\(\int (a+b \csc ^{-1}(c x))^2 \, dx\) [18]

Optimal result

Integrand size = 10, antiderivative size = 84 \[ \int \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=x \left (a+b \csc ^{-1}(c x)\right )^2+\frac {4 b \left (a+b \csc ^{-1}(c x)\right ) \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {2 i b^2 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c} \] Output:

x*(a+b*arccsc(c*x))^2+4*b*(a+b*arccsc(c*x))*arctanh(I/c/x+(1-1/c^2/x^2)^(1 
/2))/c-2*I*b^2*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))/c+2*I*b^2*polylog(2,I 
/c/x+(1-1/c^2/x^2)^(1/2))/c
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.75 \[ \int \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\frac {a^2 c x+2 a b c x \csc ^{-1}(c x)+b^2 c x \csc ^{-1}(c x)^2-2 b^2 \csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right )+2 b^2 \csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right )+2 a b \log \left (\cos \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )-2 a b \log \left (\sin \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )-2 i b^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )+2 i b^2 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5740, 4910, 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.

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

\(\Big \downarrow \) 5740

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

\(\Big \downarrow \) 4910

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((-(c*x*(a + b*ArcCsc[c*x])^2) + 2*b*(-2*(a + b*ArcCsc[c*x])*ArcTanh[E^(I 
*ArcCsc[c*x])] + I*b*PolyLog[2, -E^(I*ArcCsc[c*x])] - I*b*PolyLog[2, E^(I* 
ArcCsc[c*x])]))/c)
 

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 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 5740

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-c^(-1)   Su 
bst[Int[(a + b*x)^n*Csc[x]*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, 
c, n}, x] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.10

method	result	size
derivativedivides	\(\frac {a^{2} c x +b^{2} \left (\operatorname {arccsc}\left (c x \right )^{2} c x -2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 i \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {dilog}\left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+2 a b \left (\operatorname {arccsc}\left (c x \right ) c x +\ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}\)	\(176\)
default	\(\frac {a^{2} c x +b^{2} \left (\operatorname {arccsc}\left (c x \right )^{2} c x -2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 i \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {dilog}\left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+2 a b \left (\operatorname {arccsc}\left (c x \right ) c x +\ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}\)	\(176\)
parts	\(a^{2} x +\frac {b^{2} \left (\operatorname {arccsc}\left (c x \right )^{2} c x -2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 i \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {dilog}\left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}+2 \,\operatorname {arccsc}\left (c x \right ) a b x +\frac {2 a b \ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c}\)	\(178\)

Input:

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

Output:

1/c*(a^2*c*x+b^2*(arccsc(c*x)^2*c*x-2*arccsc(c*x)*ln(1-I/c/x-(1-1/c^2/x^2) 
^(1/2))+2*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))-2*I*dilog(1+I/c/x+(1 
-1/c^2/x^2)^(1/2))+2*I*dilog(1-I/c/x-(1-1/c^2/x^2)^(1/2)))+2*a*b*(arccsc(c 
*x)*c*x+ln(c*x+c*x*(1-1/c^2/x^2)^(1/2))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/4*(2*c^2*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3)*log(c)^2 - 4*c 
^2*integrate(x^2*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) + 8*c^2*integrate(x 
^2*log(x)/(c^2*x^2 - 1), x)*log(c) - 4*x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x 
 - 1))^2 - 4*c^2*integrate(x^2*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) + 4*c 
^2*integrate(x^2*log(x)^2/(c^2*x^2 - 1), x) - 4*c^2*integrate(x^2*log(c^2* 
x^2)/(c^2*x^2 - 1), x) + x*log(c^2*x^2)^2 + 2*(log(c*x + 1)/c - log(c*x - 
1)/c)*log(c)^2 + 4*integrate(log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) - 8*int 
egrate(log(x)/(c^2*x^2 - 1), x)*log(c) - 8*integrate(sqrt(c*x + 1)*sqrt(c* 
x - 1)*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^2 - 1), x) + 4*integ 
rate(log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) - 4*integrate(log(x)^2/(c^2*x^2 
 - 1), x) + 4*integrate(log(c^2*x^2)/(c^2*x^2 - 1), x))*b^2 + a^2*x + (2*c 
*x*arccsc(c*x) + log(sqrt(-1/(c^2*x^2) + 1) + 1) - log(-sqrt(-1/(c^2*x^2) 
+ 1) + 1))*a*b/c
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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