3.100 \(\int \frac{a+b \csc ^{-1}(c x)}{d+e x^2} \, dx\)

Optimal. Leaf size=529 \[ -\frac{i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}} \]

[Out]

-((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[
e]) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*
Sqrt[e]) - ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*Sqrt
[-d]*Sqrt[e]) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2
*Sqrt[-d]*Sqrt[e]) - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(Sq
rt[-d]*Sqrt[e]) + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(Sqrt[-d]
*Sqrt[e]) - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(Sqrt[-d]*Sq
rt[e]) + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e])

________________________________________________________________________________________

Rubi [A]  time = 0.870293, antiderivative size = 529, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {5231, 4667, 4741, 4519, 2190, 2279, 2391} \[ -\frac{i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])/(d + e*x^2),x]

[Out]

-((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[
e]) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*
Sqrt[e]) - ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*Sqrt
[-d]*Sqrt[e]) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2
*Sqrt[-d]*Sqrt[e]) - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(Sq
rt[-d]*Sqrt[e]) + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(Sqrt[-d]
*Sqrt[e]) - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(Sqrt[-d]*Sq
rt[e]) + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e])

Rule 5231

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Subst[Int[((e + d*x^2
)^p*(a + b*ArcSin[x/c])^n)/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && Integer
Q[p]

Rule 4667

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4741

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Cos[x])/
(c*d + e*Sin[x]), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 4519

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(I*(c + d*x)))/(a - Rt[a^2 - b^2, 2] - I*b
*E^(I*(c + d*x))), x] + Int[((e + f*x)^m*E^(I*(c + d*x)))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2391

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]

Rubi steps

\begin{align*} \int \frac{a+b \csc ^{-1}(c x)}{d+e x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{2 \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{2 \sqrt{e}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{2 \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{2 \sqrt{e}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}-i \sqrt{-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}-i \sqrt{-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}+i \sqrt{-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}+i \sqrt{-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 \sqrt{e}}\\ &=-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{i \sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{i \sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{i \sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{i \sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 \sqrt{-d} \sqrt{e}}\\ &=-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i \sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i \sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i \sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i \sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 \sqrt{-d} \sqrt{e}}\\ &=-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{i b \text{Li}_2\left (-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \text{Li}_2\left (\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{i b \text{Li}_2\left (-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \text{Li}_2\left (\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}\\ \end{align*}

Mathematica [B]  time = 0.380951, size = 1068, normalized size = 2.02 \[ -\frac{i \left (4 i a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )+8 i b \sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right ) \tan ^{-1}\left (\frac{\left (\sqrt{e}-i c \sqrt{d}\right ) \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(c x)+\pi \right )\right )}{\sqrt{d c^2+e}}\right )-8 i b \sin ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right ) \tan ^{-1}\left (\frac{\left (i \sqrt{d} c+\sqrt{e}\right ) \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(c x)+\pi \right )\right )}{\sqrt{d c^2+e}}\right )-2 b \csc ^{-1}(c x) \log \left (\frac{e^{-i \csc ^{-1}(c x)} \left (\sqrt{e}-\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )+4 b \sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right ) \log \left (\frac{e^{-i \csc ^{-1}(c x)} \left (\sqrt{e}-\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )+b \pi \log \left (\frac{e^{-i \csc ^{-1}(c x)} \left (\sqrt{e}-\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )+2 b \csc ^{-1}(c x) \log \left (\frac{e^{-i \csc ^{-1}(c x)} \left (\sqrt{d c^2+e}-\sqrt{e}\right )}{c \sqrt{d}}+1\right )-4 b \sin ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right ) \log \left (\frac{e^{-i \csc ^{-1}(c x)} \left (\sqrt{d c^2+e}-\sqrt{e}\right )}{c \sqrt{d}}+1\right )-b \pi \log \left (\frac{e^{-i \csc ^{-1}(c x)} \left (\sqrt{d c^2+e}-\sqrt{e}\right )}{c \sqrt{d}}+1\right )+2 b \csc ^{-1}(c x) \log \left (1-\frac{\left (\sqrt{e}+\sqrt{d c^2+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right )+4 b \sin ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right ) \log \left (1-\frac{\left (\sqrt{e}+\sqrt{d c^2+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right )-b \pi \log \left (1-\frac{\left (\sqrt{e}+\sqrt{d c^2+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right )-2 b \csc ^{-1}(c x) \log \left (\frac{e^{-i \csc ^{-1}(c x)} \left (\sqrt{e}+\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )-4 b \sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right ) \log \left (\frac{e^{-i \csc ^{-1}(c x)} \left (\sqrt{e}+\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )+b \pi \log \left (\frac{e^{-i \csc ^{-1}(c x)} \left (\sqrt{e}+\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )+b \pi \log \left (\sqrt{e}-\frac{i \sqrt{d}}{x}\right )-b \pi \log \left (\frac{i \sqrt{d}}{x}+\sqrt{e}\right )+2 i b \text{PolyLog}\left (2,\frac{\left (\sqrt{e}-\sqrt{d c^2+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right )-2 i b \text{PolyLog}\left (2,\frac{\left (\sqrt{d c^2+e}-\sqrt{e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right )-2 i b \text{PolyLog}\left (2,-\frac{\left (\sqrt{e}+\sqrt{d c^2+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right )+2 i b \text{PolyLog}\left (2,\frac{\left (\sqrt{e}+\sqrt{d c^2+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right )\right )}{4 \sqrt{d} \sqrt{e}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCsc[c*x])/(d + e*x^2),x]

[Out]

((-I/4)*((4*I)*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + (8*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTa
n[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] - (8*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[
e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + b*Pi
*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[
c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sq
rt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - b*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]
*E^(I*ArcCsc[c*x]))] + 2*b*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 4
*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcC
sc[c*x]))] - b*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 2*b*ArcCsc[c*x]*Log[1 -
(Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqr
t[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + b*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d
+ e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*Arc
Csc[c*x]))] - 4*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqr
t[d]*E^(I*ArcCsc[c*x]))] + b*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - b*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + (2*I)*b*Pol
yLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (2*I)*b*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d
+ e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (2*I)*b*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCs
c[c*x])))] + (2*I)*b*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))]))/(Sqrt[d]*Sqrt[e])

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Maple [C]  time = 0.94, size = 272, normalized size = 0.5 \begin{align*}{a\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{cb}{2}\sum _{{\it \_R1}={\it RootOf} \left ({c}^{2}d{{\it \_Z}}^{4}+ \left ( -2\,{c}^{2}d-4\,e \right ){{\it \_Z}}^{2}+{c}^{2}d \right ) }{\frac{1}{{\it \_R1}\, \left ({{\it \_R1}}^{2}{c}^{2}d-{c}^{2}d-2\,e \right ) } \left ( i{\rm arccsc} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \right ) \right ) }}-{\frac{cb}{2}\sum _{{\it \_R1}={\it RootOf} \left ({c}^{2}d{{\it \_Z}}^{4}+ \left ( -2\,{c}^{2}d-4\,e \right ){{\it \_Z}}^{2}+{c}^{2}d \right ) }{\frac{{\it \_R1}}{{{\it \_R1}}^{2}{c}^{2}d-{c}^{2}d-2\,e} \left ( i{\rm arccsc} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsc(c*x))/(e*x^2+d),x)

[Out]

a/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-1/2*c*b*sum(1/_R1/(_R1^2*c^2*d-c^2*d-2*e)*(I*arccsc(c*x)*ln((_R1-I/c/x-(
1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2
+c^2*d))-1/2*c*b*sum(_R1/(_R1^2*c^2*d-c^2*d-2*e)*(I*arccsc(c*x)*ln((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)+dilog(
(_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2+c^2*d))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arccsc}\left (c x\right ) + a}{e x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arccsc(c*x) + a)/(e*x^2 + d), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{acsc}{\left (c x \right )}}{d + e x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acsc(c*x))/(d + e*x**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arccsc}\left (c x\right ) + a}{e x^{2} + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccsc(c*x) + a)/(e*x^2 + d), x)