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

Optimal result

Integrand size = 10, antiderivative size = 123 \[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {\csc ^{-1}(a+b x)}{2 x^2}+\frac {\left (1-2 a^2\right ) b^2 \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}} \]

[Out]

1/2*b^2*arccsc(b*x+a)/a^2-1/2*arccsc(b*x+a)/x^2+(-2*a^2+1)*b^2*arctan((a-tan(1/2*arccsc(b*x+a)))/(-a^2+1)^(1/2
))/a^2/(-a^2+1)^(3/2)-1/2*b*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/a/(-a^2+1)/x

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5367, 4512, 3870, 4004, 3916, 2739, 632, 210} \[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=\frac {\left (1-2 a^2\right ) b^2 \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}-\frac {\csc ^{-1}(a+b x)}{2 x^2} \]

[In]

Int[ArcCsc[a + b*x]/x^3,x]

[Out]

-1/2*(b*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(a*(1 - a^2)*x) + (b^2*ArcCsc[a + b*x])/(2*a^2) - ArcCsc[a + b*x]/
(2*x^2) + ((1 - 2*a^2)*b^2*ArcTan[(a - Tan[ArcCsc[a + b*x]/2])/Sqrt[1 - a^2]])/(a^2*(1 - a^2)^(3/2))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 3870

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[c + d*x]*((a + b*Csc[c + d*x])^(n +
1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Dist[1/(a*(n + 1)*(a^2 - b^2)), Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^
2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 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]

Rubi steps \begin{align*} \text {integral}& = -\left (b^2 \text {Subst}\left (\int \frac {x \cot (x) \csc (x)}{(-a+\csc (x))^3} \, dx,x,\csc ^{-1}(a+b x)\right )\right ) \\ & = -\frac {\csc ^{-1}(a+b x)}{2 x^2}+\frac {1}{2} b^2 \text {Subst}\left (\int \frac {1}{(-a+\csc (x))^2} \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = -\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}-\frac {\csc ^{-1}(a+b x)}{2 x^2}-\frac {b^2 \text {Subst}\left (\int \frac {1-a^2-a \csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{2 a \left (1-a^2\right )} \\ & = -\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {\csc ^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1-2 a^2\right ) b^2\right ) \text {Subst}\left (\int \frac {\csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{2 a^2 \left (1-a^2\right )} \\ & = -\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {\csc ^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1-2 a^2\right ) b^2\right ) \text {Subst}\left (\int \frac {1}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{2 a^2 \left (1-a^2\right )} \\ & = -\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {\csc ^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1-2 a^2\right ) b^2\right ) \text {Subst}\left (\int \frac {1}{1-2 a x+x^2} \, dx,x,\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )\right )}{a^2 \left (1-a^2\right )} \\ & = -\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {\csc ^{-1}(a+b x)}{2 x^2}+\frac {\left (2 \left (1-2 a^2\right ) b^2\right ) \text {Subst}\left (\int \frac {1}{-4 \left (1-a^2\right )-x^2} \, dx,x,-2 a+2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )\right )}{a^2 \left (1-a^2\right )} \\ & = -\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {\csc ^{-1}(a+b x)}{2 x^2}+\frac {\left (1-2 a^2\right ) b^2 \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.55 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.62 \[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=\frac {\frac {b x (a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}}{a \left (-1+a^2\right )}-\csc ^{-1}(a+b x)+\frac {b^2 x^2 \arcsin \left (\frac {1}{a+b x}\right )}{a^2}+\frac {i \left (-1+2 a^2\right ) b^2 x^2 \log \left (\frac {4 (-1+a) a^2 (1+a) \left (\frac {i \left (-1+a^2+a b x\right )}{\sqrt {1-a^2}}+(a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )}{\left (-1+2 a^2\right ) b^2 x}\right )}{a^2 \left (1-a^2\right )^{3/2}}}{2 x^2} \]

[In]

Integrate[ArcCsc[a + b*x]/x^3,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(318\) vs. \(2(109)=218\).

Time = 0.71 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.59

method	result	size
parts	\(-\frac {\operatorname {arccsc}\left (b x +a \right )}{2 x^{2}}+\frac {b \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (\left (a^{2}-1\right )^{\frac {3}{2}} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) a^{2} b x -2 \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{4} b x -b \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) x \left (a^{2}-1\right )^{\frac {3}{2}}+\left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a +3 \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{2} b x -b \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) x \right )}{2 \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}-1\right )^{\frac {5}{2}} x}\)	\(319\)
derivativedivides	\(b^{2} \left (-\frac {\operatorname {arccsc}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} a^{3}-\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} a^{2} \left (b x +a \right )-2 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{5}+2 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{4} \left (b x +a \right )-\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} a +\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} \left (b x +a \right )-\left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {\left (b x +a \right )^{2}-1}\, a +3 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{3}-3 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{2} \left (b x +a \right )-a \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right )+\ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) \left (b x +a \right )\right )}{2 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}-1\right )^{\frac {5}{2}} b x}\right )\)	\(457\)
default	\(b^{2} \left (-\frac {\operatorname {arccsc}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} a^{3}-\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} a^{2} \left (b x +a \right )-2 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{5}+2 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{4} \left (b x +a \right )-\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} a +\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} \left (b x +a \right )-\left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {\left (b x +a \right )^{2}-1}\, a +3 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{3}-3 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{2} \left (b x +a \right )-a \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right )+\ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) \left (b x +a \right )\right )}{2 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}-1\right )^{\frac {5}{2}} b x}\right )\)	\(457\)

[In]

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

[Out]

-1/2*arccsc(b*x+a)/x^2+1/2*b*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*((a^2-1)^(3/2)*arctan(1/(b^2*x^2+2*a*b*x+a^2-1)^(1/
2))*a^2*b*x-2*ln(2*(a*b*x+(a^2-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x)*a^4*b*x-b*arctan(1/(b^2*x^2+2*
a*b*x+a^2-1)^(1/2))*x*(a^2-1)^(3/2)+(a^2-1)^(3/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*a+3*ln(2*(a*b*x+(a^2-1)^(1/2)*
(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x)*a^2*b*x-b*ln(2*(a*b*x+(a^2-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-
1)/x)*x)/((b^2*x^2+2*a*b*x+a^2-1)/(b*x+a)^2)^(1/2)/(b*x+a)/a^2/(a^2-1)^(5/2)/x

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 428, normalized size of antiderivative = 3.48 \[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=\left [\frac {{\left (2 \, a^{2} - 1\right )} \sqrt {a^{2} - 1} b^{2} x^{2} \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - \sqrt {a^{2} - 1} a - 1\right )} - {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (a^{3} - a\right )} b^{2} x^{2} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{3} - a\right )} b x - {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \operatorname {arccsc}\left (b x + a\right )}{2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}}, \frac {2 \, {\left (2 \, a^{2} - 1\right )} \sqrt {-a^{2} + 1} b^{2} x^{2} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) - 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (a^{3} - a\right )} b^{2} x^{2} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{3} - a\right )} b x - {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \operatorname {arccsc}\left (b x + a\right )}{2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}}\right ] \]

[In]

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

[Out]

[1/2*((2*a^2 - 1)*sqrt(a^2 - 1)*b^2*x^2*log((a^2*b*x + a^3 + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(a^2 - sqrt(a^2
- 1)*a - 1) - (a*b*x + a^2 - 1)*sqrt(a^2 - 1) - a)/x) - 2*(a^4 - 2*a^2 + 1)*b^2*x^2*arctan(-b*x - a + sqrt(b^
2*x^2 + 2*a*b*x + a^2 - 1)) + (a^3 - a)*b^2*x^2 + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(a^3 - a)*b*x - (a^6 - 2*a
^4 + a^2)*arccsc(b*x + a))/((a^6 - 2*a^4 + a^2)*x^2), 1/2*(2*(2*a^2 - 1)*sqrt(-a^2 + 1)*b^2*x^2*arctan(-(sqrt(
-a^2 + 1)*b*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*sqrt(-a^2 + 1))/(a^2 - 1)) - 2*(a^4 - 2*a^2 + 1)*b^2*x^2*arc
tan(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (a^3 - a)*b^2*x^2 + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(a^3
- a)*b*x - (a^6 - 2*a^4 + a^2)*arccsc(b*x + a))/((a^6 - 2*a^4 + a^2)*x^2)]

Sympy [F]

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

[In]

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

[Out]

Integral(acsc(a + b*x)/x**3, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (108) = 216\).

Time = 0.35 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.76 \[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=\frac {1}{2} \, b {\left (\frac {2 \, {\left (2 \, a^{2} b - b\right )} \arctan \left (\frac {{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + a}{\sqrt {-a^{2} + 1}}\right )}{{\left (a^{4} - a^{2}\right )} \sqrt {-a^{2} + 1}} + \frac {2 \, {\left ({\left (b x + a\right )} a b {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + b\right )}}{{\left ({\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 2 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 1\right )} {\left (a^{3} - a\right )}} - \frac {{\left (\frac {2 \, a b}{b x + a} - b\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{a^{2} {\left (\frac {a}{b x + a} - 1\right )}^{2}}\right )} \]

[In]

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

[Out]

1/2*b*(2*(2*a^2*b - b)*arctan(((b*x + a)*(sqrt(-1/(b*x + a)^2 + 1) - 1) + a)/sqrt(-a^2 + 1))/((a^4 - a^2)*sqrt
(-a^2 + 1)) + 2*((b*x + a)*a*b*(sqrt(-1/(b*x + a)^2 + 1) - 1) + b)/(((b*x + a)^2*(sqrt(-1/(b*x + a)^2 + 1) - 1
)^2 + 2*(b*x + a)*a*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 1)*(a^3 - a)) - (2*a*b/(b*x + a) - b)*arcsin(-1/((b*x + a
)*(a/(b*x + a) - 1) - a))/(a^2*(a/(b*x + a) - 1)^2))

Mupad [F(-1)]

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

[In]

int(asin(1/(a + b*x))/x^3,x)

[Out]

int(asin(1/(a + b*x))/x^3, x)