3.1.37 \(\int \frac {\csc ^{-1}(a+b x)^3}{x^2} \, dx\) [37]

Optimal. Leaf size=378 \[ -\frac {b \csc ^{-1}(a+b x)^3}{a}-\frac {\csc ^{-1}(a+b x)^3}{x}-\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 b \csc ^{-1}(a+b x) \text {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 b \csc ^{-1}(a+b x) \text {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 i b \text {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 i b \text {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}} \]

[Out]

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

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Rubi [A]
time = 0.42, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5367, 4512, 4276, 3404, 2296, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {6 b \csc ^{-1}(a+b x) \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 b \csc ^{-1}(a+b x) \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}-\frac {6 i b \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 i b \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}-\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}-\frac {b \csc ^{-1}(a+b x)^3}{a}-\frac {\csc ^{-1}(a+b x)^3}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCsc[a + b*x]^3/x^2,x]

[Out]

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

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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

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 3404

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[(c + d*x)^m*(E
^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4276

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 4512

Int[Cot[(c_.) + (d_.)*(x_)]*Csc[(c_.) + (d_.)*(x_)]*(Csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.)*((e_.) + (f_.
)*(x_))^(m_.), x_Symbol] :> Simp[(-(e + f*x)^m)*((a + b*Csc[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Dist[f*(m/(
b*d*(n + 1))), Int[(e + f*x)^(m - 1)*(a + b*Csc[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &
& IGtQ[m, 0] && NeQ[n, -1]

Rule 5367

Int[((a_.) + ArcCsc[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[-(d^(m + 1))
^(-1), Subst[Int[(a + b*x)^p*Csc[x]*Cot[x]*(d*e - c*f + f*Csc[x])^m, x], x, ArcCsc[c + d*x]], x] /; FreeQ[{a,
b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]

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*} \int \frac {\csc ^{-1}(a+b x)^3}{x^2} \, dx &=-\left (b \text {Subst}\left (\int \frac {x^3 \cot (x) \csc (x)}{(-a+\csc (x))^2} \, dx,x,\csc ^{-1}(a+b x)\right )\right )\\ &=-\frac {\csc ^{-1}(a+b x)^3}{x}+(3 b) \text {Subst}\left (\int \frac {x^2}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=-\frac {\csc ^{-1}(a+b x)^3}{x}+(3 b) \text {Subst}\left (\int \left (-\frac {x^2}{a}+\frac {x^2}{a (1-a \sin (x))}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=-\frac {b \csc ^{-1}(a+b x)^3}{a}-\frac {\csc ^{-1}(a+b x)^3}{x}+\frac {(3 b) \text {Subst}\left (\int \frac {x^2}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{a}\\ &=-\frac {b \csc ^{-1}(a+b x)^3}{a}-\frac {\csc ^{-1}(a+b x)^3}{x}+\frac {(6 b) \text {Subst}\left (\int \frac {e^{i x} x^2}{-i a+2 e^{i x}+i a e^{2 i x}} \, dx,x,\csc ^{-1}(a+b x)\right )}{a}\\ &=-\frac {b \csc ^{-1}(a+b x)^3}{a}-\frac {\csc ^{-1}(a+b x)^3}{x}+\frac {(6 i b) \text {Subst}\left (\int \frac {e^{i x} x^2}{2-2 \sqrt {1-a^2}+2 i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}-\frac {(6 i b) \text {Subst}\left (\int \frac {e^{i x} x^2}{2+2 \sqrt {1-a^2}+2 i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\\ &=-\frac {b \csc ^{-1}(a+b x)^3}{a}-\frac {\csc ^{-1}(a+b x)^3}{x}-\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {(6 i b) \text {Subst}\left (\int x \log \left (1+\frac {2 i a e^{i x}}{2-2 \sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{a \sqrt {1-a^2}}-\frac {(6 i b) \text {Subst}\left (\int x \log \left (1+\frac {2 i a e^{i x}}{2+2 \sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{a \sqrt {1-a^2}}\\ &=-\frac {b \csc ^{-1}(a+b x)^3}{a}-\frac {\csc ^{-1}(a+b x)^3}{x}-\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 b \csc ^{-1}(a+b x) \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 b \csc ^{-1}(a+b x) \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {(6 b) \text {Subst}\left (\int \text {Li}_2\left (-\frac {2 i a e^{i x}}{2-2 \sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{a \sqrt {1-a^2}}-\frac {(6 b) \text {Subst}\left (\int \text {Li}_2\left (-\frac {2 i a e^{i x}}{2+2 \sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{a \sqrt {1-a^2}}\\ &=-\frac {b \csc ^{-1}(a+b x)^3}{a}-\frac {\csc ^{-1}(a+b x)^3}{x}-\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 b \csc ^{-1}(a+b x) \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 b \csc ^{-1}(a+b x) \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {(6 i b) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i a x}{-1+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{a \sqrt {1-a^2}}+\frac {(6 i b) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {i a x}{1+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{a \sqrt {1-a^2}}\\ &=-\frac {b \csc ^{-1}(a+b x)^3}{a}-\frac {\csc ^{-1}(a+b x)^3}{x}-\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 b \csc ^{-1}(a+b x) \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 b \csc ^{-1}(a+b x) \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 i b \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 i b \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 289, normalized size = 0.76 \begin {gather*} -\frac {\frac {(a+b x) \csc ^{-1}(a+b x)^3}{x}+\frac {3 i b \left (\csc ^{-1}(a+b x)^2 \log \left (1-\frac {i a e^{i \csc ^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \text {PolyLog}\left (2,\frac {i a e^{i \csc ^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )+2 i \csc ^{-1}(a+b x) \text {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+2 \text {PolyLog}\left (3,\frac {i a e^{i \csc ^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )-2 \text {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )\right )}{\sqrt {1-a^2}}}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCsc[a + b*x]^3/x^2,x]

[Out]

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

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Maple [F]
time = 2.86, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccsc}\left (b x +a \right )^{3}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccsc(b*x+a)^3/x^2,x)

[Out]

int(arccsc(b*x+a)^3/x^2,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(b*x+a)^3/x^2,x, algorithm="maxima")

[Out]

-1/4*(4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^3 - 3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))*
log(b^2*x^2 + 2*a*b*x + a^2)^2 - 4*x*integrate(-3/4*(4*(b^3*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)
) + 3*a*b^2*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + a^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x +
a - 1)) + (3*a^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a -
1)))*b*x - a*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*log(b*x + a)^2 + (4*b*x*arctan2(1, sqrt(b*x + a
+ 1)*sqrt(b*x + a - 1))^2 - b*x*log(b^2*x^2 + 2*a*b*x + a^2)^2)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + 4*(b^3*
x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 2*a*b^2*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a -
1)) + (a^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*
b*x - (b^3*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 3*a*b^2*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt
(b*x + a - 1)) + a^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + (3*a^2*arctan2(1, sqrt(b*x + a + 1)*sqr
t(b*x + a - 1)) - arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*b*x - a*arctan2(1, sqrt(b*x + a + 1)*sqrt(b
*x + a - 1)))*log(b*x + a))*log(b^2*x^2 + 2*a*b*x + a^2))/(b^3*x^5 + 3*a*b^2*x^4 + (3*a^2 - 1)*b*x^3 + (a^3 -
a)*x^2), x))/x

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(b*x+a)^3/x^2,x, algorithm="fricas")

[Out]

integral(arccsc(b*x + a)^3/x^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acsc}^{3}{\left (a + b x \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acsc(b*x+a)**3/x**2,x)

[Out]

Integral(acsc(a + b*x)**3/x**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(b*x+a)^3/x^2,x, algorithm="giac")

[Out]

integrate(arccsc(b*x + a)^3/x^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^3}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(1/(a + b*x))^3/x^2,x)

[Out]

int(asin(1/(a + b*x))^3/x^2, x)

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