Integrand size = 8, antiderivative size = 86 \[ \int \csc ^{-1}(a+b x)^2 \, dx=\frac {(a+b x) \csc ^{-1}(a+b x)^2}{b}+\frac {4 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}-\frac {2 i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac {2 i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b} \]
(b*x+a)*arccsc(b*x+a)^2/b+4*arccsc(b*x+a)*arctanh(I/(b*x+a)+(1-1/(b*x+a)^2 )^(1/2))/b-2*I*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b+2*I*polylog(2 ,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b
Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.15 \[ \int \csc ^{-1}(a+b x)^2 \, dx=\frac {\csc ^{-1}(a+b x) \left ((a+b x) \csc ^{-1}(a+b x)-2 \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )+2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )\right )-2 i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )+2 i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b} \]
(ArcCsc[a + b*x]*((a + b*x)*ArcCsc[a + b*x] - 2*Log[1 - E^(I*ArcCsc[a + b* x])] + 2*Log[1 + E^(I*ArcCsc[a + b*x])]) - (2*I)*PolyLog[2, -E^(I*ArcCsc[a + b*x])] + (2*I)*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b
Time = 0.42 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5776, 5740, 4245, 3042, 4671, 2715, 2838}
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.
\(\displaystyle \int \csc ^{-1}(a+b x)^2 \, dx\) |
\(\Big \downarrow \) 5776 |
\(\displaystyle \frac {\int \csc ^{-1}(a+b x)^2d(a+b x)}{b}\) |
\(\Big \downarrow \) 5740 |
\(\displaystyle -\frac {\int (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2d\csc ^{-1}(a+b x)}{b}\) |
\(\Big \downarrow \) 4245 |
\(\displaystyle -\frac {2 \int (a+b x) \csc ^{-1}(a+b x)d\csc ^{-1}(a+b x)-(a+b x) \csc ^{-1}(a+b x)^2}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \int \csc ^{-1}(a+b x) \csc \left (\csc ^{-1}(a+b x)\right )d\csc ^{-1}(a+b x)-(a+b x) \csc ^{-1}(a+b x)^2}{b}\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle -\frac {-(a+b x) \csc ^{-1}(a+b x)^2+2 \left (-\int \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )d\csc ^{-1}(a+b x)+\int \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )d\csc ^{-1}(a+b x)-2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )\right )}{b}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {-(a+b x) \csc ^{-1}(a+b x)^2+2 \left (-i \int e^{-i \csc ^{-1}(a+b x)} \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )de^{i \csc ^{-1}(a+b x)}+i \int e^{-i \csc ^{-1}(a+b x)} \log (-a-b x+1)de^{i \csc ^{-1}(a+b x)}-2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )\right )}{b}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {-(a+b x) \csc ^{-1}(a+b x)^2+2 \left (-2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )+i \operatorname {PolyLog}(2,-a-b x)-i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )\right )}{b}\) |
-((-((a + b*x)*ArcCsc[a + b*x]^2) + 2*(-2*ArcCsc[a + b*x]*ArcTanh[E^(I*Arc Csc[a + b*x])] - I*PolyLog[2, E^(I*ArcCsc[a + b*x])] + I*PolyLog[2, -a - b *x]))/b)
3.1.30.3.1 Defintions of rubi rules used
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[Cot[(a_.) + (b_.)*(x_)^(n_.)]^(q_.)*Csc[(a_.) + (b_.)*(x_)^(n_.)]^(p_.) *(x_)^(m_.), x_Symbol] :> Simp[(-x^(m - n + 1))*(Csc[a + b*x^n]^p/(b*n*p)), x] + Simp[(m - n + 1)/(b*n*p) Int[x^(m - n)*Csc[a + b*x^n]^p, x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m, n] && EqQ[q, 1]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-c^(-1) Su bst[Int[(a + b*x)^n*Csc[x]*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]
Int[((a_.) + ArcCsc[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCsc[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
Time = 0.52 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.74
method | result | size |
derivativedivides | \(\frac {\operatorname {arccsc}\left (b x +a \right )^{2} \left (b x +a \right )-2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-2 i \operatorname {dilog}\left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 i \operatorname {dilog}\left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b}\) | \(150\) |
default | \(\frac {\operatorname {arccsc}\left (b x +a \right )^{2} \left (b x +a \right )-2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-2 i \operatorname {dilog}\left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 i \operatorname {dilog}\left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b}\) | \(150\) |
1/b*(arccsc(b*x+a)^2*(b*x+a)-2*arccsc(b*x+a)*ln(1-I/(b*x+a)-(1-1/(b*x+a)^2 )^(1/2))+2*arccsc(b*x+a)*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-2*I*dilog(1 +I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+2*I*dilog(1-I/(b*x+a)-(1-1/(b*x+a)^2)^(1 /2)))
\[ \int \csc ^{-1}(a+b x)^2 \, dx=\int { \operatorname {arccsc}\left (b x + a\right )^{2} \,d x } \]
\[ \int \csc ^{-1}(a+b x)^2 \, dx=\int \operatorname {acsc}^{2}{\left (a + b x \right )}\, dx \]
\[ \int \csc ^{-1}(a+b x)^2 \, dx=\int { \operatorname {arccsc}\left (b x + a\right )^{2} \,d x } \]
x*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 - 1/4*x*log(b^2*x^2 + 2*a*b*x + a^2)^2 + integrate((2*sqrt(b*x + a + 1)*sqrt(b*x + a - 1)*b*x*ar ctan2(1, sqrt(b*x + a + 1)*sqrt(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^3 + 3*a*b^2*x^2 + a^3 + (3*a^2 - 1) *b*x - a), x)
\[ \int \csc ^{-1}(a+b x)^2 \, dx=\int { \operatorname {arccsc}\left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int \csc ^{-1}(a+b x)^2 \, dx=\int {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \]