3.7.49 \(\int x^3 \csc ^{-1}(x)^2 \, dx\) [649]

3.7.49.1 Optimal result

Integrand size = 8, antiderivative size = 63 \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\frac {x^2}{12}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)+\frac {1}{6} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)+\frac {1}{4} x^4 \csc ^{-1}(x)^2+\frac {\log (x)}{3} \]

output

1/12*x^2+1/4*x^4*arccsc(x)^2+1/3*ln(x)+1/3*x*arccsc(x)*(1-1/x^2)^(1/2)+1/6 
*x^3*arccsc(x)*(1-1/x^2)^(1/2)
 

3.7.49.2 Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.67 \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\frac {1}{12} \left (x^2+2 \sqrt {1-\frac {1}{x^2}} x \left (2+x^2\right ) \csc ^{-1}(x)+3 x^4 \csc ^{-1}(x)^2+4 \log (x)\right ) \]

input

Integrate[x^3*ArcCsc[x]^2,x]
 

output

(x^2 + 2*Sqrt[1 - x^(-2)]*x*(2 + x^2)*ArcCsc[x] + 3*x^4*ArcCsc[x]^2 + 4*Lo 
g[x])/12
 

3.7.49.3 Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {5746, 4245, 3042, 4673, 3042, 4672, 3042, 25, 3956}

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.

\(\displaystyle \int x^3 \csc ^{-1}(x)^2 \, dx\)

\(\Big \downarrow \) 5746

\(\displaystyle -\int \sqrt {1-\frac {1}{x^2}} x^5 \csc ^{-1}(x)^2d\csc ^{-1}(x)\)

\(\Big \downarrow \) 4245

\(\displaystyle \frac {1}{4} x^4 \csc ^{-1}(x)^2-\frac {1}{2} \int x^4 \csc ^{-1}(x)d\csc ^{-1}(x)\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} x^4 \csc ^{-1}(x)^2-\frac {1}{2} \int \csc ^{-1}(x) \csc \left (\csc ^{-1}(x)\right )^4d\csc ^{-1}(x)\)

\(\Big \downarrow \) 4673

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \int x^2 \csc ^{-1}(x)d\csc ^{-1}(x)+\frac {x^2}{6}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)\right )+\frac {1}{4} x^4 \csc ^{-1}(x)^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \int \csc ^{-1}(x) \csc \left (\csc ^{-1}(x)\right )^2d\csc ^{-1}(x)+\frac {x^2}{6}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)\right )+\frac {1}{4} x^4 \csc ^{-1}(x)^2\)

\(\Big \downarrow \) 4672

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (\int \sqrt {1-\frac {1}{x^2}} xd\csc ^{-1}(x)-\sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)\right )+\frac {x^2}{6}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)\right )+\frac {1}{4} x^4 \csc ^{-1}(x)^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (\int -\tan \left (\csc ^{-1}(x)+\frac {\pi }{2}\right )d\csc ^{-1}(x)-\sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)\right )+\frac {x^2}{6}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)\right )+\frac {1}{4} x^4 \csc ^{-1}(x)^2\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (-\int \tan \left (\csc ^{-1}(x)+\frac {\pi }{2}\right )d\csc ^{-1}(x)-\sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)\right )+\frac {x^2}{6}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)\right )+\frac {1}{4} x^4 \csc ^{-1}(x)^2\)

\(\Big \downarrow \) 3956

\(\displaystyle \frac {1}{4} x^4 \csc ^{-1}(x)^2+\frac {1}{2} \left (\frac {x^2}{6}-\frac {2}{3} \left (\log \left (\frac {1}{x}\right )-\sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)\right )+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)\right )\)

input

Int[x^3*ArcCsc[x]^2,x]
 

output

(x^4*ArcCsc[x]^2)/4 + (x^2/6 + (Sqrt[1 - x^(-2)]*x^3*ArcCsc[x])/3 - (2*(-( 
Sqrt[1 - x^(-2)]*x*ArcCsc[x]) + Log[x^(-1)]))/3)/2
 

3.7.49.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 4245

Int[Cot[(a_.) + (b_.)*(x_)^(n_.)]^(q_.)*Csc[(a_.) + (b_.)*(x_)^(n_.)]^(p_.) 
*(x_)^(m_.), x_Symbol] :> Simp[(-x^(m - n + 1))*(Csc[a + b*x^n]^p/(b*n*p)), 
 x] + Simp[(m - n + 1)/(b*n*p)   Int[x^(m - n)*Csc[a + b*x^n]^p, x], x] /; 
FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m, n] && EqQ[q, 1]
 

rule 4672

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1) 
*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 4673

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> 
 Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), 
 x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S 
imp[b^2*((n - 2)/(n - 1))   Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) 
 /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
 

rule 5746

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- 
(c^(m + 1))^(-1)   Subst[Int[(a + b*x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcC 
sc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n 
, 0] || LtQ[m, -1])
 

3.7.49.4 Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {x^{4} \operatorname {arccsc}\left (x \right )^{2}}{4}+\frac {x^{3} \operatorname {arccsc}\left (x \right ) \sqrt {\frac {x^{2}-1}{x^{2}}}}{6}+\frac {x^{2}}{12}+\frac {\operatorname {arccsc}\left (x \right ) \sqrt {\frac {x^{2}-1}{x^{2}}}\, x}{3}-\frac {\ln \left (\frac {1}{x}\right )}{3}\)	\(56\)

input

int(x^3*arccsc(x)^2,x,method=_RETURNVERBOSE)
 

output

1/4*x^4*arccsc(x)^2+1/6*x^3*arccsc(x)*((x^2-1)/x^2)^(1/2)+1/12*x^2+1/3*arc 
csc(x)*((x^2-1)/x^2)^(1/2)*x-1/3*ln(1/x)
 

3.7.49.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.56 \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\frac {1}{4} \, x^{4} \operatorname {arccsc}\left (x\right )^{2} + \frac {1}{6} \, {\left (x^{2} + 2\right )} \sqrt {x^{2} - 1} \operatorname {arccsc}\left (x\right ) + \frac {1}{12} \, x^{2} + \frac {1}{3} \, \log \left (x\right ) \]

input

integrate(x^3*arccsc(x)^2,x, algorithm="fricas")
 

output

1/4*x^4*arccsc(x)^2 + 1/6*(x^2 + 2)*sqrt(x^2 - 1)*arccsc(x) + 1/12*x^2 + 1 
/3*log(x)
 

3.7.49.6 Sympy [F]

\[ \int x^3 \csc ^{-1}(x)^2 \, dx=\int x^{3} \operatorname {acsc}^{2}{\left (x \right )}\, dx \]

input

integrate(x**3*acsc(x)**2,x)
 

output

Integral(x**3*acsc(x)**2, x)
 

3.7.49.7 Maxima [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.51 \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\frac {1}{4} \, x^{4} \operatorname {arccsc}\left (x\right )^{2} + \frac {2 \, x^{4} \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right ) + 2 \, x^{2} \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right ) + {\left (x^{2} + 2 \, \log \left (x^{2}\right )\right )} \sqrt {x + 1} \sqrt {x - 1} - 4 \, \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right )}{12 \, \sqrt {x + 1} \sqrt {x - 1}} \]

input

integrate(x^3*arccsc(x)^2,x, algorithm="maxima")
 

output

1/4*x^4*arccsc(x)^2 + 1/12*(2*x^4*arctan2(1, sqrt(x + 1)*sqrt(x - 1)) + 2* 
x^2*arctan2(1, sqrt(x + 1)*sqrt(x - 1)) + (x^2 + 2*log(x^2))*sqrt(x + 1)*s 
qrt(x - 1) - 4*arctan2(1, sqrt(x + 1)*sqrt(x - 1)))/(sqrt(x + 1)*sqrt(x - 
1))
 

3.7.49.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (49) = 98\).

Time = 0.31 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.68 \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\frac {1}{4} \, x^{4} \arcsin \left (\frac {1}{x}\right )^{2} + \frac {1}{12} \, x^{2} {\left (\frac {2}{x^{2}} + 1\right )} + \frac {1}{48} \, {\left (x^{3} {\left (\sqrt {-\frac {1}{x^{2}} + 1} - 1\right )}^{3} + 9 \, x {\left (\sqrt {-\frac {1}{x^{2}} + 1} - 1\right )} - \frac {9 \, x^{2} {\left (\sqrt {-\frac {1}{x^{2}} + 1} - 1\right )}^{2} + 1}{x^{3} {\left (\sqrt {-\frac {1}{x^{2}} + 1} - 1\right )}^{3}}\right )} \arcsin \left (\frac {1}{x}\right ) - \frac {1}{6} \, \log \left (\frac {1}{x^{2}}\right ) \]

input

integrate(x^3*arccsc(x)^2,x, algorithm="giac")
 

output

1/4*x^4*arcsin(1/x)^2 + 1/12*x^2*(2/x^2 + 1) + 1/48*(x^3*(sqrt(-1/x^2 + 1) 
 - 1)^3 + 9*x*(sqrt(-1/x^2 + 1) - 1) - (9*x^2*(sqrt(-1/x^2 + 1) - 1)^2 + 1 
)/(x^3*(sqrt(-1/x^2 + 1) - 1)^3))*arcsin(1/x) - 1/6*log(x^(-2))
 

3.7.49.9 Mupad [F(-1)]

Timed out. \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\int x^3\,{\mathrm {asin}\left (\frac {1}{x}\right )}^2 \,d x \]

input

int(x^3*asin(1/x)^2,x)
 

output

int(x^3*asin(1/x)^2, x)
 

3.7.49.10 Reduce [F]

\[ \int x^3 \csc ^{-1}(x)^2 \, dx=\int \mathit {acsc} \left (x \right )^{2} x^{3}d x \]

input

int(acsc(x)**2*x**3,x)
 

output

int(acsc(x)**2*x**3,x)