3.1.26 \(\int x (a+b \csc ^{-1}(c x))^3 \, dx\) [26]

3.1.26.1 Optimal result

Integrand size = 12, antiderivative size = 126 \[ \int x \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {3 i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^2}+\frac {3 i b^3 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 c^2} \]

output

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

3.1.26.2 Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.44 \[ \int x \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {3 b^2 \left (a c^2 x^2+b \left (i+c \sqrt {1-\frac {1}{c^2 x^2}} x\right )\right ) \csc ^{-1}(c x)^2+b^3 c^2 x^2 \csc ^{-1}(c x)^3+3 b \csc ^{-1}(c x) \left (a c x \left (2 b \sqrt {1-\frac {1}{c^2 x^2}}+a c x\right )-2 b^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right )+a \left (a c x \left (3 b \sqrt {1-\frac {1}{c^2 x^2}}+a c x\right )-6 b^2 \log \left (\frac {1}{c x}\right )\right )+3 i b^3 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 c^2} \]

input

Integrate[x*(a + b*ArcCsc[c*x])^3,x]
 

output

(3*b^2*(a*c^2*x^2 + b*(I + c*Sqrt[1 - 1/(c^2*x^2)]*x))*ArcCsc[c*x]^2 + b^3 
*c^2*x^2*ArcCsc[c*x]^3 + 3*b*ArcCsc[c*x]*(a*c*x*(2*b*Sqrt[1 - 1/(c^2*x^2)] 
 + a*c*x) - 2*b^2*Log[1 - E^((2*I)*ArcCsc[c*x])]) + a*(a*c*x*(3*b*Sqrt[1 - 
 1/(c^2*x^2)] + a*c*x) - 6*b^2*Log[1/(c*x)]) + (3*I)*b^3*PolyLog[2, E^((2* 
I)*ArcCsc[c*x])])/(2*c^2)
 

3.1.26.3 Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5746, 4910, 3042, 4672, 3042, 25, 4200, 25, 2620, 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.

\(\displaystyle \int x \left (a+b \csc ^{-1}(c x)\right )^3 \, dx\)

\(\Big \downarrow \) 5746

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

\(\Big \downarrow \) 4910

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4672

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {3}{2} b \left (-c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-2 b \left (2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b}\right )\right )}{c^2}\)

input

Int[x*(a + b*ArcCsc[c*x])^3,x]
 

output

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

3.1.26.3.1 Defintions of rubi rules used

rule 25

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

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

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 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^ 
m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

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 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])
 

3.1.26.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (142 ) = 284\).

Time = 1.59 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.40

method	result	size
derivativedivides	\(\frac {\frac {c^{2} x^{2} a^{3}}{2}+b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right )^{2} \left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )+3 x c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-3 i\right )}{2}-3 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-3 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i \operatorname {arccsc}\left (c x \right )^{2}+3 i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}}{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-\ln \left (\frac {1}{c x}\right )\right )+3 a^{2} b \left (\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )}{2}+\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\)	\(303\)
default	\(\frac {\frac {c^{2} x^{2} a^{3}}{2}+b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right )^{2} \left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )+3 x c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-3 i\right )}{2}-3 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-3 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i \operatorname {arccsc}\left (c x \right )^{2}+3 i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}}{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-\ln \left (\frac {1}{c x}\right )\right )+3 a^{2} b \left (\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )}{2}+\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\)	\(303\)
parts	\(\frac {a^{3} x^{2}}{2}+\frac {b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right )^{2} \left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )+3 x c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-3 i\right )}{2}-3 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-3 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i \operatorname {arccsc}\left (c x \right )^{2}+3 i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c^{2}}+\frac {3 a \,b^{2} \left (\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}}{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-\ln \left (\frac {1}{c x}\right )\right )}{c^{2}}+\frac {3 a^{2} b \left (\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )}{2}+\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\)	\(305\)

input

int(x*(a+b*arccsc(c*x))^3,x,method=_RETURNVERBOSE)
 

output

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

3.1.26.5 Fricas [F]

\[ \int x \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} x \,d x } \]

input

integrate(x*(a+b*arccsc(c*x))^3,x, algorithm="fricas")
 

output

integral(b^3*x*arccsc(c*x)^3 + 3*a*b^2*x*arccsc(c*x)^2 + 3*a^2*b*x*arccsc( 
c*x) + a^3*x, x)
 

3.1.26.6 Sympy [F]

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

input

integrate(x*(a+b*acsc(c*x))**3,x)
 

output

Integral(x*(a + b*acsc(c*x))**3, x)
 

3.1.26.7 Maxima [F]

\[ \int x \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} x \,d x } \]

input

integrate(x*(a+b*arccsc(c*x))^3,x, algorithm="maxima")
 

output

3/2*a*b^2*x^2*arccsc(c*x)^2 + 1/2*a^3*x^2 + 3/2*(x^2*arccsc(c*x) + x*sqrt( 
-1/(c^2*x^2) + 1)/c)*a^2*b + 3*(x*sqrt(-1/(c^2*x^2) + 1)*arccsc(c*x)/c + l 
og(x)/c^2)*a*b^2 + 1/8*(4*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 
3*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2 - 8*integrate 
(3/8*(8*c^2*x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c)^2 - 8*x*arc 
tan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c)^2 + 8*(c^2*x^3*arctan2(1, sqrt 
(c*x + 1)*sqrt(c*x - 1)) - x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log( 
x)^2 - (4*x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^2 - x*log(c^2*x^2)^2)* 
sqrt(c*x + 1)*sqrt(c*x - 1) - 4*((2*c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x 
- 1))*log(c) + c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*x^3 - (2*arcta 
n2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c) + arctan2(1, sqrt(c*x + 1)*sqrt( 
c*x - 1)))*x + 2*(c^2*x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) - x*arct 
an2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x))*log(c^2*x^2) + 16*(c^2*x^3*ar 
ctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c) - x*arctan2(1, sqrt(c*x + 1)* 
sqrt(c*x - 1))*log(c))*log(x))/(c^2*x^2 - 1), x))*b^3
 

3.1.26.8 Giac [F]

\[ \int x \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} x \,d x } \]

input

integrate(x*(a+b*arccsc(c*x))^3,x, algorithm="giac")
 

output

integrate((b*arccsc(c*x) + a)^3*x, x)
 

3.1.26.9 Mupad [F(-1)]

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

input

int(x*(a + b*asin(1/(c*x)))^3,x)
 

output

int(x*(a + b*asin(1/(c*x)))^3, x)