Integrand size = 8, antiderivative size = 79 \[ \int x \csc ^{-1}(a+b x) \, dx=\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 b^2}-\frac {a^2 \csc ^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \csc ^{-1}(a+b x)-\frac {a \text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^2} \]
-1/2*a^2*arccsc(b*x+a)/b^2+1/2*x^2*arccsc(b*x+a)-a*arctanh((1-1/(b*x+a)^2) ^(1/2))/b^2+1/2*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^2
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.39 \[ \int x \csc ^{-1}(a+b x) \, dx=\frac {(a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}+b^2 x^2 \csc ^{-1}(a+b x)-a^2 \arcsin \left (\frac {1}{a+b x}\right )-2 a \log \left ((a+b x) \left (1+\sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{2 b^2} \]
((a + b*x)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2] + b^2*x^2*ArcC sc[a + b*x] - a^2*ArcSin[(a + b*x)^(-1)] - 2*a*Log[(a + b*x)*(1 + Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2])])/(2*b^2)
Time = 0.41 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {5782, 25, 4927, 3042, 4260, 3042, 4254, 24, 4257}
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.
\(\displaystyle \int x \csc ^{-1}(a+b x) \, dx\) |
\(\Big \downarrow \) 5782 |
\(\displaystyle -\frac {\int b x (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)d\csc ^{-1}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int -b x (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)d\csc ^{-1}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 4927 |
\(\displaystyle -\frac {\frac {1}{2} \int b^2 x^2d\csc ^{-1}(a+b x)-\frac {1}{2} b^2 x^2 \csc ^{-1}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{2} \int \left (a-\csc \left (\csc ^{-1}(a+b x)\right )\right )^2d\csc ^{-1}(a+b x)-\frac {1}{2} b^2 x^2 \csc ^{-1}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 4260 |
\(\displaystyle -\frac {\frac {1}{2} \left (-2 a \int (a+b x)d\csc ^{-1}(a+b x)+\int (a+b x)^2d\csc ^{-1}(a+b x)+a^2 \csc ^{-1}(a+b x)\right )-\frac {1}{2} b^2 x^2 \csc ^{-1}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{2} \left (-2 a \int \csc \left (\csc ^{-1}(a+b x)\right )d\csc ^{-1}(a+b x)+\int \csc \left (\csc ^{-1}(a+b x)\right )^2d\csc ^{-1}(a+b x)+a^2 \csc ^{-1}(a+b x)\right )-\frac {1}{2} b^2 x^2 \csc ^{-1}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -\frac {\frac {1}{2} \left (-\int 1d\left ((a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-2 a \int \csc \left (\csc ^{-1}(a+b x)\right )d\csc ^{-1}(a+b x)+a^2 \csc ^{-1}(a+b x)\right )-\frac {1}{2} b^2 x^2 \csc ^{-1}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\frac {1}{2} \left (-2 a \int \csc \left (\csc ^{-1}(a+b x)\right )d\csc ^{-1}(a+b x)+a^2 \csc ^{-1}(a+b x)-(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-\frac {1}{2} b^2 x^2 \csc ^{-1}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\frac {\frac {1}{2} \left (a^2 \csc ^{-1}(a+b x)+2 a \text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )-(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-\frac {1}{2} b^2 x^2 \csc ^{-1}(a+b x)}{b^2}\) |
-((-1/2*(b^2*x^2*ArcCsc[a + b*x]) + (-((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) + a^2*ArcCsc[a + b*x] + 2*a*ArcTanh[Sqrt[1 - (a + b*x)^(-2)]])/2)/b^2)
3.1.20.3.1 Defintions of rubi rules used
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Simp[a^2*x, x] + (Simp[2*a*b Int[Csc[c + d*x], x], x] + Simp[b^2 Int[Csc[c + d*x]^2, x] , x]) /; FreeQ[{a, b, c, d}, x]
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 = 0.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.34
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {arccsc}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\operatorname {arccsc}\left (b x +a \right ) a \left (b x +a \right )+\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (-2 a \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )+\sqrt {\left (b x +a \right )^{2}-1}\right )}{2 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{2}}\) | \(106\) |
default | \(\frac {\frac {\operatorname {arccsc}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\operatorname {arccsc}\left (b x +a \right ) a \left (b x +a \right )+\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (-2 a \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )+\sqrt {\left (b x +a \right )^{2}-1}\right )}{2 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{2}}\) | \(106\) |
parts | \(\frac {x^{2} \operatorname {arccsc}\left (b x +a \right )}{2}+\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (-a^{2} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) \sqrt {b^{2}}-2 a \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) b +\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}\right )}{2 b^{2} \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \sqrt {b^{2}}}\) | \(177\) |
1/b^2*(1/2*arccsc(b*x+a)*(b*x+a)^2-arccsc(b*x+a)*a*(b*x+a)+1/2/(((b*x+a)^2 -1)/(b*x+a)^2)^(1/2)/(b*x+a)*((b*x+a)^2-1)^(1/2)*(-2*a*ln(b*x+a+((b*x+a)^2 -1)^(1/2))+((b*x+a)^2-1)^(1/2)))
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.29 \[ \int x \csc ^{-1}(a+b x) \, dx=\frac {b^{2} x^{2} \operatorname {arccsc}\left (b x + a\right ) + 2 \, a^{2} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 2 \, a \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{2 \, b^{2}} \]
1/2*(b^2*x^2*arccsc(b*x + a) + 2*a^2*arctan(-b*x - a + sqrt(b^2*x^2 + 2*a* b*x + a^2 - 1)) + 2*a*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/b^2
\[ \int x \csc ^{-1}(a+b x) \, dx=\int x \operatorname {acsc}{\left (a + b x \right )}\, dx \]
\[ \int x \csc ^{-1}(a+b x) \, dx=\int { x \operatorname {arccsc}\left (b x + a\right ) \,d x } \]
1/2*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + integrate(1/2*(b ^2*x^3 + a*b*x^2)*e^(1/2*log(b*x + a + 1) + 1/2*log(b*x + a - 1))/(b^2*x^2 + 2*a*b*x + a^2 + (b^2*x^2 + 2*a*b*x + a^2 - 1)*e^(log(b*x + a + 1) + log (b*x + a - 1)) - 1), x)
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.70 \[ \int x \csc ^{-1}(a+b x) \, dx=-\frac {1}{4} \, b {\left (\frac {2 \, {\left (b x + a\right )}^{2} {\left (\frac {2 \, a}{b x + a} - 1\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{3}} - \frac {{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 4 \, a \log \left (-{\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} {\left | b x + a \right |}\right ) - \frac {1}{{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}}}{b^{3}}\right )} \]
-1/4*b*(2*(b*x + a)^2*(2*a/(b*x + a) - 1)*arcsin(-1/((b*x + a)*(a/(b*x + a ) - 1) - a))/b^3 - ((b*x + a)*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 4*a*log(-(s qrt(-1/(b*x + a)^2 + 1) - 1)*abs(b*x + a)) - 1/((b*x + a)*(sqrt(-1/(b*x + a)^2 + 1) - 1)))/b^3)
Timed out. \[ \int x \csc ^{-1}(a+b x) \, dx=\int x\,\mathrm {asin}\left (\frac {1}{a+b\,x}\right ) \,d x \]