\(\int \csc ^2(x)^{7/2} \, dx\) [39]

Optimal result

Integrand size = 8, antiderivative size = 50 \[ \int \csc ^2(x)^{7/2} \, dx=-\frac {5}{16} \text {arcsinh}(\cot (x))-\frac {5}{16} \cot (x) \sqrt {\csc ^2(x)}-\frac {5}{24} \cot (x) \csc ^2(x)^{3/2}-\frac {1}{6} \cot (x) \csc ^2(x)^{5/2} \]

[Out]

-5/16*arcsinh(cot(x))-5/24*cot(x)*(csc(x)^2)^(3/2)-1/6*cot(x)*(csc(x)^2)^(5/2)-5/16*cot(x)*(csc(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4207, 201, 221} \[ \int \csc ^2(x)^{7/2} \, dx=-\frac {5}{16} \text {arcsinh}(\cot (x))-\frac {1}{6} \cot (x) \csc ^2(x)^{5/2}-\frac {5}{24} \cot (x) \csc ^2(x)^{3/2}-\frac {5}{16} \cot (x) \sqrt {\csc ^2(x)} \]

[In]

Int[(Csc[x]^2)^(7/2),x]

[Out]

(-5*ArcSinh[Cot[x]])/16 - (5*Cot[x]*Sqrt[Csc[x]^2])/16 - (5*Cot[x]*(Csc[x]^2)^(3/2))/24 - (Cot[x]*(Csc[x]^2)^(
5/2))/6

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.84 \[ \int \csc ^2(x)^{7/2} \, dx=\frac {1}{384} \sqrt {\csc ^2(x)} \left (-30 \csc ^2\left (\frac {x}{2}\right )-6 \csc ^4\left (\frac {x}{2}\right )-\csc ^6\left (\frac {x}{2}\right )-120 \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )+30 \sec ^2\left (\frac {x}{2}\right )+6 \sec ^4\left (\frac {x}{2}\right )+\sec ^6\left (\frac {x}{2}\right )\right ) \sin (x) \]

[In]

Integrate[(Csc[x]^2)^(7/2),x]

[Out]

(Sqrt[Csc[x]^2]*(-30*Csc[x/2]^2 - 6*Csc[x/2]^4 - Csc[x/2]^6 - 120*(Log[Cos[x/2]] - Log[Sin[x/2]]) + 30*Sec[x/2
]^2 + 6*Sec[x/2]^4 + Sec[x/2]^6)*Sin[x])/384

Maple [C] (warning: unable to verify)

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

Time = 0.80 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {\csc \left (x \right )^{6} \left (-15 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right ) \sin \left (x \right )^{6}+15 \cos \left (x \right )^{5}-40 \cos \left (x \right )^{3}+33 \cos \left (x \right )\right ) \operatorname {csgn}\left (\csc \left (x \right )\right ) \sqrt {4}}{96}\)	\(44\)
risch	\(-\frac {i \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left (15 \,{\mathrm e}^{10 i x}-85 \,{\mathrm e}^{8 i x}+198 \,{\mathrm e}^{6 i x}+198 \,{\mathrm e}^{4 i x}-85 \,{\mathrm e}^{2 i x}+15\right )}{24 \left ({\mathrm e}^{2 i x}-1\right )^{5}}+\frac {5 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )}{8}-\frac {5 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )}{8}\)	\(129\)

[In]

int((csc(x)^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/96*csc(x)^6*(-15*ln(csc(x)-cot(x))*sin(x)^6+15*cos(x)^5-40*cos(x)^3+33*cos(x))*csgn(csc(x))*4^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (36) = 72\).

Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.86 \[ \int \csc ^2(x)^{7/2} \, dx=\frac {30 \, \cos \left (x\right )^{5} - 80 \, \cos \left (x\right )^{3} - 15 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 66 \, \cos \left (x\right )}{96 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \]

[In]

integrate((csc(x)^2)^(7/2),x, algorithm="fricas")

[Out]

1/96*(30*cos(x)^5 - 80*cos(x)^3 - 15*(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*log(1/2*cos(x) + 1/2) + 15*(cos(
x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*log(-1/2*cos(x) + 1/2) + 66*cos(x))/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 -
1)

Sympy [F(-1)]

Timed out. \[ \int \csc ^2(x)^{7/2} \, dx=\text {Timed out} \]

[In]

integrate((csc(x)**2)**(7/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1669 vs. \(2 (36) = 72\).

Time = 0.35 (sec) , antiderivative size = 1669, normalized size of antiderivative = 33.38 \[ \int \csc ^2(x)^{7/2} \, dx=\text {Too large to display} \]

[In]

integrate((csc(x)^2)^(7/2),x, algorithm="maxima")

[Out]

-1/96*(4*(15*cos(11*x) - 85*cos(9*x) + 198*cos(7*x) + 198*cos(5*x) - 85*cos(3*x) + 15*cos(x))*cos(12*x) - 60*(
6*cos(10*x) - 15*cos(8*x) + 20*cos(6*x) - 15*cos(4*x) + 6*cos(2*x) - 1)*cos(11*x) + 24*(85*cos(9*x) - 198*cos(
7*x) - 198*cos(5*x) + 85*cos(3*x) - 15*cos(x))*cos(10*x) - 340*(15*cos(8*x) - 20*cos(6*x) + 15*cos(4*x) - 6*co
s(2*x) + 1)*cos(9*x) + 60*(198*cos(7*x) + 198*cos(5*x) - 85*cos(3*x) + 15*cos(x))*cos(8*x) - 792*(20*cos(6*x)
- 15*cos(4*x) + 6*cos(2*x) - 1)*cos(7*x) - 80*(198*cos(5*x) - 85*cos(3*x) + 15*cos(x))*cos(6*x) + 792*(15*cos(
4*x) - 6*cos(2*x) + 1)*cos(5*x) - 300*(17*cos(3*x) - 3*cos(x))*cos(4*x) + 340*(6*cos(2*x) - 1)*cos(3*x) - 360*
cos(2*x)*cos(x) + 15*(2*(6*cos(10*x) - 15*cos(8*x) + 20*cos(6*x) - 15*cos(4*x) + 6*cos(2*x) - 1)*cos(12*x) - c
os(12*x)^2 + 12*(15*cos(8*x) - 20*cos(6*x) + 15*cos(4*x) - 6*cos(2*x) + 1)*cos(10*x) - 36*cos(10*x)^2 + 30*(20
*cos(6*x) - 15*cos(4*x) + 6*cos(2*x) - 1)*cos(8*x) - 225*cos(8*x)^2 + 40*(15*cos(4*x) - 6*cos(2*x) + 1)*cos(6*
x) - 400*cos(6*x)^2 + 30*(6*cos(2*x) - 1)*cos(4*x) - 225*cos(4*x)^2 - 36*cos(2*x)^2 + 2*(6*sin(10*x) - 15*sin(
8*x) + 20*sin(6*x) - 15*sin(4*x) + 6*sin(2*x))*sin(12*x) - sin(12*x)^2 + 12*(15*sin(8*x) - 20*sin(6*x) + 15*si
n(4*x) - 6*sin(2*x))*sin(10*x) - 36*sin(10*x)^2 + 30*(20*sin(6*x) - 15*sin(4*x) + 6*sin(2*x))*sin(8*x) - 225*s
in(8*x)^2 + 120*(5*sin(4*x) - 2*sin(2*x))*sin(6*x) - 400*sin(6*x)^2 - 225*sin(4*x)^2 + 180*sin(4*x)*sin(2*x) -
36*sin(2*x)^2 + 12*cos(2*x) - 1)*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) - 15*(2*(6*cos(10*x) - 15*cos(8*x) +
20*cos(6*x) - 15*cos(4*x) + 6*cos(2*x) - 1)*cos(12*x) - cos(12*x)^2 + 12*(15*cos(8*x) - 20*cos(6*x) + 15*cos(
4*x) - 6*cos(2*x) + 1)*cos(10*x) - 36*cos(10*x)^2 + 30*(20*cos(6*x) - 15*cos(4*x) + 6*cos(2*x) - 1)*cos(8*x) -
225*cos(8*x)^2 + 40*(15*cos(4*x) - 6*cos(2*x) + 1)*cos(6*x) - 400*cos(6*x)^2 + 30*(6*cos(2*x) - 1)*cos(4*x) -
225*cos(4*x)^2 - 36*cos(2*x)^2 + 2*(6*sin(10*x) - 15*sin(8*x) + 20*sin(6*x) - 15*sin(4*x) + 6*sin(2*x))*sin(1
2*x) - sin(12*x)^2 + 12*(15*sin(8*x) - 20*sin(6*x) + 15*sin(4*x) - 6*sin(2*x))*sin(10*x) - 36*sin(10*x)^2 + 30
*(20*sin(6*x) - 15*sin(4*x) + 6*sin(2*x))*sin(8*x) - 225*sin(8*x)^2 + 120*(5*sin(4*x) - 2*sin(2*x))*sin(6*x) -
400*sin(6*x)^2 - 225*sin(4*x)^2 + 180*sin(4*x)*sin(2*x) - 36*sin(2*x)^2 + 12*cos(2*x) - 1)*log(cos(x)^2 + sin
(x)^2 - 2*cos(x) + 1) + 4*(15*sin(11*x) - 85*sin(9*x) + 198*sin(7*x) + 198*sin(5*x) - 85*sin(3*x) + 15*sin(x))
*sin(12*x) - 60*(6*sin(10*x) - 15*sin(8*x) + 20*sin(6*x) - 15*sin(4*x) + 6*sin(2*x))*sin(11*x) + 24*(85*sin(9*
x) - 198*sin(7*x) - 198*sin(5*x) + 85*sin(3*x) - 15*sin(x))*sin(10*x) - 340*(15*sin(8*x) - 20*sin(6*x) + 15*si
n(4*x) - 6*sin(2*x))*sin(9*x) + 60*(198*sin(7*x) + 198*sin(5*x) - 85*sin(3*x) + 15*sin(x))*sin(8*x) - 792*(20*
sin(6*x) - 15*sin(4*x) + 6*sin(2*x))*sin(7*x) - 80*(198*sin(5*x) - 85*sin(3*x) + 15*sin(x))*sin(6*x) + 2376*(5
*sin(4*x) - 2*sin(2*x))*sin(5*x) - 300*(17*sin(3*x) - 3*sin(x))*sin(4*x) + 2040*sin(3*x)*sin(2*x) - 360*sin(2*
x)*sin(x) + 60*cos(x))/(2*(6*cos(10*x) - 15*cos(8*x) + 20*cos(6*x) - 15*cos(4*x) + 6*cos(2*x) - 1)*cos(12*x) -
cos(12*x)^2 + 12*(15*cos(8*x) - 20*cos(6*x) + 15*cos(4*x) - 6*cos(2*x) + 1)*cos(10*x) - 36*cos(10*x)^2 + 30*(
20*cos(6*x) - 15*cos(4*x) + 6*cos(2*x) - 1)*cos(8*x) - 225*cos(8*x)^2 + 40*(15*cos(4*x) - 6*cos(2*x) + 1)*cos(
6*x) - 400*cos(6*x)^2 + 30*(6*cos(2*x) - 1)*cos(4*x) - 225*cos(4*x)^2 - 36*cos(2*x)^2 + 2*(6*sin(10*x) - 15*si
n(8*x) + 20*sin(6*x) - 15*sin(4*x) + 6*sin(2*x))*sin(12*x) - sin(12*x)^2 + 12*(15*sin(8*x) - 20*sin(6*x) + 15*
sin(4*x) - 6*sin(2*x))*sin(10*x) - 36*sin(10*x)^2 + 30*(20*sin(6*x) - 15*sin(4*x) + 6*sin(2*x))*sin(8*x) - 225
*sin(8*x)^2 + 120*(5*sin(4*x) - 2*sin(2*x))*sin(6*x) - 400*sin(6*x)^2 - 225*sin(4*x)^2 + 180*sin(4*x)*sin(2*x)
- 36*sin(2*x)^2 + 12*cos(2*x) - 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (36) = 72\).

Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.58 \[ \int \csc ^2(x)^{7/2} \, dx=-\frac {\frac {45 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {9 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {{\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{384 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} - \frac {{\left (\frac {9 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {45 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {110 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}^{3}}{384 \, {\left (\cos \left (x\right ) - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (x\right )\right )} + \frac {5 \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )}{32 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

integrate((csc(x)^2)^(7/2),x, algorithm="giac")

[Out]

-1/384*(45*(cos(x) - 1)/(cos(x) + 1) - 9*(cos(x) - 1)^2/(cos(x) + 1)^2 + (cos(x) - 1)^3/(cos(x) + 1)^3)/sgn(si
n(x)) - 1/384*(9*(cos(x) - 1)/(cos(x) + 1) - 45*(cos(x) - 1)^2/(cos(x) + 1)^2 + 110*(cos(x) - 1)^3/(cos(x) + 1
)^3 - 1)*(cos(x) + 1)^3/((cos(x) - 1)^3*sgn(sin(x))) + 5/32*log(-(cos(x) - 1)/(cos(x) + 1))/sgn(sin(x))

Mupad [F(-1)]

Timed out. \[ \int \csc ^2(x)^{7/2} \, dx=\int {\left (\frac {1}{{\sin \left (x\right )}^2}\right )}^{7/2} \,d x \]

[In]

int((1/sin(x)^2)^(7/2),x)

[Out]

int((1/sin(x)^2)^(7/2), x)