3.39 \(\int \csc ^2(x)^{7/2} \, dx\)

Optimal. Leaf size=50 \[ -\frac{1}{6} \cot (x) \csc ^2(x)^{5/2}-\frac{5}{24} \cot (x) \csc ^2(x)^{3/2}-\frac{5}{16} \cot (x) \sqrt{\csc ^2(x)}-\frac{5}{16} \sinh ^{-1}(\cot (x)) \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.0168905, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4122, 195, 215} \[ -\frac{1}{6} \cot (x) \csc ^2(x)^{5/2}-\frac{5}{24} \cot (x) \csc ^2(x)^{3/2}-\frac{5}{16} \cot (x) \sqrt{\csc ^2(x)}-\frac{5}{16} \sinh ^{-1}(\cot (x)) \]

Antiderivative was successfully verified.

[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 4122

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
]

Rule 195

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 215

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]

Rubi steps

\begin{align*} \int \csc ^2(x)^{7/2} \, dx &=-\operatorname{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} \operatorname{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} \operatorname{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} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac{5}{16} \sinh ^{-1}(\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]  time = 0.257745, size = 92, normalized size = 1.84 \[ \frac{1}{384} \sin (x) \sqrt{\csc ^2(x)} \left (-\csc ^6\left (\frac{x}{2}\right )-6 \csc ^4\left (\frac{x}{2}\right )-30 \csc ^2\left (\frac{x}{2}\right )+\sec ^6\left (\frac{x}{2}\right )+6 \sec ^4\left (\frac{x}{2}\right )+30 \sec ^2\left (\frac{x}{2}\right )-120 \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )\right ) \]

Antiderivative was successfully verified.

[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 [B]  time = 0.119, size = 101, normalized size = 2. \begin{align*} -{\frac{\sqrt{4}\sin \left ( x \right ) }{96} \left ( 15\, \left ( \cos \left ( x \right ) \right ) ^{6}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +15\, \left ( \cos \left ( x \right ) \right ) ^{5}-45\, \left ( \cos \left ( x \right ) \right ) ^{4}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) -40\, \left ( \cos \left ( x \right ) \right ) ^{3}+45\, \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +33\,\cos \left ( x \right ) -15\,\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) \right ) \left ( - \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-1} \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((csc(x)^2)^(7/2),x)

[Out]

-1/96*4^(1/2)*(15*cos(x)^6*ln(-(-1+cos(x))/sin(x))+15*cos(x)^5-45*cos(x)^4*ln(-(-1+cos(x))/sin(x))-40*cos(x)^3
+45*cos(x)^2*ln(-(-1+cos(x))/sin(x))+33*cos(x)-15*ln(-(-1+cos(x))/sin(x)))*sin(x)*(-1/(cos(x)^2-1))^(7/2)

________________________________________________________________________________________

Maxima [B]  time = 1.96895, size = 2253, normalized size = 45.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [B]  time = 0.486982, size = 302, normalized size = 6.04 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.29056, size = 174, normalized size = 3.48 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))