt {\csc ^2(x)} \]

[In]

Int[(Csc[x]^2)^(5/2),x]

[Out]

(-3*ArcSinh[Cot[x]])/8 - (3*Cot[x]*Sqrt[Csc[x]^2])/8 - (Cot[x]*(Csc[x]^2)^(3/2))/4

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 4207

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[b*(ff/
f), Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \left (1+x^2\right )^{3/2} \, dx,x,\cot (x)\right ) \\ & = -\frac {1}{4} \cot (x) \csc ^2(x)^{3/2}-\frac {3}{4} \text {Subst}\left (\int \sqrt {1+x^2} \, dx,x,\cot (x)\right ) \\ & = -\frac {3}{8} \cot (x) \sqrt {\csc ^2(x)}-\frac {1}{4} \cot (x) \csc ^2(x)^{3/2}-\frac {3}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {3}{8} \text {arcsinh}(\cot (x))-\frac {3}{8} \cot (x) \sqrt {\csc ^2(x)}-\frac {1}{4} \cot (x) \csc ^2(x)^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.00 \[ \int \csc ^2(x)^{5/2} \, dx=\frac {1}{64} \sqrt {\csc ^2(x)} \left (-6 \csc ^2\left (\frac {x}{2}\right )-\csc ^4\left (\frac {x}{2}\right )+24 \left (-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )\right )+6 \sec ^2\left (\frac {x}{2}\right )+\sec ^4\left (\frac {x}{2}\right )\right ) \sin (x) \]

[In]

Integrate[(Csc[x]^2)^(5/2),x]

[Out]

(Sqrt[Csc[x]^2]*(-6*Csc[x/2]^2 - Csc[x/2]^4 + 24*(-Log[Cos[x/2]] + Log[Sin[x/2]]) + 6*Sec[x/2]^2 + Sec[x/2]^4)
*Sin[x])/64

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.55 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {\operatorname {csgn}\left (\csc \left (x \right )\right ) \left (3 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+3 \cot \left (x \right )^{3} \csc \left (x \right )-5 \cot \left (x \right ) \csc \left (x \right )^{3}\right ) \sqrt {4}}{16}\)	\(36\)
risch	\(-\frac {i \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left (3 \,{\mathrm e}^{6 i x}-11 \,{\mathrm e}^{4 i x}-11 \,{\mathrm e}^{2 i x}+3\right )}{4 \left ({\mathrm e}^{2 i x}-1\right )^{3}}+\frac {3 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )}{4}-\frac {3 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )}{4}\)	\(115\)

[In]

int((csc(x)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/16*csgn(csc(x))*(3*ln(csc(x)-cot(x))+3*cot(x)^3*csc(x)-5*cot(x)*csc(x)^3)*4^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (26) = 52\).

Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.92 \[ \int \csc ^2(x)^{5/2} \, dx=\frac {6 \, \cos \left (x\right )^{3} - 3 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 10 \, \cos \left (x\right )}{16 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} \]

[In]

integrate((csc(x)^2)^(5/2),x, algorithm="fricas")

[Out]

1/16*(6*cos(x)^3 - 3*(cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*cos(x) + 1/2) + 3*(cos(x)^4 - 2*cos(x)^2 + 1)*log(-1/
2*cos(x) + 1/2) - 10*cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1)

Sympy [F]

\[ \int \csc ^2(x)^{5/2} \, dx=\int \left (\csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \]

[In]

integrate((csc(x)**2)**(5/2),x)

[Out]

Integral((csc(x)**2)**(5/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 869 vs. \(2 (26) = 52\).

Time = 0.32 (sec) , antiderivative size = 869, normalized size of antiderivative = 24.14 \[ \int \csc ^2(x)^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((csc(x)^2)^(5/2),x, algorithm="maxima")

[Out]

-1/16*(4*(3*cos(7*x) - 11*cos(5*x) - 11*cos(3*x) + 3*cos(x))*cos(8*x) - 12*(4*cos(6*x) - 6*cos(4*x) + 4*cos(2*
x) - 1)*cos(7*x) + 16*(11*cos(5*x) + 11*cos(3*x) - 3*cos(x))*cos(6*x) - 44*(6*cos(4*x) - 4*cos(2*x) + 1)*cos(5
*x) - 24*(11*cos(3*x) - 3*cos(x))*cos(4*x) + 44*(4*cos(2*x) - 1)*cos(3*x) - 48*cos(2*x)*cos(x) + 3*(2*(4*cos(6
*x) - 6*cos(4*x) + 4*cos(2*x) - 1)*cos(8*x) - cos(8*x)^2 + 8*(6*cos(4*x) - 4*cos(2*x) + 1)*cos(6*x) - 16*cos(6
*x)^2 + 12*(4*cos(2*x) - 1)*cos(4*x) - 36*cos(4*x)^2 - 16*cos(2*x)^2 + 4*(2*sin(6*x) - 3*sin(4*x) + 2*sin(2*x)
)*sin(8*x) - sin(8*x)^2 + 16*(3*sin(4*x) - 2*sin(2*x))*sin(6*x) - 16*sin(6*x)^2 - 36*sin(4*x)^2 + 48*sin(4*x)*
sin(2*x) - 16*sin(2*x)^2 + 8*cos(2*x) - 1)*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) - 3*(2*(4*cos(6*x) - 6*cos(
4*x) + 4*cos(2*x) - 1)*cos(8*x) - cos(8*x)^2 + 8*(6*cos(4*x) - 4*cos(2*x) + 1)*cos(6*x) - 16*cos(6*x)^2 + 12*(
4*cos(2*x) - 1)*cos(4*x) - 36*cos(4*x)^2 - 16*cos(2*x)^2 + 4*(2*sin(6*x) - 3*sin(4*x) + 2*sin(2*x))*sin(8*x) -
sin(8*x)^2 + 16*(3*sin(4*x) - 2*sin(2*x))*sin(6*x) - 16*sin(6*x)^2 - 36*sin(4*x)^2 + 48*sin(4*x)*sin(2*x) - 1
6*sin(2*x)^2 + 8*cos(2*x) - 1)*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) + 4*(3*sin(7*x) - 11*sin(5*x) - 11*sin(
3*x) + 3*sin(x))*sin(8*x) - 24*(2*sin(6*x) - 3*sin(4*x) + 2*sin(2*x))*sin(7*x) + 16*(11*sin(5*x) + 11*sin(3*x)
- 3*sin(x))*sin(6*x) - 88*(3*sin(4*x) - 2*sin(2*x))*sin(5*x) - 24*(11*sin(3*x) - 3*sin(x))*sin(4*x) + 176*sin
(3*x)*sin(2*x) - 48*sin(2*x)*sin(x) + 12*cos(x))/(2*(4*cos(6*x) - 6*cos(4*x) + 4*cos(2*x) - 1)*cos(8*x) - cos(
8*x)^2 + 8*(6*cos(4*x) - 4*cos(2*x) + 1)*cos(6*x) - 16*cos(6*x)^2 + 12*(4*cos(2*x) - 1)*cos(4*x) - 36*cos(4*x)
^2 - 16*cos(2*x)^2 + 4*(2*sin(6*x) - 3*sin(4*x) + 2*sin(2*x))*sin(8*x) - sin(8*x)^2 + 16*(3*sin(4*x) - 2*sin(2
*x))*sin(6*x) - 16*sin(6*x)^2 - 36*sin(4*x)^2 + 48*sin(4*x)*sin(2*x) - 16*sin(2*x)^2 + 8*cos(2*x) - 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.78 \[ \int \csc ^2(x)^{5/2} \, dx=-\frac {{\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{8 \, {\left (\cos \left (x\right ) + 1\right )}} + \frac {{\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (x\right )\right )}{64 \, {\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {{\left (\frac {8 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {18 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}^{2}}{64 \, {\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (x\right )\right )} + \frac {3 \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )}{16 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

integrate((csc(x)^2)^(5/2),x, algorithm="giac")

[Out]

-1/8*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) + 1/64*(cos(x) - 1)^2*sgn(sin(x))/(cos(x) + 1)^2 + 1/64*(8*(cos(x)
- 1)/(cos(x) + 1) - 18*(cos(x) - 1)^2/(cos(x) + 1)^2 - 1)*(cos(x) + 1)^2/((cos(x) - 1)^2*sgn(sin(x))) + 3/16*l
og(-(cos(x) - 1)/(cos(x) + 1))/sgn(sin(x))

Mupad [F(-1)]

Timed out. \[ \int \csc ^2(x)^{5/2} \, dx=\int {\left (\frac {1}{{\sin \left (x\right )}^2}\right )}^{5/2} \,d x \]

[In]

int((1/sin(x)^2)^(5/2),x)

[Out]

int((1/sin(x)^2)^(5/2), x)