\(\int (a+b \csc ^{-1}(c x))^3 \, dx\) [27]

Optimal result

Integrand size = 10, antiderivative size = 144 \[ \int \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=x \left (a+b \csc ^{-1}(c x)\right )^3+\frac {6 b \left (a+b \csc ^{-1}(c x)\right )^2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {6 b^3 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {6 b^3 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5325, 4495, 4268, 2611, 2320, 6724} \[ \int \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {6 b \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2}{c}-\frac {6 i b^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac {6 i b^2 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c}+x \left (a+b \csc ^{-1}(c x)\right )^3+\frac {6 b^3 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {6 b^3 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c} \]

[In]

Int[(a + b*ArcCsc[c*x])^3,x]

[Out]

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

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4268

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] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

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 5325

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

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int (a+b x)^3 \cot (x) \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{c} \\ & = x \left (a+b \csc ^{-1}(c x)\right )^3-\frac {(3 b) \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{c} \\ & = x \left (a+b \csc ^{-1}(c x)\right )^3+\frac {6 b \left (a+b \csc ^{-1}(c x)\right )^2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c}-\frac {\left (6 b^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c} \\ & = x \left (a+b \csc ^{-1}(c x)\right )^3+\frac {6 b \left (a+b \csc ^{-1}(c x)\right )^2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {\left (6 i b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c}-\frac {\left (6 i b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c} \\ & = x \left (a+b \csc ^{-1}(c x)\right )^3+\frac {6 b \left (a+b \csc ^{-1}(c x)\right )^2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {\left (6 b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {\left (6 b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c} \\ & = x \left (a+b \csc ^{-1}(c x)\right )^3+\frac {6 b \left (a+b \csc ^{-1}(c x)\right )^2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {6 b^3 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {6 b^3 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.84 \[ \int \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {a^3 c x+3 a^2 b c x \csc ^{-1}(c x)+3 a b^2 c x \csc ^{-1}(c x)^2+b^3 c x \csc ^{-1}(c x)^3-6 a b^2 \csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right )-3 b^3 \csc ^{-1}(c x)^2 \log \left (1-e^{i \csc ^{-1}(c x)}\right )+6 a b^2 \csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right )+3 b^3 \csc ^{-1}(c x)^2 \log \left (1+e^{i \csc ^{-1}(c x)}\right )+3 a^2 b \log \left (c \left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )-6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )+6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )+6 b^3 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )-6 b^3 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c} \]

[In]

Integrate[(a + b*ArcCsc[c*x])^3,x]

[Out]

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

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.62

method	result	size
derivativedivides	\(\frac {c x \,a^{3}+b^{3} \left (\operatorname {arccsc}\left (c x \right )^{3} c x -3 \operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 \operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccsc}\left (c x \right )^{2} c x -2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 i \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {dilog}\left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a^{2} b \left (\operatorname {arccsc}\left (c x \right ) c x +\ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}\)	\(378\)
default	\(\frac {c x \,a^{3}+b^{3} \left (\operatorname {arccsc}\left (c x \right )^{3} c x -3 \operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 \operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccsc}\left (c x \right )^{2} c x -2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 i \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {dilog}\left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a^{2} b \left (\operatorname {arccsc}\left (c x \right ) c x +\ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}\)	\(378\)
parts	\(a^{3} x +\frac {b^{3} \left (\operatorname {arccsc}\left (c x \right )^{3} c x -3 \operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 \operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}+\frac {3 a \,b^{2} \left (\operatorname {arccsc}\left (c x \right )^{2} c x -2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 i \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {dilog}\left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}+3 a^{2} b x \,\operatorname {arccsc}\left (c x \right )+\frac {3 a^{2} b \ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c}\)	\(385\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*acsc(c*x))**3, x)

Maxima [F]

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

[In]

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

[Out]

-3/2*a*b^2*c^2*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3)*log(c)^2 - 12*b^3*c^2*integrate(1/4*x^2*arctan(
1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^2 - 1), x)*log(c)^2 + b^3*x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3
- 3/4*b^3*x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2 + 12*b^3*c^2*integrate(1/4*x^2*arctan(1/(sq
rt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) - 24*b^3*c^2*integrate(1/4*x^2*arctan(1/(sqr
t(c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^2 - 1), x)*log(c) + 12*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)/(c^2*
x^2 - 1), x)*log(c) - 24*a*b^2*c^2*integrate(1/4*x^2*log(x)/(c^2*x^2 - 1), x)*log(c) + 12*b^3*c^2*integrate(1/
4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) - 12*b^3*c^2*integrate(1/4
*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^2 - 1), x) + 12*a*b^2*c^2*integrate(1/4*x^2*arcta
n(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^2 - 1), x) + 12*b^3*c^2*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*
sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x) - 3*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)^2/(c^2*x^2 - 1), x
) + 12*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) - 12*a*b^2*c^2*integrate(1/4*x^2*log(
x)^2/(c^2*x^2 - 1), x) - 3/2*a*b^2*(log(c*x + 1)/c - log(c*x - 1)/c)*log(c)^2 + 12*b^3*integrate(1/4*arctan(1/
(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^2 - 1), x)*log(c)^2 - 12*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(
c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) + 24*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1))
)*log(x)/(c^2*x^2 - 1), x)*log(c) - 12*a*b^2*integrate(1/4*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) + 24*a*b^2*in
tegrate(1/4*log(x)/(c^2*x^2 - 1), x)*log(c) + a^3*x + 12*b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*x - 1)*arctan(
1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^2 - 1), x) - 3*b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*x - 1)*log(c^2
*x^2)^2/(c^2*x^2 - 1), x) - 12*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)*log(x)/(
c^2*x^2 - 1), x) + 12*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^2 - 1), x) - 1
2*a*b^2*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^2 - 1), x) - 12*b^3*integrate(1/4*arcta
n(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x) + 3*a*b^2*integrate(1/4*log(c^2*x^2)^2/(c^2*
x^2 - 1), x) - 12*a*b^2*integrate(1/4*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) + 12*a*b^2*integrate(1/4*log(x)^2/
(c^2*x^2 - 1), x) + 3/2*(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^2*b/c

Giac [F]

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

[In]

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

[Out]

integrate((b*arccsc(c*x) + a)^3, x)

Mupad [F(-1)]

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

[In]

int((a + b*asin(1/(c*x)))^3,x)

[Out]

int((a + b*asin(1/(c*x)))^3, x)