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

Optimal. Leaf size=1157 \[ \text{result too large to display} \]

[Out]

((-2*I)*b^2*x)/(a^2*(a^2 - b^2)*d) + (2*x^(3/2))/(3*a^2) + (4*b^2*Sqrt[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) + (4*b^2*Sqrt[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*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*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqr
t[-a^2 + b^2]*d) + ((2*I)*b^3*x*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*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) -
((4*I)*b^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((4*I)*b
^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (4*b^3*Sqrt[x]*Po
lyLog[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) + (8*b*Sqrt[x]*Poly
Log[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (4*b^3*Sqrt[x]*PolyLo
g[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) - (8*b*Sqrt[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) - ((4*I)*b^3*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) + ((8*I)*b*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) + ((4*I)*b^3*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) - ((8*I)*b*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) - (2*b^2*x*Cos[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b
+ a*Sin[c + d*Sqrt[x]]))

________________________________________________________________________________________

Rubi [A]  time = 2.14046, antiderivative size = 1157, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {4205, 4191, 3324, 3323, 2264, 2190, 2531, 2282, 6589, 4521, 2279, 2391} \[ -\frac{2 i x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac{2 i x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac{4 \sqrt{x} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}+\frac{4 \sqrt{x} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac{4 i \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac{4 i \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac{2 i x b^2}{a^2 \left (a^2-b^2\right ) d}+\frac{4 \sqrt{x} \log \left (\frac{e^{i \left (c+d \sqrt{x}\right )} a}{i b-\sqrt{a^2-b^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac{4 \sqrt{x} \log \left (\frac{e^{i \left (c+d \sqrt{x}\right )} a}{i b+\sqrt{a^2-b^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac{4 i \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b-\sqrt{a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}-\frac{4 i \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{i b+\sqrt{a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}-\frac{2 x \cos \left (c+d \sqrt{x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt{x}\right )\right )}+\frac{4 i x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d}-\frac{4 i x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d}+\frac{8 \sqrt{x} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^2}-\frac{8 \sqrt{x} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{8 i \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{8 i \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^3}+\frac{2 x^{3/2}}{3 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*b^2*x)/(a^2*(a^2 - b^2)*d) + (2*x^(3/2))/(3*a^2) + (4*b^2*Sqrt[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) + (4*b^2*Sqrt[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*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*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqr
t[-a^2 + b^2]*d) + ((2*I)*b^3*x*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*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) -
((4*I)*b^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((4*I)*b
^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (4*b^3*Sqrt[x]*Po
lyLog[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) + (8*b*Sqrt[x]*Poly
Log[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (4*b^3*Sqrt[x]*PolyLo
g[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) - (8*b*Sqrt[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) - ((4*I)*b^3*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) + ((8*I)*b*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) + ((4*I)*b^3*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) - ((8*I)*b*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) - (2*b^2*x*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 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]

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{\sqrt{x}}{\left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^2}{(a+b \csc (c+d x))^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^2}{a^2}+\frac{b^2 x^2}{a^2 (b+a \sin (c+d x))^2}-\frac{2 b x^2}{a^2 (b+a \sin (c+d x))}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 x^{3/2}}{3 a^2}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{x^2}{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^2}{(b+a \sin (c+d x))^2} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=\frac{2 x^{3/2}}{3 a^2}-\frac{2 b^2 x \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^2}{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^2}{b+a \sin (c+d x)} \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right )}+\frac{\left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{x \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}{a^2 \left (a^2-b^2\right ) d}+\frac{2 x^{3/2}}{3 a^2}-\frac{2 b^2 x \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^2}{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^2}{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^2}{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 (4 i b^2\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x}{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 (4 i b^2\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x}{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}{a^2 \left (a^2-b^2\right ) d}+\frac{2 x^{3/2}}{3 a^2}+\frac{4 b^2 \sqrt{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 b^2 \sqrt{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 \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 \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 \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^2}{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^2}{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 b^2\right ) \operatorname{Subst}\left (\int \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 (4 b^2\right ) \operatorname{Subst}\left (\int \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{(8 i b) \operatorname{Subst}\left (\int x \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{(8 i b) \operatorname{Subst}\left (\int x \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}{a^2 \left (a^2-b^2\right ) d}+\frac{2 x^{3/2}}{3 a^2}+\frac{4 b^2 \sqrt{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 b^2 \sqrt{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 \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 \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 \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 \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{8 b \sqrt{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{8 b \sqrt{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 \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^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\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^3}+\frac{\left (4 i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\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^3}-\frac{(8 b) \operatorname{Subst}\left (\int \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{(8 b) \operatorname{Subst}\left (\int \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 (4 i b^3\right ) \operatorname{Subst}\left (\int x \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 (4 i b^3\right ) \operatorname{Subst}\left (\int x \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}{a^2 \left (a^2-b^2\right ) d}+\frac{2 x^{3/2}}{3 a^2}+\frac{4 b^2 \sqrt{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 b^2 \sqrt{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 \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 \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 \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 \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^2 \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{4 i b^2 \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{4 b^3 \sqrt{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{8 b \sqrt{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{4 b^3 \sqrt{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{8 b \sqrt{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 \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 i b) \operatorname{Subst}\left (\int \frac{\text{Li}_2\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^3}-\frac{(8 i b) \operatorname{Subst}\left (\int \frac{\text{Li}_2\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^3}+\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \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 (4 b^3\right ) \operatorname{Subst}\left (\int \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}{a^2 \left (a^2-b^2\right ) d}+\frac{2 x^{3/2}}{3 a^2}+\frac{4 b^2 \sqrt{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 b^2 \sqrt{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 \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 \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 \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 \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^2 \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{4 i b^2 \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{4 b^3 \sqrt{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{8 b \sqrt{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{4 b^3 \sqrt{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{8 b \sqrt{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{8 i b \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{8 i b \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 \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{\text{Li}_2\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^3}+\frac{\left (4 i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\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^3}\\ &=-\frac{2 i b^2 x}{a^2 \left (a^2-b^2\right ) d}+\frac{2 x^{3/2}}{3 a^2}+\frac{4 b^2 \sqrt{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 b^2 \sqrt{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 \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 \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 \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 \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^2 \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{4 i b^2 \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{4 b^3 \sqrt{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{8 b \sqrt{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{4 b^3 \sqrt{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{8 b \sqrt{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{4 i b^3 \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{8 i b \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{4 i b^3 \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{8 i b \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 \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 = 8.33583, size = 846, normalized size = 0.73 \[ \frac{\csc ^2\left (c+d \sqrt{x}\right ) \left (b+a \sin \left (c+d \sqrt{x}\right )\right ) \left (\frac{6 x \csc (c) \left (b \cos (c)+a \sin \left (d \sqrt{x}\right )\right ) b^2}{(a-b) (a+b) d}-\frac{6 i \left (\frac{2 b e^{2 i c} x d^2}{-1+e^{2 i c}}+\frac{2 \left (-2 d e^{i c} \sqrt{x} a^2+b \sqrt{\left (a^2-b^2\right ) e^{2 i c}}+b^2 d e^{i c} \sqrt{x}\right ) \text{PolyLog}\left (2,\frac{i a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+i \sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right )+2 \left (2 d e^{i c} \sqrt{x} a^2+b \sqrt{\left (a^2-b^2\right ) e^{2 i c}}-b^2 d e^{i c} \sqrt{x}\right ) \text{PolyLog}\left (2,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{i e^{i c} b+\sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right )+i \left (d \sqrt{x} \left (\left (-2 d e^{i c} \sqrt{x} a^2+2 b \sqrt{\left (a^2-b^2\right ) e^{2 i c}}+b^2 d e^{i c} \sqrt{x}\right ) \log \left (\frac{e^{i \left (2 c+d \sqrt{x}\right )} a}{i b e^{i c}-\sqrt{\left (a^2-b^2\right ) e^{2 i c}}}+1\right )+\left (2 d e^{i c} \sqrt{x} a^2+2 b \sqrt{\left (a^2-b^2\right ) e^{2 i c}}-b^2 d e^{i c} \sqrt{x}\right ) \log \left (\frac{e^{i \left (2 c+d \sqrt{x}\right )} a}{i e^{i c} b+\sqrt{\left (a^2-b^2\right ) e^{2 i c}}}+1\right )\right )-2 \left (2 a^2-b^2\right ) e^{i c} \text{PolyLog}\left (3,\frac{i a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+i \sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right )+2 \left (2 a^2-b^2\right ) e^{i c} \text{PolyLog}\left (3,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{i e^{i c} b+\sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right )\right )}{\sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right ) \left (b+a \sin \left (c+d \sqrt{x}\right )\right ) b}{\left (a^2-b^2\right ) d^3}+2 x^{3/2} \left (b+a \sin \left (c+d \sqrt{x}\right )\right )\right )}{3 a^2 \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[c + d*Sqrt[x]]^2*(b + a*Sin[c + d*Sqrt[x]])*(2*x^(3/2)*(b + a*Sin[c + d*Sqrt[x]]) - ((6*I)*b*((2*b*d^2*E^
((2*I)*c)*x)/(-1 + E^((2*I)*c)) + (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])*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)])] + 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])*PolyLog[2, -((a*E^(I*(2*c + d*Sqr
t[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + I*(d*Sqrt[x]*((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])*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*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])*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)])]) - 2*(2*a^2 - b^2)*E^(I*c)*P
olyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 2*(2*a^2 - b^2)*E^(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)]))]))/Sqrt[(a^2 - b^2)
*E^((2*I)*c)])*(b + a*Sin[c + d*Sqrt[x]]))/((a^2 - b^2)*d^3) + (6*b^2*x*Csc[c]*(b*Cos[c] + a*Sin[d*Sqrt[x]]))/
((a - b)*(a + b)*d)))/(3*a^2*(a + b*Csc[c + d*Sqrt[x]])^2)

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Maple [F]  time = 0.174, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{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^(1/2)/(a+b*csc(c+d*x^(1/2)))^2,x)

[Out]

int(x^(1/2)/(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^(1/2)/(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{\sqrt{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^(1/2)/(a+b*csc(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

integral(sqrt(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{\sqrt{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**(1/2)/(a+b*csc(c+d*x**(1/2)))**2,x)

[Out]

Integral(sqrt(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{\sqrt{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^(1/2)/(a+b*csc(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

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