\(\int x^2 (a+b \csc ^{-1}(c x))^2 \, dx\) [16]

Optimal result

Integrand size = 14, antiderivative size = 139 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\frac {b^2 x}{3 c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^2+\frac {2 b \left (a+b \csc ^{-1}(c x)\right ) \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{3 c^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{3 c^3} \] Output:

1/3*b^2*x/c^2+1/3*b*(1-1/c^2/x^2)^(1/2)*x^2*(a+b*arccsc(c*x))/c+1/3*x^3*(a 
+b*arccsc(c*x))^2+2/3*b*(a+b*arccsc(c*x))*arctanh(I/c/x+(1-1/c^2/x^2)^(1/2 
))/c^3-1/3*I*b^2*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))/c^3+1/3*I*b^2*polyl 
og(2,I/c/x+(1-1/c^2/x^2)^(1/2))/c^3
 

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.53 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\frac {1}{3} \left (a^2 x^3+2 a b x^3 \csc ^{-1}(c x)+\frac {a b \left (-c x+c^3 x^3-\sqrt {-1+c^2 x^2} \log \left (-c x+\sqrt {-1+c^2 x^2}\right )\right )}{c^4 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {i b^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^2 \left (c x+c^3 x^3 \csc ^{-1}(c x)^2+\csc ^{-1}(c x) \left (c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2-\log \left (1-e^{i \csc ^{-1}(c x)}\right )+\log \left (1+e^{i \csc ^{-1}(c x)}\right )\right )+i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )\right )}{c^3}\right ) \] Input:

Integrate[x^2*(a + b*ArcCsc[c*x])^2,x]
 

Output:

(a^2*x^3 + 2*a*b*x^3*ArcCsc[c*x] + (a*b*(-(c*x) + c^3*x^3 - Sqrt[-1 + c^2* 
x^2]*Log[-(c*x) + Sqrt[-1 + c^2*x^2]]))/(c^4*Sqrt[1 - 1/(c^2*x^2)]*x) - (I 
*b^2*PolyLog[2, -E^(I*ArcCsc[c*x])])/c^3 + (b^2*(c*x + c^3*x^3*ArcCsc[c*x] 
^2 + ArcCsc[c*x]*(c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2 - Log[1 - E^(I*ArcCsc[c*x] 
)] + Log[1 + E^(I*ArcCsc[c*x])]) + I*PolyLog[2, E^(I*ArcCsc[c*x])]))/c^3)/ 
3
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5746, 4910, 3042, 4673, 3042, 4671, 2715, 2838}

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.

\(\displaystyle \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx\)

\(\Big \downarrow \) 5746

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

\(\Big \downarrow \) 4910

\(\displaystyle -\frac {\frac {2}{3} b \int c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )d\csc ^{-1}(c x)-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{c^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {2}{3} b \int \left (a+b \csc ^{-1}(c x)\right ) \csc \left (\csc ^{-1}(c x)\right )^3d\csc ^{-1}(c x)-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{c^3}\)

\(\Big \downarrow \) 4673

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

\(\displaystyle -\frac {-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^2+\frac {2}{3} b \left (\frac {1}{2} \left (-b \int \log \left (1-e^{i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)+b \int \log \left (1+e^{i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)-2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )\right )-\frac {1}{2} c^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac {b c x}{2}\right )}{c^3}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^2+\frac {2}{3} b \left (\frac {1}{2} \left (i b \int e^{-i \csc ^{-1}(c x)} \log \left (1-e^{i \csc ^{-1}(c x)}\right )de^{i \csc ^{-1}(c x)}-i b \int e^{-i \csc ^{-1}(c x)} \log \left (1+e^{i \csc ^{-1}(c x)}\right )de^{i \csc ^{-1}(c x)}-2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )\right )-\frac {1}{2} c^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac {b c x}{2}\right )}{c^3}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^2+\frac {2}{3} b \left (\frac {1}{2} \left (-2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+i b \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )\right )-\frac {1}{2} c^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac {b c x}{2}\right )}{c^3}\)

Input:

Int[x^2*(a + b*ArcCsc[c*x])^2,x]
 

Output:

-((-1/3*(c^3*x^3*(a + b*ArcCsc[c*x])^2) + (2*b*(-1/2*(b*c*x) - (c^2*Sqrt[1 
 - 1/(c^2*x^2)]*x^2*(a + b*ArcCsc[c*x]))/2 + (-2*(a + b*ArcCsc[c*x])*ArcTa 
nh[E^(I*ArcCsc[c*x])] + I*b*PolyLog[2, -E^(I*ArcCsc[c*x])] - I*b*PolyLog[2 
, E^(I*ArcCsc[c*x])])/2))/3)/c^3)
 

Defintions of rubi rules used

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[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 4910

Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d 
_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Csc[a + b*x]^n/(b*n)), x 
] + Simp[d*(m/(b*n))   Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Free 
Q[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
 

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])
 

Maple [A] (verified)

Time = 0.96 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.93

method	result	size
parts	\(\frac {a^{2} x^{3}}{3}+\frac {b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}\right )}{c^{3}}+\frac {2 a b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\)	\(268\)
derivativedivides	\(\frac {\frac {c^{3} x^{3} a^{2}}{3}+b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}\right )+2 a b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\)	\(269\)
default	\(\frac {\frac {c^{3} x^{3} a^{2}}{3}+b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}\right )+2 a b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\)	\(269\)

Input:

int(x^2*(a+b*arccsc(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

1/3*a^2*x^3+b^2/c^3*(1/3*(c^2*x^2*arccsc(c*x)^2+arccsc(c*x)*c*x*((c^2*x^2- 
1)/c^2/x^2)^(1/2)+1)*c*x+1/3*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))-1 
/3*I*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))-1/3*arccsc(c*x)*ln(1-I/c/x-(1-1 
/c^2/x^2)^(1/2))+1/3*I*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2)))+2*a*b/c^3*(1/ 
3*c^3*x^3*arccsc(c*x)+1/6*(c^2*x^2-1)^(1/2)*(c*x*(c^2*x^2-1)^(1/2)+ln(c*x+ 
(c^2*x^2-1)^(1/2)))/((c^2*x^2-1)/c^2/x^2)^(1/2)/c/x)
 

Fricas [F]

\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:

integrate(x^2*(a+b*arccsc(c*x))^2,x, algorithm="fricas")
 

Output:

integral(b^2*x^2*arccsc(c*x)^2 + 2*a*b*x^2*arccsc(c*x) + a^2*x^2, x)
 

Sympy [F]

\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}\, dx \] Input:

integrate(x**2*(a+b*acsc(c*x))**2,x)
 

Output:

Integral(x**2*(a + b*acsc(c*x))**2, x)
 

Maxima [F]

\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:

integrate(x^2*(a+b*arccsc(c*x))^2,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

1/3*a^2*x^3 + 1/6*(4*x^3*arccsc(c*x) + (2*sqrt(-1/(c^2*x^2) + 1)/(c^2*(1/( 
c^2*x^2) - 1) + c^2) + log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 - log(sqrt(-1/( 
c^2*x^2) + 1) - 1)/c^2)/c)*a*b + 1/12*(4*x^3*arctan2(1, sqrt(c*x + 1)*sqrt 
(c*x - 1))^2 - x^3*log(c^2*x^2)^2 - 2*c^2*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c 
*x + 1)/c^5 + 3*log(c*x - 1)/c^5)*log(c)^2 + 36*c^2*integrate(1/3*x^4*log( 
c^2*x^2)/(c^2*x^2 - 1), x)*log(c) - 72*c^2*integrate(1/3*x^4*log(x)/(c^2*x 
^2 - 1), x)*log(c) + 36*c^2*integrate(1/3*x^4*log(c^2*x^2)*log(x)/(c^2*x^2 
 - 1), x) - 36*c^2*integrate(1/3*x^4*log(x)^2/(c^2*x^2 - 1), x) + 12*c^2*i 
ntegrate(1/3*x^4*log(c^2*x^2)/(c^2*x^2 - 1), x) + 6*(2*x/c^2 - log(c*x + 1 
)/c^3 + log(c*x - 1)/c^3)*log(c)^2 - 36*integrate(1/3*x^2*log(c^2*x^2)/(c^ 
2*x^2 - 1), x)*log(c) + 72*integrate(1/3*x^2*log(x)/(c^2*x^2 - 1), x)*log( 
c) + 24*integrate(1/3*sqrt(c*x + 1)*sqrt(c*x - 1)*x^2*arctan(1/(sqrt(c*x + 
 1)*sqrt(c*x - 1)))/(c^2*x^2 - 1), x) - 36*integrate(1/3*x^2*log(c^2*x^2)* 
log(x)/(c^2*x^2 - 1), x) + 36*integrate(1/3*x^2*log(x)^2/(c^2*x^2 - 1), x) 
 - 12*integrate(1/3*x^2*log(c^2*x^2)/(c^2*x^2 - 1), x))*b^2
 

Giac [F(-2)]

Exception generated. \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^2*(a+b*arccsc(c*x))^2,x, algorithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \] Input:

int(x^2*(a + b*asin(1/(c*x)))^2,x)
 

Output:

int(x^2*(a + b*asin(1/(c*x)))^2, x)
 

Reduce [F]

\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=2 \left (\int \mathit {acsc} \left (c x \right ) x^{2}d x \right ) a b +\left (\int \mathit {acsc} \left (c x \right )^{2} x^{2}d x \right ) b^{2}+\frac {a^{2} x^{3}}{3} \] Input:

int(x^2*(a+b*acsc(c*x))^2,x)
 

Output:

(6*int(acsc(c*x)*x**2,x)*a*b + 3*int(acsc(c*x)**2*x**2,x)*b**2 + a**2*x**3 
)/3