\(\int x^3 (a+b \csc ^{-1}(c x))^3 \, dx\) [24]

Optimal result

Integrand size = 14, antiderivative size = 207 \[ \int x^3 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^4}+\frac {i b^3 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 c^4} \]

[Out]

1/4*b^2*x^2*(a+b*arccsc(c*x))/c^2+1/2*I*b*(a+b*arccsc(c*x))^2/c^4+1/4*x^4*(a+b*arccsc(c*x))^3-b^2*(a+b*arccsc(
c*x))*ln(1-(I/c/x+(1-1/c^2/x^2)^(1/2))^2)/c^4+1/2*I*b^3*polylog(2,(I/c/x+(1-1/c^2/x^2)^(1/2))^2)/c^4+1/4*b^3*x
*(1-1/c^2/x^2)^(1/2)/c^3+1/2*b*x*(a+b*arccsc(c*x))^2*(1-1/c^2/x^2)^(1/2)/c^3+1/4*b*x^3*(a+b*arccsc(c*x))^2*(1-
1/c^2/x^2)^(1/2)/c

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5331, 4495, 4271, 3852, 8, 4269, 3798, 2221, 2317, 2438} \[ \int x^3 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=-\frac {b^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^4}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {i b^3 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 c^4}+\frac {b^3 x \sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3} \]

[In]

Int[x^3*(a + b*ArcCsc[c*x])^3,x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2221

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]
- Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[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 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[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 4271

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

Rule 4495

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] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /;
FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5331

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int (a+b x)^3 \cot (x) \csc ^4(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^4} \\ & = \frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {(3 b) \text {Subst}\left (\int (a+b x)^2 \csc ^4(x) \, dx,x,\csc ^{-1}(c x)\right )}{4 c^4} \\ & = \frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b \text {Subst}\left (\int (a+b x)^2 \csc ^2(x) \, dx,x,\csc ^{-1}(c x)\right )}{2 c^4}-\frac {b^3 \text {Subst}\left (\int \csc ^2(x) \, dx,x,\csc ^{-1}(c x)\right )}{4 c^4} \\ & = \frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b^2 \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\csc ^{-1}(c x)\right )}{c^4}+\frac {b^3 \text {Subst}\left (\int 1 \, dx,x,c \sqrt {1-\frac {1}{c^2 x^2}} x\right )}{4 c^4} \\ & = \frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )}{c^4} \\ & = \frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^4}+\frac {b^3 \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^4} \\ & = \frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^4}-\frac {\left (i b^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )}{2 c^4} \\ & = \frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^4}+\frac {i b^3 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.38 \[ \int x^3 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {2 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x+b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x+a b^2 c^2 x^2+a^2 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3+a^3 c^4 x^4+b^2 \left (3 a c^4 x^4+b \left (2 i+2 c \sqrt {1-\frac {1}{c^2 x^2}} x+c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3\right )\right ) \csc ^{-1}(c x)^2+b^3 c^4 x^4 \csc ^{-1}(c x)^3+b \csc ^{-1}(c x) \left (c x \left (b^2 c x+3 a^2 c^3 x^3+2 a b \sqrt {1-\frac {1}{c^2 x^2}} \left (2+c^2 x^2\right )\right )-4 b^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right )-4 a b^2 \log \left (\frac {1}{c x}\right )+2 i b^3 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{4 c^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.01

method	result	size
derivativedivides	\(\frac {\frac {a^{3} c^{4} x^{4}}{4}+b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right )^{3} c^{4} x^{4}}{4}+\frac {\operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{4}+\frac {\operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}{2}+\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}+\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )}{4}+\frac {x c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{4}-\frac {i}{4}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {\operatorname {arccsc}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\operatorname {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{6}+\frac {c^{2} x^{2}}{12}+\frac {\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}-\frac {\ln \left (\frac {1}{c x}\right )}{3}\right )+3 a^{2} b \left (\frac {c^{4} x^{4} \operatorname {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\)	\(417\)
default	\(\frac {\frac {a^{3} c^{4} x^{4}}{4}+b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right )^{3} c^{4} x^{4}}{4}+\frac {\operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{4}+\frac {\operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}{2}+\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}+\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )}{4}+\frac {x c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{4}-\frac {i}{4}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {\operatorname {arccsc}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\operatorname {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{6}+\frac {c^{2} x^{2}}{12}+\frac {\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}-\frac {\ln \left (\frac {1}{c x}\right )}{3}\right )+3 a^{2} b \left (\frac {c^{4} x^{4} \operatorname {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\)	\(417\)
parts	\(\frac {a^{3} x^{4}}{4}+\frac {b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right )^{3} c^{4} x^{4}}{4}+\frac {\operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{4}+\frac {\operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}{2}+\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}+\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )}{4}+\frac {x c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{4}-\frac {i}{4}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c^{4}}+\frac {3 a \,b^{2} \left (\frac {\operatorname {arccsc}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\operatorname {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{6}+\frac {c^{2} x^{2}}{12}+\frac {\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}-\frac {\ln \left (\frac {1}{c x}\right )}{3}\right )}{c^{4}}+\frac {3 a^{2} b \left (\frac {c^{4} x^{4} \operatorname {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\)	\(419\)

[In]

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

[Out]

1/c^4*(1/4*a^3*c^4*x^4+b^3*(1/4*arccsc(c*x)^3*c^4*x^4+1/4*arccsc(c*x)^2*((c^2*x^2-1)/c^2/x^2)^(1/2)*c^3*x^3+1/
2*arccsc(c*x)^2*((c^2*x^2-1)/c^2/x^2)^(1/2)*c*x+1/2*I*arccsc(c*x)^2+1/4*c^2*x^2*arccsc(c*x)+1/4*x*c*((c^2*x^2-
1)/c^2/x^2)^(1/2)-1/4*I-arccsc(c*x)*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))-arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2)
)+I*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))+I*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2)))+3*a*b^2*(1/4*arccsc(c*x)^2*c
^4*x^4+1/6*arccsc(c*x)*((c^2*x^2-1)/c^2/x^2)^(1/2)*c^3*x^3+1/12*c^2*x^2+1/3*arccsc(c*x)*c*x*((c^2*x^2-1)/c^2/x
^2)^(1/2)-1/3*ln(1/c/x))+3*a^2*b*(1/4*c^4*x^4*arccsc(c*x)+1/12*(c^2*x^2-1)*(c^2*x^2+2)/((c^2*x^2-1)/c^2/x^2)^(
1/2)/c/x))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

3/4*a*b^2*x^4*arccsc(c*x)^2 + 1/4*a^3*x^4 + 1/4*(3*x^4*arccsc(c*x) + (c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 3*x*s
qrt(-1/(c^2*x^2) + 1))/c^3)*a^2*b + 1/16*(4*x^4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 3*x^4*arctan2(1, s
qrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2 - 16*integrate(3/16*(16*c^2*x^5*arctan2(1, sqrt(c*x + 1)*sqrt(c*x -
1))*log(c)^2 - 16*x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c)^2 + 16*(c^2*x^5*arctan2(1, sqrt(c*x + 1)
*sqrt(c*x - 1)) - x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2 - (4*x^3*arctan2(1, sqrt(c*x + 1)*sqrt
(c*x - 1))^2 - x^3*log(c^2*x^2)^2)*sqrt(c*x + 1)*sqrt(c*x - 1) - 4*((4*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^5 - (4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*lo
g(c) + arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*x^3 + 4*(c^2*x^5*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) - x^3
*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x))*log(c^2*x^2) + 32*(c^2*x^5*arctan2(1, sqrt(c*x + 1)*sqrt(c*x
- 1))*log(c) - x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c))*log(x))/(c^2*x^2 - 1), x))*b^3 + 1/4*(2*c^
4*x^4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 2*c^2*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + (c^2*x^2 +
2*log(x^2))*sqrt(c*x + 1)*sqrt(c*x - 1) - 4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*a*b^2/(sqrt(c*x + 1)*sqr
t(c*x - 1)*c^4)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x^3*(a + b*asin(1/(c*x)))^3,x)

[Out]

int(x^3*(a + b*asin(1/(c*x)))^3, x)