\(\int \frac {x^3}{a+b \csc (c+d \sqrt {x})} \, dx\) [41]

Optimal result

Integrand size = 20, antiderivative size = 1075 \[ \int \frac {x^3}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {5040 b x \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}-\frac {5040 b x \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {10080 i b \sqrt {x} \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {10080 i b \sqrt {x} \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {10080 b \operatorname {PolyLog}\left (8,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^8}+\frac {10080 b \operatorname {PolyLog}\left (8,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^8} \]

[Out]

1/4*x^4/a-84*I*b*x^(5/2)*polylog(3,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a/d^3/(-a^2+b^2)^(1/2)-1680*
I*b*x^(3/2)*polylog(5,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a/d^5/(-a^2+b^2)^(1/2)+14*b*x^3*polylog(2
,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a/d^2/(-a^2+b^2)^(1/2)-14*b*x^3*polylog(2,I*a*exp(I*(c+d*x^(1/
2)))/(b+(-a^2+b^2)^(1/2)))/a/d^2/(-a^2+b^2)^(1/2)+10080*I*b*polylog(7,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(
1/2)))*x^(1/2)/a/d^7/(-a^2+b^2)^(1/2)+2*I*b*x^(7/2)*ln(1-I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a/d/(-
a^2+b^2)^(1/2)-420*b*x^2*polylog(4,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a/d^4/(-a^2+b^2)^(1/2)+420*b
*x^2*polylog(4,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a/d^4/(-a^2+b^2)^(1/2)-2*I*b*x^(7/2)*ln(1-I*a*ex
p(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a/d/(-a^2+b^2)^(1/2)+1680*I*b*x^(3/2)*polylog(5,I*a*exp(I*(c+d*x^(1/2
)))/(b+(-a^2+b^2)^(1/2)))/a/d^5/(-a^2+b^2)^(1/2)+5040*b*x*polylog(6,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/
2)))/a/d^6/(-a^2+b^2)^(1/2)-5040*b*x*polylog(6,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a/d^6/(-a^2+b^2)
^(1/2)-10080*b*polylog(8,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a/d^8/(-a^2+b^2)^(1/2)+10080*b*polylog
(8,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a/d^8/(-a^2+b^2)^(1/2)-10080*I*b*polylog(7,I*a*exp(I*(c+d*x^
(1/2)))/(b+(-a^2+b^2)^(1/2)))*x^(1/2)/a/d^7/(-a^2+b^2)^(1/2)+84*I*b*x^(5/2)*polylog(3,I*a*exp(I*(c+d*x^(1/2)))
/(b-(-a^2+b^2)^(1/2)))/a/d^3/(-a^2+b^2)^(1/2)

Rubi [A] (verified)

Time = 1.92 (sec) , antiderivative size = 1075, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {4290, 4276, 3404, 2296, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {x^3}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\frac {x^4}{4 a}+\frac {2 i b \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{7/2}}{a \sqrt {b^2-a^2} d}-\frac {2 i b \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{7/2}}{a \sqrt {b^2-a^2} d}+\frac {14 b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^3}{a \sqrt {b^2-a^2} d^2}-\frac {14 b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^3}{a \sqrt {b^2-a^2} d^2}+\frac {84 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{5/2}}{a \sqrt {b^2-a^2} d^3}-\frac {84 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{5/2}}{a \sqrt {b^2-a^2} d^3}-\frac {420 b \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a \sqrt {b^2-a^2} d^4}+\frac {420 b \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a \sqrt {b^2-a^2} d^4}-\frac {1680 i b \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{3/2}}{a \sqrt {b^2-a^2} d^5}+\frac {1680 i b \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{3/2}}{a \sqrt {b^2-a^2} d^5}+\frac {5040 b \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x}{a \sqrt {b^2-a^2} d^6}-\frac {5040 b \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x}{a \sqrt {b^2-a^2} d^6}+\frac {10080 i b \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a \sqrt {b^2-a^2} d^7}-\frac {10080 i b \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a \sqrt {b^2-a^2} d^7}-\frac {10080 b \operatorname {PolyLog}\left (8,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a \sqrt {b^2-a^2} d^8}+\frac {10080 b \operatorname {PolyLog}\left (8,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a \sqrt {b^2-a^2} d^8} \]

[In]

Int[x^3/(a + b*Csc[c + d*Sqrt[x]]),x]

[Out]

x^4/(4*a) + ((2*I)*b*x^(7/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*
d) - ((2*I)*b*x^(7/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (1
4*b*x^3*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) - (14*b*x^3*P
olyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) + ((84*I)*b*x^(5/2)*Po
lyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - ((84*I)*b*x^(5/2)*Pol
yLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - (420*b*x^2*PolyLog[4,
(I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) + (420*b*x^2*PolyLog[4, (I*a*E^(
I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) - ((1680*I)*b*x^(3/2)*PolyLog[5, (I*a*E^
(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^5) + ((1680*I)*b*x^(3/2)*PolyLog[5, (I*a*E
^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^5) + (5040*b*x*PolyLog[6, (I*a*E^(I*(c +
d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^6) - (5040*b*x*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x]
)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^6) + ((10080*I)*b*Sqrt[x]*PolyLog[7, (I*a*E^(I*(c + d*Sqrt[
x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^7) - ((10080*I)*b*Sqrt[x]*PolyLog[7, (I*a*E^(I*(c + d*Sqr
t[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^7) - (10080*b*PolyLog[8, (I*a*E^(I*(c + d*Sqrt[x])))/(b
- Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^8) + (10080*b*PolyLog[8, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a
^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^8)

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2320

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 2611

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 3404

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 4276

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 4290

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 6724

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 6744

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]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^7}{a+b \csc (c+d x)} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {x^7}{a}-\frac {b x^7}{a (b+a \sin (c+d x))}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {x^4}{4 a}-\frac {(2 b) \text {Subst}\left (\int \frac {x^7}{b+a \sin (c+d x)} \, dx,x,\sqrt {x}\right )}{a} \\ & = \frac {x^4}{4 a}-\frac {(4 b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^7}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a} \\ & = \frac {x^4}{4 a}+\frac {(4 i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^7}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^2+b^2}}-\frac {(4 i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^7}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^2+b^2}} \\ & = \frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {(14 i b) \text {Subst}\left (\int x^6 \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 \sqrt {-a^2+b^2} d}+\frac {(14 i b) \text {Subst}\left (\int x^6 \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 \sqrt {-a^2+b^2} d} \\ & = \frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {(84 b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {(84 b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^2} \\ & = \frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {(420 i b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^3}+\frac {(420 i b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^3} \\ & = \frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {(1680 b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {(1680 b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^4} \\ & = \frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {(5040 i b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^5}-\frac {(5040 i b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^5} \\ & = \frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {5040 b x \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}-\frac {5040 b x \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}-\frac {(10080 b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {(10080 b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^6} \\ & = \frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {5040 b x \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}-\frac {5040 b x \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {10080 i b \sqrt {x} \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {10080 i b \sqrt {x} \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {(10080 i b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^7}+\frac {(10080 i b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^7} \\ & = \frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {5040 b x \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}-\frac {5040 b x \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {10080 i b \sqrt {x} \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {10080 i b \sqrt {x} \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {(10080 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (7,\frac {i a x}{b-\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{a \sqrt {-a^2+b^2} d^8}+\frac {(10080 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (7,\frac {i a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{a \sqrt {-a^2+b^2} d^8} \\ & = \frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {5040 b x \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}-\frac {5040 b x \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {10080 i b \sqrt {x} \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {10080 i b \sqrt {x} \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {10080 b \operatorname {PolyLog}\left (8,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^8}+\frac {10080 b \operatorname {PolyLog}\left (8,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.08 (sec) , antiderivative size = 850, normalized size of antiderivative = 0.79 \[ \int \frac {x^3}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\frac {\sqrt {a^2-b^2} d^8 x^4-8 b d^7 x^{7/2} \log \left (1-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-i b+\sqrt {a^2-b^2}}\right )+8 b d^7 x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )+56 i b d^6 x^3 \operatorname {PolyLog}\left (2,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-i b+\sqrt {a^2-b^2}}\right )-56 i b d^6 x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )-336 b d^5 x^{5/2} \operatorname {PolyLog}\left (3,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-i b+\sqrt {a^2-b^2}}\right )+336 b d^5 x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )-1680 i b d^4 x^2 \operatorname {PolyLog}\left (4,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-i b+\sqrt {a^2-b^2}}\right )+1680 i b d^4 x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )+6720 b d^3 x^{3/2} \operatorname {PolyLog}\left (5,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-i b+\sqrt {a^2-b^2}}\right )-6720 b d^3 x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )+20160 i b d^2 x \operatorname {PolyLog}\left (6,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-i b+\sqrt {a^2-b^2}}\right )-20160 i b d^2 x \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )-40320 b d \sqrt {x} \operatorname {PolyLog}\left (7,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-i b+\sqrt {a^2-b^2}}\right )+40320 b d \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )-40320 i b \operatorname {PolyLog}\left (8,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-i b+\sqrt {a^2-b^2}}\right )+40320 i b \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )}{4 a \sqrt {a^2-b^2} d^8} \]

[In]

Integrate[x^3/(a + b*Csc[c + d*Sqrt[x]]),x]

[Out]

(Sqrt[a^2 - b^2]*d^8*x^4 - 8*b*d^7*x^(7/2)*Log[1 - (a*E^(I*(c + d*Sqrt[x])))/((-I)*b + Sqrt[a^2 - b^2])] + 8*b
*d^7*x^(7/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2])] + (56*I)*b*d^6*x^3*PolyLog[2, (a*E^(I*
(c + d*Sqrt[x])))/((-I)*b + Sqrt[a^2 - b^2])] - (56*I)*b*d^6*x^3*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b +
Sqrt[a^2 - b^2]))] - 336*b*d^5*x^(5/2)*PolyLog[3, (a*E^(I*(c + d*Sqrt[x])))/((-I)*b + Sqrt[a^2 - b^2])] + 336
*b*d^5*x^(5/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))] - (1680*I)*b*d^4*x^2*PolyLog[4
, (a*E^(I*(c + d*Sqrt[x])))/((-I)*b + Sqrt[a^2 - b^2])] + (1680*I)*b*d^4*x^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[
x])))/(I*b + Sqrt[a^2 - b^2]))] + 6720*b*d^3*x^(3/2)*PolyLog[5, (a*E^(I*(c + d*Sqrt[x])))/((-I)*b + Sqrt[a^2 -
b^2])] - 6720*b*d^3*x^(3/2)*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))] + (20160*I)*b*d^
2*x*PolyLog[6, (a*E^(I*(c + d*Sqrt[x])))/((-I)*b + Sqrt[a^2 - b^2])] - (20160*I)*b*d^2*x*PolyLog[6, -((a*E^(I*
(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))] - 40320*b*d*Sqrt[x]*PolyLog[7, (a*E^(I*(c + d*Sqrt[x])))/((-I)*b +
Sqrt[a^2 - b^2])] + 40320*b*d*Sqrt[x]*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))] - (403
20*I)*b*PolyLog[8, (a*E^(I*(c + d*Sqrt[x])))/((-I)*b + Sqrt[a^2 - b^2])] + (40320*I)*b*PolyLog[8, -((a*E^(I*(c
+ d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(4*a*Sqrt[a^2 - b^2]*d^8)

Maple [F]

\[\int \frac {x^{3}}{a +b \csc \left (c +d \sqrt {x}\right )}d x\]

[In]

int(x^3/(a+b*csc(c+d*x^(1/2))),x)

[Out]

int(x^3/(a+b*csc(c+d*x^(1/2))),x)

Fricas [F]

\[ \int \frac {x^3}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x^{3}}{b \csc \left (d \sqrt {x} + c\right ) + a} \,d x } \]

[In]

integrate(x^3/(a+b*csc(c+d*x^(1/2))),x, algorithm="fricas")

[Out]

integral(x^3/(b*csc(d*sqrt(x) + c) + a), x)

Sympy [F]

\[ \int \frac {x^3}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\int \frac {x^{3}}{a + b \csc {\left (c + d \sqrt {x} \right )}}\, dx \]

[In]

integrate(x**3/(a+b*csc(c+d*x**(1/2))),x)

[Out]

Integral(x**3/(a + b*csc(c + d*sqrt(x))), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^3/(a+b*csc(c+d*x^(1/2))),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

Giac [F]

\[ \int \frac {x^3}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x^{3}}{b \csc \left (d \sqrt {x} + c\right ) + a} \,d x } \]

[In]

integrate(x^3/(a+b*csc(c+d*x^(1/2))),x, algorithm="giac")

[Out]

integrate(x^3/(b*csc(d*sqrt(x) + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\int \frac {x^3}{a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}} \,d x \]

[In]

int(x^3/(a + b/sin(c + d*x^(1/2))),x)

[Out]

int(x^3/(a + b/sin(c + d*x^(1/2))), x)