\(\int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx\) [23]

Optimal result

Integrand size = 10, antiderivative size = 69 \[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=-\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x}-\frac {2 b \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5367, 4512, 3868, 2739, 632, 210} \[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=-\frac {2 b \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x} \]

[In]

Int[ArcCsc[a + b*x]/x^2,x]

[Out]

-((b*ArcCsc[a + b*x])/a) - ArcCsc[a + b*x]/x - (2*b*ArcTan[(a - Tan[ArcCsc[a + b*x]/2])/Sqrt[1 - a^2]])/(a*Sqr
t[1 - a^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 3868

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(-1), x_Symbol] :> Simp[x/a, x] - Dist[1/a, Int[1/(1 + (a/b)*Sin[c
+ d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 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 \text {Subst}\left (\int \frac {x \cot (x) \csc (x)}{(-a+\csc (x))^2} \, dx,x,\csc ^{-1}(a+b x)\right )\right ) \\ & = -\frac {\csc ^{-1}(a+b x)}{x}+b \text {Subst}\left (\int \frac {1}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = -\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x}+\frac {b \text {Subst}\left (\int \frac {1}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{a} \\ & = -\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x}+\frac {(2 b) \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} \\ & = -\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x}-\frac {(4 b) \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} \\ & = -\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x}-\frac {2 b \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.67 \[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=-\frac {\csc ^{-1}(a+b x)}{x}+\frac {b \left (-\arcsin \left (\frac {1}{a+b x}\right )+\frac {i \log \left (\frac {2 \left (-\frac {i a \left (-1+a^2+a b x\right )}{\sqrt {1-a^2}}-a (a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )}{b x}\right )}{\sqrt {1-a^2}}\right )}{a} \]

[In]

Integrate[ArcCsc[a + b*x]/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.84

method	result	size
derivativedivides	\(b \left (-\frac {\operatorname {arccsc}\left (b x +a \right )}{b x}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \sqrt {a^{2}-1}-\ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right )\right )}{\sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \sqrt {a^{2}-1}}\right )\)	\(127\)
default	\(b \left (-\frac {\operatorname {arccsc}\left (b x +a \right )}{b x}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \sqrt {a^{2}-1}-\ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right )\right )}{\sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \sqrt {a^{2}-1}}\right )\)	\(127\)
parts	\(-\frac {\operatorname {arccsc}\left (b x +a \right )}{x}-\frac {b \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) \sqrt {a^{2}-1}-\ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right )\right )}{\sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \sqrt {a^{2}-1}}\)	\(152\)

[In]

int(arccsc(b*x+a)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (63) = 126\).

Time = 0.32 (sec) , antiderivative size = 278, normalized size of antiderivative = 4.03 \[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=\left [\frac {2 \, {\left (a^{2} - 1\right )} b x \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {a^{2} - 1} b x \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} + \sqrt {a^{2} - 1} a - 1\right )} + {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - {\left (a^{3} - a\right )} \operatorname {arccsc}\left (b x + a\right )}{{\left (a^{3} - a\right )} x}, \frac {2 \, {\left (a^{2} - 1\right )} b x \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, \sqrt {-a^{2} + 1} b x \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) - {\left (a^{3} - a\right )} \operatorname {arccsc}\left (b x + a\right )}{{\left (a^{3} - a\right )} x}\right ] \]

[In]

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

[Out]

[(2*(a^2 - 1)*b*x*arctan(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + sqrt(a^2 - 1)*b*x*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) - (a
^3 - a)*arccsc(b*x + a))/((a^3 - a)*x), (2*(a^2 - 1)*b*x*arctan(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))
- 2*sqrt(-a^2 + 1)*b*x*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)) - (a^3 - a)*arccsc(b*x + a))/((a^3 - a)*x)]

Sympy [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=\int \frac {\operatorname {acsc}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

Integral(acsc(a + b*x)/x**2, x)

Maxima [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.39 \[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=-b {\left (\frac {2 \, \arctan \left (\frac {{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + a}{\sqrt {-a^{2} + 1}}\right )}{\sqrt {-a^{2} + 1} a} - \frac {\arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{a {\left (\frac {a}{b x + a} - 1\right )}}\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=\int \frac {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}{x^2} \,d x \]

[In]

int(asin(1/(a + b*x))/x^2,x)

[Out]

int(asin(1/(a + b*x))/x^2, x)