Integrand size = 10, antiderivative size = 69 \[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=-\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x}-\frac {2 b \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}} \]
-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)
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.67 \[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=-\frac {\csc ^{-1}(a+b x)}{x}+\frac {b \left (-\arcsin \left (\frac {1}{a+b x}\right )+\frac {i \log \left (\frac {2 \left (-\frac {i a \left (-1+a^2+a b x\right )}{\sqrt {1-a^2}}-a (a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )}{b x}\right )}{\sqrt {1-a^2}}\right )}{a} \]
-(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
Time = 0.42 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5782, 4927, 3042, 4270, 3042, 3139, 1083, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx\) |
\(\Big \downarrow \) 5782 |
\(\displaystyle -b \int \frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^2 x^2}d\csc ^{-1}(a+b x)\) |
\(\Big \downarrow \) 4927 |
\(\displaystyle -b \left (\int -\frac {1}{b x}d\csc ^{-1}(a+b x)+\frac {\csc ^{-1}(a+b x)}{b x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -b \left (\int \frac {1}{a-\csc \left (\csc ^{-1}(a+b x)\right )}d\csc ^{-1}(a+b x)+\frac {\csc ^{-1}(a+b x)}{b x}\right )\) |
\(\Big \downarrow \) 4270 |
\(\displaystyle -b \left (-\frac {\int \frac {1}{1-\frac {a}{a+b x}}d\csc ^{-1}(a+b x)}{a}+\frac {\csc ^{-1}(a+b x)}{a}+\frac {\csc ^{-1}(a+b x)}{b x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -b \left (-\frac {\int \frac {1}{1-a \sin \left (\csc ^{-1}(a+b x)\right )}d\csc ^{-1}(a+b x)}{a}+\frac {\csc ^{-1}(a+b x)}{a}+\frac {\csc ^{-1}(a+b x)}{b x}\right )\) |
\(\Big \downarrow \) 3139 |
\(\displaystyle -b \left (-\frac {2 \int \frac {1}{\tan ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+1}d\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{a}+\frac {\csc ^{-1}(a+b x)}{a}+\frac {\csc ^{-1}(a+b x)}{b x}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle -b \left (\frac {4 \int \frac {1}{-\left (2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a\right )^2-4 \left (1-a^2\right )}d\left (2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a\right )}{a}+\frac {\csc ^{-1}(a+b x)}{a}+\frac {\csc ^{-1}(a+b x)}{b x}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -b \left (-\frac {2 \arctan \left (\frac {2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a}{2 \sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {\csc ^{-1}(a+b x)}{a}+\frac {\csc ^{-1}(a+b x)}{b x}\right )\) |
-(b*(ArcCsc[a + b*x]/a + ArcCsc[a + b*x]/(b*x) - (2*ArcTan[(-2*a + 2*Tan[A rcCsc[a + b*x]/2])/(2*Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2])))
3.1.23.3.1 Defintions of rubi rules used
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(-1), x_Symbol] :> Simp[x/a, x] - Simp[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]
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] + Simp[f*(m/(b*d*( n + 1))) Int[(e + f*x)^(m - 1)*(a + b*Csc[c + d*x])^(n + 1), x], x] /; Fr eeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[((a_.) + ArcCsc[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[-(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]
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 | \(-\frac {\operatorname {arccsc}\left (b x +a \right )}{x}-\frac {b \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) \sqrt {a^{2}-1}-\ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right )\right )}{\sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \sqrt {a^{2}-1}}\) | \(152\) |
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))
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 \[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=\left [\frac {2 \, {\left (a^{2} - 1\right )} b x \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {a^{2} - 1} b x \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} + \sqrt {a^{2} - 1} a - 1\right )} + {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - {\left (a^{3} - a\right )} \operatorname {arccsc}\left (b x + a\right )}{{\left (a^{3} - a\right )} x}, \frac {2 \, {\left (a^{2} - 1\right )} b x \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, \sqrt {-a^{2} + 1} b x \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) - {\left (a^{3} - a\right )} \operatorname {arccsc}\left (b x + a\right )}{{\left (a^{3} - a\right )} x}\right ] \]
[(2*(a^2 - 1)*b*x*arctan(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + s qrt(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 + s qrt(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)]
\[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=\int \frac {\operatorname {acsc}{\left (a + b x \right )}}{x^{2}}\, dx \]
\[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )}{x^{2}} \,d x } \]
-(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
Time = 0.31 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.39 \[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=-b {\left (\frac {2 \, \arctan \left (\frac {{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + a}{\sqrt {-a^{2} + 1}}\right )}{\sqrt {-a^{2} + 1} a} - \frac {\arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{a {\left (\frac {a}{b x + a} - 1\right )}}\right )} \]
-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)))
Timed out. \[ \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx=\int \frac {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}{x^2} \,d x \]