\(\int (1+\csc ^2(x))^{3/2} \, dx\) [16]

Optimal result

Integrand size = 10, antiderivative size = 44 \[ \int \left (1+\csc ^2(x)\right )^{3/2} \, dx=-2 \text {arcsinh}\left (\frac {\cot (x)}{\sqrt {2}}\right )-\arctan \left (\frac {\cot (x)}{\sqrt {2+\cot ^2(x)}}\right )-\frac {1}{2} \cot (x) \sqrt {2+\cot ^2(x)} \] Output:

-2*arcsinh(1/2*cot(x)*2^(1/2))-arctan(cot(x)/(2+cot(x)^2)^(1/2))-1/2*cot(x 
)*(2+cot(x)^2)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(94\) vs. \(2(44)=88\).

Time = 0.17 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.14 \[ \int \left (1+\csc ^2(x)\right )^{3/2} \, dx=\frac {\left (1+\csc ^2(x)\right )^{3/2} \left (-4 \sqrt {2} \arctan \left (\frac {\sqrt {2} \cos (x)}{\sqrt {-3+\cos (2 x)}}\right )+\sqrt {-3+\cos (2 x)} \cot (x) \csc (x)-2 \sqrt {2} \log \left (\sqrt {2} \cos (x)+\sqrt {-3+\cos (2 x)}\right )\right ) \sin ^3(x)}{(-3+\cos (2 x))^{3/2}} \] Input:

Integrate[(1 + Csc[x]^2)^(3/2),x]
 

Output:

((1 + Csc[x]^2)^(3/2)*(-4*Sqrt[2]*ArcTan[(Sqrt[2]*Cos[x])/Sqrt[-3 + Cos[2* 
x]]] + Sqrt[-3 + Cos[2*x]]*Cot[x]*Csc[x] - 2*Sqrt[2]*Log[Sqrt[2]*Cos[x] + 
Sqrt[-3 + Cos[2*x]]])*Sin[x]^3)/(-3 + Cos[2*x])^(3/2)
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4616, 318, 27, 398, 222, 291, 216}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (\csc ^2(x)+1\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\sec \left (x+\frac {\pi }{2}\right )^2+1\right )^{3/2}dx\)

\(\Big \downarrow \) 4616

\(\displaystyle -\int \frac {\left (\cot ^2(x)+2\right )^{3/2}}{\cot ^2(x)+1}d\cot (x)\)

\(\Big \downarrow \) 318

\(\displaystyle -\frac {1}{2} \int \frac {2 \left (2 \cot ^2(x)+3\right )}{\left (\cot ^2(x)+1\right ) \sqrt {\cot ^2(x)+2}}d\cot (x)-\frac {1}{2} \sqrt {\cot ^2(x)+2} \cot (x)\)

\(\Big \downarrow \) 27

\(\displaystyle -\int \frac {2 \cot ^2(x)+3}{\left (\cot ^2(x)+1\right ) \sqrt {\cot ^2(x)+2}}d\cot (x)-\frac {1}{2} \sqrt {\cot ^2(x)+2} \cot (x)\)

\(\Big \downarrow \) 398

\(\displaystyle -2 \int \frac {1}{\sqrt {\cot ^2(x)+2}}d\cot (x)-\int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {\cot ^2(x)+2}}d\cot (x)-\frac {1}{2} \sqrt {\cot ^2(x)+2} \cot (x)\)

\(\Big \downarrow \) 222

\(\displaystyle -\int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {\cot ^2(x)+2}}d\cot (x)-2 \text {arcsinh}\left (\frac {\cot (x)}{\sqrt {2}}\right )-\frac {1}{2} \cot (x) \sqrt {\cot ^2(x)+2}\)

\(\Big \downarrow \) 291

\(\displaystyle -\int \frac {1}{\frac {\cot ^2(x)}{\cot ^2(x)+2}+1}d\frac {\cot (x)}{\sqrt {\cot ^2(x)+2}}-2 \text {arcsinh}\left (\frac {\cot (x)}{\sqrt {2}}\right )-\frac {1}{2} \cot (x) \sqrt {\cot ^2(x)+2}\)

\(\Big \downarrow \) 216

\(\displaystyle -2 \text {arcsinh}\left (\frac {\cot (x)}{\sqrt {2}}\right )-\arctan \left (\frac {\cot (x)}{\sqrt {\cot ^2(x)+2}}\right )-\frac {1}{2} \cot (x) \sqrt {\cot ^2(x)+2}\)

Input:

Int[(1 + Csc[x]^2)^(3/2),x]
 

Output:

-2*ArcSinh[Cot[x]/Sqrt[2]] - ArcTan[Cot[x]/Sqrt[2 + Cot[x]^2]] - (Cot[x]*S 
qrt[2 + Cot[x]^2])/2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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

rule 222

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 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4616

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = 
FreeFactors[Tan[e + f*x], x]}, Simp[ff/f   Subst[Int[(a + b + b*ff^2*x^2)^p 
/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] 
&& NeQ[a + b, 0] && NeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(189\) vs. \(2(37)=74\).

Time = 0.41 (sec) , antiderivative size = 190, normalized size of antiderivative = 4.32

method	result	size
default	\(\frac {\sqrt {4}\, \sin \left (x \right ) \left (\left (2 \cos \left (x \right )-2\right ) \ln \left (\frac {2 \sqrt {-\frac {\cos \left (x \right )^{2}-2}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )+2 \cos \left (x \right )+2 \sqrt {-\frac {\cos \left (x \right )^{2}-2}{\left (\cos \left (x \right )+1\right )^{2}}}+4}{\cos \left (x \right )+1}\right )+\left (2 \cos \left (x \right )-2\right ) \operatorname {arctanh}\left (\frac {\cos \left (x \right )-2}{\left (\cos \left (x \right )+1\right ) \sqrt {-\frac {\cos \left (x \right )^{2}-2}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )+\left (-2 \cos \left (x \right )+2\right ) \arctan \left (\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right ) \sqrt {-\frac {\cos \left (x \right )^{2}-2}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )+\sqrt {-\frac {\cos \left (x \right )^{2}-2}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )\right ) \left (-\cot \left (x \right )^{2}+2 \csc \left (x \right )^{2}\right )^{\frac {3}{2}}}{4 \left (\cos \left (x \right )^{2}-2\right ) \sqrt {-\frac {\cos \left (x \right )^{2}-2}{\left (\cos \left (x \right )+1\right )^{2}}}}\)	\(190\)

Input:

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

Output:

1/4*4^(1/2)*sin(x)*((2*cos(x)-2)*ln(2*((-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)* 
cos(x)+(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)+cos(x)+2)/(cos(x)+1))+(2*cos(x)- 
2)*arctanh((cos(x)-2)/(cos(x)+1)/(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2))+(-2*c 
os(x)+2)*arctan(cos(x)/(cos(x)+1)/(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2))+(-(c 
os(x)^2-2)/(cos(x)+1)^2)^(1/2)*cos(x))*(-cot(x)^2+2*csc(x)^2)^(3/2)/(cos(x 
)^2-2)/(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (37) = 74\).

Time = 0.09 (sec) , antiderivative size = 193, normalized size of antiderivative = 4.39 \[ \int \left (1+\csc ^2(x)\right )^{3/2} \, dx=\frac {\arctan \left (\frac {{\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sqrt {\frac {\cos \left (x\right )^{2} - 2}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right ) - \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} + 1}\right ) \sin \left (x\right ) - \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right ) \sin \left (x\right ) - 2 \, \log \left (-\cos \left (x\right )^{2} + \cos \left (x\right ) \sin \left (x\right ) - {\left (\cos \left (x\right )^{2} - \cos \left (x\right ) \sin \left (x\right ) - 1\right )} \sqrt {\frac {\cos \left (x\right )^{2} - 2}{\cos \left (x\right )^{2} - 1}} + 2\right ) \sin \left (x\right ) + 2 \, \log \left (-\cos \left (x\right )^{2} - \cos \left (x\right ) \sin \left (x\right ) - {\left (\cos \left (x\right )^{2} + \cos \left (x\right ) \sin \left (x\right ) - 1\right )} \sqrt {\frac {\cos \left (x\right )^{2} - 2}{\cos \left (x\right )^{2} - 1}} + 2\right ) \sin \left (x\right ) - \sqrt {\frac {\cos \left (x\right )^{2} - 2}{\cos \left (x\right )^{2} - 1}} \cos \left (x\right )}{2 \, \sin \left (x\right )} \] Input:

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

Output:

1/2*(arctan(((cos(x)^3 - cos(x))*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1))*sin(x 
) - cos(x)*sin(x))/(cos(x)^4 - 3*cos(x)^2 + 1))*sin(x) - arctan(sin(x)/cos 
(x))*sin(x) - 2*log(-cos(x)^2 + cos(x)*sin(x) - (cos(x)^2 - cos(x)*sin(x) 
- 1)*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1)) + 2)*sin(x) + 2*log(-cos(x)^2 - c 
os(x)*sin(x) - (cos(x)^2 + cos(x)*sin(x) - 1)*sqrt((cos(x)^2 - 2)/(cos(x)^ 
2 - 1)) + 2)*sin(x) - sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1))*cos(x))/sin(x)
 

Sympy [F]

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

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

Output:

Integral((csc(x)**2 + 1)**(3/2), x)
 

Maxima [F]

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

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

Output:

integrate((csc(x)^2 + 1)^(3/2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (37) = 74\).

Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 5.27 \[ \int \left (1+\csc ^2(x)\right )^{3/2} \, dx=2 \, \arctan \left (-\frac {1}{2} \, \tan \left (\frac {1}{2} \, x\right )^{2} + \frac {1}{2} \, \sqrt {\tan \left (\frac {1}{2} \, x\right )^{4} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1} - \frac {1}{2}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) - \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} - \sqrt {\tan \left (\frac {1}{2} \, x\right )^{4} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1} + 3\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) - \log \left (-\tan \left (\frac {1}{2} \, x\right )^{2} + \sqrt {\tan \left (\frac {1}{2} \, x\right )^{4} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1} + 1\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) + \log \left (-\tan \left (\frac {1}{2} \, x\right )^{2} + \sqrt {\tan \left (\frac {1}{2} \, x\right )^{4} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1} - 1\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) + \frac {1}{8} \, \sqrt {\tan \left (\frac {1}{2} \, x\right )^{4} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1} \mathrm {sgn}\left (\sin \left (x\right )\right ) + \frac {3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - \sqrt {\tan \left (\frac {1}{2} \, x\right )^{4} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) + \mathrm {sgn}\left (\sin \left (x\right )\right )}{4 \, {\left ({\left (\tan \left (\frac {1}{2} \, x\right )^{2} - \sqrt {\tan \left (\frac {1}{2} \, x\right )^{4} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )}^{2} - 1\right )}} \] Input:

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

Output:

2*arctan(-1/2*tan(1/2*x)^2 + 1/2*sqrt(tan(1/2*x)^4 + 6*tan(1/2*x)^2 + 1) - 
 1/2)*sgn(sin(x)) - log(tan(1/2*x)^2 - sqrt(tan(1/2*x)^4 + 6*tan(1/2*x)^2 
+ 1) + 3)*sgn(sin(x)) - log(-tan(1/2*x)^2 + sqrt(tan(1/2*x)^4 + 6*tan(1/2* 
x)^2 + 1) + 1)*sgn(sin(x)) + log(-tan(1/2*x)^2 + sqrt(tan(1/2*x)^4 + 6*tan 
(1/2*x)^2 + 1) - 1)*sgn(sin(x)) + 1/8*sqrt(tan(1/2*x)^4 + 6*tan(1/2*x)^2 + 
 1)*sgn(sin(x)) + 1/4*(3*(tan(1/2*x)^2 - sqrt(tan(1/2*x)^4 + 6*tan(1/2*x)^ 
2 + 1))*sgn(sin(x)) + sgn(sin(x)))/((tan(1/2*x)^2 - sqrt(tan(1/2*x)^4 + 6* 
tan(1/2*x)^2 + 1))^2 - 1)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (1+\csc ^2(x)\right )^{3/2} \, dx=\int \sqrt {\csc \left (x \right )^{2}+1}d x +\int \sqrt {\csc \left (x \right )^{2}+1}\, \csc \left (x \right )^{2}d x \] Input:

int((1+csc(x)^2)^(3/2),x)
 

Output:

int(sqrt(csc(x)**2 + 1),x) + int(sqrt(csc(x)**2 + 1)*csc(x)**2,x)