3.48 \(\int \frac{x}{(a+b \csc (c+d \sqrt{x}))^2} \, dx\)

Optimal. Leaf size=1565 \[ \text{result too large to display} \]

[Out]

((-2*I)*b^2*x^(3/2))/(a^2*(a^2 - b^2)*d) + x^2/(2*a^2) + (6*b^2*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqr
t[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (6*b^2*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2])])/(
a^2*(a^2 - b^2)*d^2) - ((2*I)*b^3*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-
a^2 + b^2)^(3/2)*d) + ((4*I)*b*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[
-a^2 + b^2]*d) + ((2*I)*b^3*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 +
b^2)^(3/2)*d) - ((4*I)*b*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 +
b^2]*d) - ((12*I)*b^2*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b
^2)*d^3) - ((12*I)*b^2*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b
^2)*d^3) - (6*b^3*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^
2) + (12*b*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (6*b
^3*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (12*b*x*Po
lyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (12*b^2*PolyLog[3,
-((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (12*b^2*PolyLog[3, -((a*E^(I*(c
+ d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) - ((12*I)*b^3*Sqrt[x]*PolyLog[3, (I*a*E^(I*(c
+ d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((24*I)*b*Sqrt[x]*PolyLog[3, (I*a*E^(I
*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((12*I)*b^3*Sqrt[x]*PolyLog[3, (I*a*E
^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((24*I)*b*Sqrt[x]*PolyLog[3, (I*
a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (12*b^3*PolyLog[4, (I*a*E^(I*(c
+ d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (24*b*PolyLog[4, (I*a*E^(I*(c + d*Sqrt
[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (12*b^3*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b
+ Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (24*b*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-
a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (2*b^2*x^(3/2)*Cos[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Sin[c +
d*Sqrt[x]]))

________________________________________________________________________________________

Rubi [A]  time = 2.69939, antiderivative size = 1565, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {4205, 4191, 3324, 3323, 2264, 2190, 2531, 6609, 2282, 6589, 4521} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Csc[c + d*Sqrt[x]])^2,x]

[Out]

((-2*I)*b^2*x^(3/2))/(a^2*(a^2 - b^2)*d) + x^2/(2*a^2) + (6*b^2*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqr
t[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (6*b^2*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2])])/(
a^2*(a^2 - b^2)*d^2) - ((2*I)*b^3*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-
a^2 + b^2)^(3/2)*d) + ((4*I)*b*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[
-a^2 + b^2]*d) + ((2*I)*b^3*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 +
b^2)^(3/2)*d) - ((4*I)*b*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 +
b^2]*d) - ((12*I)*b^2*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b
^2)*d^3) - ((12*I)*b^2*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b
^2)*d^3) - (6*b^3*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^
2) + (12*b*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (6*b
^3*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (12*b*x*Po
lyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (12*b^2*PolyLog[3,
-((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (12*b^2*PolyLog[3, -((a*E^(I*(c
+ d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) - ((12*I)*b^3*Sqrt[x]*PolyLog[3, (I*a*E^(I*(c
+ d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((24*I)*b*Sqrt[x]*PolyLog[3, (I*a*E^(I
*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((12*I)*b^3*Sqrt[x]*PolyLog[3, (I*a*E
^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((24*I)*b*Sqrt[x]*PolyLog[3, (I*
a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (12*b^3*PolyLog[4, (I*a*E^(I*(c
+ d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (24*b*PolyLog[4, (I*a*E^(I*(c + d*Sqrt
[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (12*b^3*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b
+ Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (24*b*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-
a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (2*b^2*x^(3/2)*Cos[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Sin[c +
d*Sqrt[x]]))

Rule 4205

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 4191

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*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 3324

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

Rule 3323

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6609

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

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 4521

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] + (Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a - Rt[-a^2 + b^
2, 2] + b*E^(I*(c + d*x))), x], x] + Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a + Rt[-a^2 + b^2, 2] + b*E^
(I*(c + d*x))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

Rubi steps

\begin{align*} \int \frac{x}{\left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{(a+b \csc (c+d x))^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^3}{a^2}+\frac{b^2 x^3}{a^2 (b+a \sin (c+d x))^2}-\frac{2 b x^3}{a^2 (b+a \sin (c+d x))}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{x^2}{2 a^2}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{x^3}{b+a \sin (c+d x)} \, dx,x,\sqrt{x}\right )}{a^2}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{(b+a \sin (c+d x))^2} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=\frac{x^2}{2 a^2}-\frac{2 b^2 x^{3/2} \cos \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt{x}\right )\right )}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^3}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,\sqrt{x}\right )}{a^2}-\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{b+a \sin (c+d x)} \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right )}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2 \cos (c+d x)}{b+a \sin (c+d x)} \, dx,x,\sqrt{x}\right )}{a \left (a^2-b^2\right ) d}\\ &=-\frac{2 i b^2 x^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}-\frac{2 b^2 x^{3/2} \cos \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt{x}\right )\right )}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^3}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right )}+\frac{(8 i b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^3}{2 b-2 \sqrt{-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2}}-\frac{(8 i b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^3}{2 b+2 \sqrt{-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2}}+\frac{\left (6 i b^2\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{i b-\sqrt{a^2-b^2}+a e^{i (c+d x)}} \, dx,x,\sqrt{x}\right )}{a \left (a^2-b^2\right ) d}+\frac{\left (6 i b^2\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{i b+\sqrt{a^2-b^2}+a e^{i (c+d x)}} \, dx,x,\sqrt{x}\right )}{a \left (a^2-b^2\right ) d}\\ &=-\frac{2 i b^2 x^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}+\frac{6 b^2 x \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{6 b^2 x \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{4 i b x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{4 i b x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{2 b^2 x^{3/2} \cos \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt{x}\right )\right )}-\frac{\left (4 i b^3\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^3}{2 b-2 \sqrt{-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt{x}\right )}{a \left (-a^2+b^2\right )^{3/2}}+\frac{\left (4 i b^3\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^3}{2 b+2 \sqrt{-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt{x}\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{a e^{i (c+d x)}}{i b-\sqrt{a^2-b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{a e^{i (c+d x)}}{i b+\sqrt{a^2-b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{(12 i b) \operatorname{Subst}\left (\int x^2 \log \left (1-\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{(12 i b) \operatorname{Subst}\left (\int x^2 \log \left (1-\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d}\\ &=-\frac{2 i b^2 x^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}+\frac{6 b^2 x \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{6 b^2 x \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 i b^3 x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 i b x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{2 i b^3 x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 i b x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{12 i b^2 \sqrt{x} \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{12 i b^2 \sqrt{x} \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{12 b x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{12 b x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{2 b^2 x^{3/2} \cos \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt{x}\right )\right )}+\frac{\left (12 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{a e^{i (c+d x)}}{i b-\sqrt{a^2-b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{\left (12 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{a e^{i (c+d x)}}{i b+\sqrt{a^2-b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{(24 b) \operatorname{Subst}\left (\int x \text{Li}_2\left (\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{(24 b) \operatorname{Subst}\left (\int x \text{Li}_2\left (\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{\left (6 i b^3\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{\left (6 i b^3\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}\\ &=-\frac{2 i b^2 x^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}+\frac{6 b^2 x \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{6 b^2 x \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 i b^3 x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 i b x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{2 i b^3 x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 i b x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{12 i b^2 \sqrt{x} \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{12 i b^2 \sqrt{x} \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{6 b^3 x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{12 b x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{6 b^3 x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{12 b x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{24 i b \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{24 i b \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{2 b^2 x^{3/2} \cos \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt{x}\right )\right )}+\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{-i b+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{a x}{i b+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{(24 i b) \operatorname{Subst}\left (\int \text{Li}_3\left (\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{(24 i b) \operatorname{Subst}\left (\int \text{Li}_3\left (\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{\left (12 b^3\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{\left (12 b^3\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}\\ &=-\frac{2 i b^2 x^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}+\frac{6 b^2 x \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{6 b^2 x \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 i b^3 x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 i b x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{2 i b^3 x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 i b x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{12 i b^2 \sqrt{x} \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{12 i b^2 \sqrt{x} \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{6 b^3 x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{12 b x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{6 b^3 x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{12 b x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{12 b^2 \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{12 b^2 \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{12 i b^3 \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{24 i b \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{12 i b^3 \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{24 i b \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{2 b^2 x^{3/2} \cos \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt{x}\right )\right )}-\frac{(24 b) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{i a x}{b-\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{(24 b) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{i a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{\left (12 i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{\left (12 i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}\\ &=-\frac{2 i b^2 x^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}+\frac{6 b^2 x \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{6 b^2 x \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 i b^3 x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 i b x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{2 i b^3 x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 i b x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{12 i b^2 \sqrt{x} \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{12 i b^2 \sqrt{x} \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{6 b^3 x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{12 b x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{6 b^3 x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{12 b x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{12 b^2 \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{12 b^2 \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{12 i b^3 \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{24 i b \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{12 i b^3 \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{24 i b \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{24 b \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{24 b \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{2 b^2 x^{3/2} \cos \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt{x}\right )\right )}+\frac{\left (12 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{i a x}{b-\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac{\left (12 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{i a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}\\ &=-\frac{2 i b^2 x^{3/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^2}{2 a^2}+\frac{6 b^2 x \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{6 b^2 x \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 i b^3 x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 i b x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{2 i b^3 x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 i b x^{3/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{12 i b^2 \sqrt{x} \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{12 i b^2 \sqrt{x} \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{6 b^3 x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{12 b x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{6 b^3 x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{12 b x \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{12 b^2 \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{12 b^2 \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{12 i b^3 \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{24 i b \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{12 i b^3 \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{24 i b \sqrt{x} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{12 b^3 \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac{24 b \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{12 b^3 \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{24 b \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{2 b^2 x^{3/2} \cos \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt{x}\right )\right )}\\ \end{align*}

Mathematica [A]  time = 14.4632, size = 1729, normalized size = 1.1 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Csc[c + d*Sqrt[x]]^2*(b + a*Sin[c + d*Sqrt[x]])*(x^2*(b + a*Sin[c + d*Sqrt[x]]) - ((4*I)*b*((2*b*d^3*E^((2*I)
*c)*x^(3/2))/(-1 + E^((2*I)*c)) + ((3*I)*b*d^2*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x
])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (2*I)*a^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*S
qrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + I*b^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*
Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (3*I)*b*d^2*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x*Log[1
+ (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^3*E^(I*c)*x^(3/2)*L
og[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - I*b^2*d^3*E^(I*c)*x^(3/2)*
Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 3*d*(2*b*Sqrt[(a^2 - b^2)
*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*Sqrt[x]*PolyLog[2, (I*a*E^(I*(2*c + d*Sqrt[x]
)))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 3*d*(2*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] + 2*a^2*d*E^(I*c)*
Sqrt[x] - b^2*d*E^(I*c)*Sqrt[x])*Sqrt[x]*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b
^2)*E^((2*I)*c)]))] + (6*I)*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c
) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (12*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/
(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (6*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[3, (I*a*E^(I*(2*c + d*Sqr
t[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (6*I)*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*PolyLog[3, -((a
*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (12*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyL
og[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (6*I)*b^2*d*E^(I*c)*Sqrt
[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 12*a^2*E^(I*c)*
PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 6*b^2*E^(I*c)*PolyLo
g[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 12*a^2*E^(I*c)*PolyLog[4,
-((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 6*b^2*E^(I*c)*PolyLog[4, -((a*
E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))])/Sqrt[(a^2 - b^2)*E^((2*I)*c)])*(b +
a*Sin[c + d*Sqrt[x]]))/((a^2 - b^2)*d^4) + (4*b^2*x^(3/2)*Csc[c]*(b*Cos[c] + a*Sin[d*Sqrt[x]]))/((a - b)*(a +
b)*d)))/(2*a^2*(a + b*Csc[c + d*Sqrt[x]])^2)

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Maple [F]  time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b\csc \left ( c+d\sqrt{x} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b*csc(c+d*x^(1/2)))^2,x)

[Out]

int(x/(a+b*csc(c+d*x^(1/2)))^2,x)

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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(x/(a+b*csc(c+d*x^(1/2)))^2,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{x}{b^{2} \csc \left (d \sqrt{x} + c\right )^{2} + 2 \, a b \csc \left (d \sqrt{x} + c\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*csc(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

integral(x/(b^2*csc(d*sqrt(x) + c)^2 + 2*a*b*csc(d*sqrt(x) + c) + a^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*csc(c+d*x**(1/2)))**2,x)

[Out]

Integral(x/(a + b*csc(c + d*sqrt(x)))**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*csc(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

integrate(x/(b*csc(d*sqrt(x) + c) + a)^2, x)