Integrand size = 12, antiderivative size = 254 \[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=-\frac {b \csc ^{-1}(a+b x)^2}{a}-\frac {\csc ^{-1}(a+b x)^2}{x}-\frac {2 i b \csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {2 i b \csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}} \]
-b*arccsc(b*x+a)^2/a-arccsc(b*x+a)^2/x-2*I*b*arccsc(b*x+a)*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)+2*I*b*arc csc(b*x+a)*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)-2*b*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)+2*b*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)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(802\) vs. \(2(254)=508\).
Time = 2.49 (sec) , antiderivative size = 802, normalized size of antiderivative = 3.16 \[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=-\frac {b \left (\frac {(a+b x) \csc ^{-1}(a+b x)^2}{b x}+\frac {2 \pi \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{\sqrt {1-a^2}}+\frac {2 \left (-2 \arccos \left (\frac {1}{a}\right ) \text {arctanh}\left (\frac {(1+a) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )+\left (\pi -2 \csc ^{-1}(a+b x)\right ) \text {arctanh}\left (\frac {(-1+a) \tan \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )+\left (\arccos \left (\frac {1}{a}\right )+2 i \left (-\text {arctanh}\left (\frac {(1+a) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )+\text {arctanh}\left (\frac {(-1+a) \tan \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )\right )\right ) \log \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-1+a^2} e^{-\frac {1}{2} i \csc ^{-1}(a+b x)}}{\sqrt {a} \sqrt {-\frac {b x}{a+b x}}}\right )+\left (\arccos \left (\frac {1}{a}\right )+2 i \text {arctanh}\left (\frac {(1+a) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )-2 i \text {arctanh}\left (\frac {(-1+a) \tan \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {-1+a^2} e^{\frac {1}{2} i \csc ^{-1}(a+b x)}}{\sqrt {a} \sqrt {-\frac {b x}{a+b x}}}\right )-\left (\arccos \left (\frac {1}{a}\right )-2 i \text {arctanh}\left (\frac {(1+a) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (\frac {(-1+a) \left (i+i a+\sqrt {-1+a^2}\right ) \left (-i+\cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}{a \left (-1+a+\sqrt {-1+a^2} \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}\right )-\left (\arccos \left (\frac {1}{a}\right )+2 i \text {arctanh}\left (\frac {(1+a) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (\frac {(-1+a) \left (-i-i a+\sqrt {-1+a^2}\right ) \left (i+\cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}{a \left (-1+a+\sqrt {-1+a^2} \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (1-i \sqrt {-1+a^2}\right ) \left (1-a+\sqrt {-1+a^2} \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}{a \left (-1+a+\sqrt {-1+a^2} \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (1+i \sqrt {-1+a^2}\right ) \left (1-a+\sqrt {-1+a^2} \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}{a \left (-1+a+\sqrt {-1+a^2} \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}\right )\right )\right )}{\sqrt {-1+a^2}}\right )}{a} \]
-((b*(((a + b*x)*ArcCsc[a + b*x]^2)/(b*x) + (2*Pi*ArcTan[(a - Tan[ArcCsc[a + b*x]/2])/Sqrt[1 - a^2]])/Sqrt[1 - a^2] + (2*(-2*ArcCos[a^(-1)]*ArcTanh[ ((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]] + (Pi - 2*ArcCsc [a + b*x])*ArcTanh[((-1 + a)*Tan[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^ 2]] + (ArcCos[a^(-1)] + (2*I)*(-ArcTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b* x])/4])/Sqrt[-1 + a^2]] + ArcTanh[((-1 + a)*Tan[(Pi + 2*ArcCsc[a + b*x])/4 ])/Sqrt[-1 + a^2]]))*Log[((1/2 + I/2)*Sqrt[-1 + a^2])/(Sqrt[a]*E^((I/2)*Ar cCsc[a + b*x])*Sqrt[-((b*x)/(a + b*x))])] + (ArcCos[a^(-1)] + (2*I)*ArcTan h[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]] - (2*I)*ArcTan h[((-1 + a)*Tan[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]])*Log[((1/2 - I/2)*Sqrt[-1 + a^2]*E^((I/2)*ArcCsc[a + b*x]))/(Sqrt[a]*Sqrt[-((b*x)/(a + b*x))])] - (ArcCos[a^(-1)] - (2*I)*ArcTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]])*Log[((-1 + a)*(I + I*a + Sqrt[-1 + a^2])*(-I + Cot[(Pi + 2*ArcCsc[a + b*x])/4]))/(a*(-1 + a + Sqrt[-1 + a^2]*Cot[(Pi + 2 *ArcCsc[a + b*x])/4]))] - (ArcCos[a^(-1)] + (2*I)*ArcTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]])*Log[((-1 + a)*(-I - I*a + Sqrt[ -1 + a^2])*(I + Cot[(Pi + 2*ArcCsc[a + b*x])/4]))/(a*(-1 + a + Sqrt[-1 + a ^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))] + I*(-PolyLog[2, ((1 - I*Sqrt[-1 + a^2])*(1 - a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))/(a*(-1 + a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))] + PolyLog[2, ((1 +...
Time = 0.75 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5782, 4927, 3042, 4679, 2009}
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.
| \(\displaystyle \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 5782 |
| \(\displaystyle -b \int \frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^2 x^2}d\csc ^{-1}(a+b x)\) |
| \(\Big \downarrow \) 4927 |
| \(\displaystyle -b \left (2 \int -\frac {\csc ^{-1}(a+b x)}{b x}d\csc ^{-1}(a+b x)+\frac {\csc ^{-1}(a+b x)^2}{b x}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -b \left (2 \int \frac {\csc ^{-1}(a+b x)}{a-\csc \left (\csc ^{-1}(a+b x)\right )}d\csc ^{-1}(a+b x)+\frac {\csc ^{-1}(a+b x)^2}{b x}\right )\) |
| \(\Big \downarrow \) 4679 |
| \(\displaystyle -b \left (2 \int \left (\frac {\csc ^{-1}(a+b x)}{a}+\frac {\csc ^{-1}(a+b x)}{a \left (\frac {a}{a+b x}-1\right )}\right )d\csc ^{-1}(a+b x)+\frac {\csc ^{-1}(a+b x)^2}{b x}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -b \left (\frac {\csc ^{-1}(a+b x)^2}{b x}+2 \left (\frac {\operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {\operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}+\frac {i \csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {i \csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}+\frac {\csc ^{-1}(a+b x)^2}{2 a}\right )\right )\) |
-(b*(ArcCsc[a + b*x]^2/(b*x) + 2*(ArcCsc[a + b*x]^2/(2*a) + (I*ArcCsc[a + b*x]*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (I*ArcCsc[a + b*x]*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[ 1 - a^2])])/(a*Sqrt[1 - a^2]) + PolyLog[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/ (1 - Sqrt[1 - a^2])]/(a*Sqrt[1 - a^2]) - PolyLog[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])]/(a*Sqrt[1 - a^2]))))
3.1.32.3.1 Defintions of rubi rules used
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*Si n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt Q[m, 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 = 1.64 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.19
| method | result | size |
| derivativedivides | \(b \left (-\frac {\left (b x +a \right ) \operatorname {arccsc}\left (b x +a \right )^{2}}{a b x}-\frac {2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}+\frac {2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i-\sqrt {a^{2}-1}}{i-\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}+\frac {2 i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}-\frac {2 i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i-\sqrt {a^{2}-1}}{i-\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}\right )\) | \(302\) |
| default | \(b \left (-\frac {\left (b x +a \right ) \operatorname {arccsc}\left (b x +a \right )^{2}}{a b x}-\frac {2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}+\frac {2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i-\sqrt {a^{2}-1}}{i-\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}+\frac {2 i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}-\frac {2 i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i-\sqrt {a^{2}-1}}{i-\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}\right )\) | \(302\) |
b*(-(b*x+a)*arccsc(b*x+a)^2/a/b/x-2/a*arccsc(b*x+a)/(a^2-1)^(1/2)*ln((-(I/ (b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a+(a^2-1)^(1/2)+I)/(I+(a^2-1)^(1/2)))+2/a*a rccsc(b*x+a)/(a^2-1)^(1/2)*ln((-(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a+I-(a^2 -1)^(1/2))/(I-(a^2-1)^(1/2)))+2*I/a/(a^2-1)^(1/2)*dilog((-(I/(b*x+a)+(1-1/ (b*x+a)^2)^(1/2))*a+(a^2-1)^(1/2)+I)/(I+(a^2-1)^(1/2)))-2*I/a/(a^2-1)^(1/2 )*dilog((-(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a+I-(a^2-1)^(1/2))/(I-(a^2-1)^ (1/2))))
\[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]
\[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=\int \frac {\operatorname {acsc}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]
\[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]
-1/4*(4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 + 4*x*integrate( (2*sqrt(b*x + a + 1)*sqrt(b*x + a - 1)*b*x*arctan2(1, sqrt(b*x + a + 1)*sq rt(b*x + a - 1)) + (b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2 - 1)*b*x - a)*log (b*x + a)^2 + (b^3*x^3 + 2*a*b^2*x^2 + (a^2 - 1)*b*x - (b^3*x^3 + 3*a*b^2* x^2 + a^3 + (3*a^2 - 1)*b*x - a)*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) - log(b ^2*x^2 + 2*a*b*x + a^2)^2)/x
\[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=\int \frac {{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^2}{x^2} \,d x \]