\(\int x^3 (a+b \csc (c+d \sqrt {x}))^2 \, dx\) [36]

Optimal result

Integrand size = 20, antiderivative size = 695 \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {315 i b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {315 b^2 \operatorname {PolyLog}\left (7,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8} \]

[Out]

28*I*a*b*x^3*polylog(2,-exp(I*(c+d*x^(1/2))))/d^2+840*I*a*b*x^2*polylog(4,exp(I*(c+d*x^(1/2))))/d^4+10080*I*a*
b*x*polylog(6,-exp(I*(c+d*x^(1/2))))/d^6+210*I*b^2*x^(3/2)*polylog(4,exp(2*I*(c+d*x^(1/2))))/d^5+20160*I*a*b*p
olylog(8,exp(I*(c+d*x^(1/2))))/d^8-8*a*b*x^(7/2)*arctanh(exp(I*(c+d*x^(1/2))))/d-168*a*b*x^(5/2)*polylog(3,-ex
p(I*(c+d*x^(1/2))))/d^3+168*a*b*x^(5/2)*polylog(3,exp(I*(c+d*x^(1/2))))/d^3+3360*a*b*x^(3/2)*polylog(5,-exp(I*
(c+d*x^(1/2))))/d^5-3360*a*b*x^(3/2)*polylog(5,exp(I*(c+d*x^(1/2))))/d^5-20160*a*b*polylog(7,-exp(I*(c+d*x^(1/
2))))*x^(1/2)/d^7+20160*a*b*polylog(7,exp(I*(c+d*x^(1/2))))*x^(1/2)/d^7-42*I*b^2*x^(5/2)*polylog(2,exp(2*I*(c+
d*x^(1/2))))/d^3-20160*I*a*b*polylog(8,-exp(I*(c+d*x^(1/2))))/d^8-315*I*b^2*polylog(6,exp(2*I*(c+d*x^(1/2))))*
x^(1/2)/d^7+1/4*a^2*x^4-28*I*a*b*x^3*polylog(2,exp(I*(c+d*x^(1/2))))/d^2-840*I*a*b*x^2*polylog(4,-exp(I*(c+d*x
^(1/2))))/d^4-10080*I*a*b*x*polylog(6,exp(I*(c+d*x^(1/2))))/d^6-2*b^2*x^(7/2)*cot(c+d*x^(1/2))/d+14*b^2*x^3*ln
(1-exp(2*I*(c+d*x^(1/2))))/d^2+105*b^2*x^2*polylog(3,exp(2*I*(c+d*x^(1/2))))/d^4-315*b^2*x*polylog(5,exp(2*I*(
c+d*x^(1/2))))/d^6-2*I*b^2*x^(7/2)/d+315/2*b^2*polylog(7,exp(2*I*(c+d*x^(1/2))))/d^8

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4290, 4275, 4268, 2611, 6744, 2320, 6724, 4269, 3798, 2221} \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {315 b^2 \operatorname {PolyLog}\left (7,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {315 i b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}-\frac {2 i b^2 x^{7/2}}{d} \]

[In]

Int[x^3*(a + b*Csc[c + d*Sqrt[x]])^2,x]

[Out]

((-2*I)*b^2*x^(7/2))/d + (a^2*x^4)/4 - (8*a*b*x^(7/2)*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d - (2*b^2*x^(7/2)*Cot[c
+ d*Sqrt[x]])/d + (14*b^2*x^3*Log[1 - E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((28*I)*a*b*x^3*PolyLog[2, -E^(I*(c +
d*Sqrt[x]))])/d^2 - ((28*I)*a*b*x^3*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - ((42*I)*b^2*x^(5/2)*PolyLog[2, E
^((2*I)*(c + d*Sqrt[x]))])/d^3 - (168*a*b*x^(5/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (168*a*b*x^(5/2)*P
olyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 + (105*b^2*x^2*PolyLog[3, E^((2*I)*(c + d*Sqrt[x]))])/d^4 - ((840*I)*a*b
*x^2*PolyLog[4, -E^(I*(c + d*Sqrt[x]))])/d^4 + ((840*I)*a*b*x^2*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4 + ((210
*I)*b^2*x^(3/2)*PolyLog[4, E^((2*I)*(c + d*Sqrt[x]))])/d^5 + (3360*a*b*x^(3/2)*PolyLog[5, -E^(I*(c + d*Sqrt[x]
))])/d^5 - (3360*a*b*x^(3/2)*PolyLog[5, E^(I*(c + d*Sqrt[x]))])/d^5 - (315*b^2*x*PolyLog[5, E^((2*I)*(c + d*Sq
rt[x]))])/d^6 + ((10080*I)*a*b*x*PolyLog[6, -E^(I*(c + d*Sqrt[x]))])/d^6 - ((10080*I)*a*b*x*PolyLog[6, E^(I*(c
+ d*Sqrt[x]))])/d^6 - ((315*I)*b^2*Sqrt[x]*PolyLog[6, E^((2*I)*(c + d*Sqrt[x]))])/d^7 - (20160*a*b*Sqrt[x]*Po
lyLog[7, -E^(I*(c + d*Sqrt[x]))])/d^7 + (20160*a*b*Sqrt[x]*PolyLog[7, E^(I*(c + d*Sqrt[x]))])/d^7 + (315*b^2*P
olyLog[7, E^((2*I)*(c + d*Sqrt[x]))])/(2*d^8) - ((20160*I)*a*b*PolyLog[8, -E^(I*(c + d*Sqrt[x]))])/d^8 + ((201
60*I)*a*b*PolyLog[8, E^(I*(c + d*Sqrt[x]))])/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 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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4275

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 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 x^7 (a+b \csc (c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^7+2 a b x^7 \csc (c+d x)+b^2 x^7 \csc ^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^4}{4}+(4 a b) \text {Subst}\left (\int x^7 \csc (c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^7 \csc ^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}-\frac {(28 a b) \text {Subst}\left (\int x^6 \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(28 a b) \text {Subst}\left (\int x^6 \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (14 b^2\right ) \text {Subst}\left (\int x^6 \cot (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(168 i a b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(168 i a b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (28 i b^2\right ) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} x^6}{1-e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {(840 a b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {(840 a b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (84 b^2\right ) \text {Subst}\left (\int x^5 \log \left (1-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {(3360 i a b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(3360 i a b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}+\frac {\left (210 i b^2\right ) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {(10080 a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5}+\frac {(10080 a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5}-\frac {\left (420 b^2\right ) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {(20160 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6}+\frac {(20160 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6}-\frac {\left (630 i b^2\right ) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (4,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {(20160 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7}-\frac {(20160 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7}+\frac {\left (630 b^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (5,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {315 i b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {(20160 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(7,-x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {(20160 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(7,x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {\left (315 i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (6,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {315 i b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {\left (315 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(6,x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{2 d^8} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {315 i b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {315 b^2 \operatorname {PolyLog}\left (7,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 21.47 (sec) , antiderivative size = 1323, normalized size of antiderivative = 1.90 \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^4 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \sin ^2\left (c+d \sqrt {x}\right )}{4 \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2}-\frac {b^2 e^{i c} \csc (c) \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \left (2 d^7 e^{-2 i c} x^{7/2}+7 i d^6 \left (1-e^{-2 i c}\right ) x^3 \log \left (1-e^{-i \left (c+d \sqrt {x}\right )}\right )+7 i d^6 \left (1-e^{-2 i c}\right ) x^3 \log \left (1+e^{-i \left (c+d \sqrt {x}\right )}\right )-42 d^5 \left (1-e^{-2 i c}\right ) x^{5/2} \operatorname {PolyLog}\left (2,-e^{-i \left (c+d \sqrt {x}\right )}\right )-42 d^5 \left (1-e^{-2 i c}\right ) x^{5/2} \operatorname {PolyLog}\left (2,e^{-i \left (c+d \sqrt {x}\right )}\right )+210 i d^4 \left (1-e^{-2 i c}\right ) x^2 \operatorname {PolyLog}\left (3,-e^{-i \left (c+d \sqrt {x}\right )}\right )+210 i d^4 \left (1-e^{-2 i c}\right ) x^2 \operatorname {PolyLog}\left (3,e^{-i \left (c+d \sqrt {x}\right )}\right )+840 d^3 \left (1-e^{-2 i c}\right ) x^{3/2} \operatorname {PolyLog}\left (4,-e^{-i \left (c+d \sqrt {x}\right )}\right )+840 d^3 \left (1-e^{-2 i c}\right ) x^{3/2} \operatorname {PolyLog}\left (4,e^{-i \left (c+d \sqrt {x}\right )}\right )-2520 i d^2 \left (1-e^{-2 i c}\right ) x \operatorname {PolyLog}\left (5,-e^{-i \left (c+d \sqrt {x}\right )}\right )-2520 i d^2 \left (1-e^{-2 i c}\right ) x \operatorname {PolyLog}\left (5,e^{-i \left (c+d \sqrt {x}\right )}\right )-5040 d \left (1-e^{-2 i c}\right ) \sqrt {x} \operatorname {PolyLog}\left (6,-e^{-i \left (c+d \sqrt {x}\right )}\right )-5040 d \left (1-e^{-2 i c}\right ) \sqrt {x} \operatorname {PolyLog}\left (6,e^{-i \left (c+d \sqrt {x}\right )}\right )+5040 i \left (1-e^{-2 i c}\right ) \operatorname {PolyLog}\left (7,-e^{-i \left (c+d \sqrt {x}\right )}\right )+5040 i \left (1-e^{-2 i c}\right ) \operatorname {PolyLog}\left (7,e^{-i \left (c+d \sqrt {x}\right )}\right )\right ) \sin ^2\left (c+d \sqrt {x}\right )}{d^8 \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2}+\frac {4 a b \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \left (d^7 x^{7/2} \log \left (1-e^{i \left (c+d \sqrt {x}\right )}\right )-d^7 x^{7/2} \log \left (1+e^{i \left (c+d \sqrt {x}\right )}\right )+7 i d^6 x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )-7 i d^6 x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )-42 d^5 x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )+42 d^5 x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )-210 i d^4 x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )+210 i d^4 x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )+840 d^3 x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )-840 d^3 x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )+2520 i d^2 x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )-2520 i d^2 x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )-5040 d \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )+5040 d \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )-5040 i \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )+5040 i \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )\right ) \sin ^2\left (c+d \sqrt {x}\right )}{d^8 \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2}+\frac {b^2 x^{7/2} \csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right ) \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \sin ^2\left (c+d \sqrt {x}\right ) \sin \left (\frac {d \sqrt {x}}{2}\right )}{d \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2}+\frac {b^2 x^{7/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right ) \sin ^2\left (c+d \sqrt {x}\right ) \sin \left (\frac {d \sqrt {x}}{2}\right )}{d \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2} \]

[In]

Integrate[x^3*(a + b*Csc[c + d*Sqrt[x]])^2,x]

[Out]

(a^2*x^4*(a + b*Csc[c + d*Sqrt[x]])^2*Sin[c + d*Sqrt[x]]^2)/(4*(b + a*Sin[c + d*Sqrt[x]])^2) - (b^2*E^(I*c)*Cs
c[c]*(a + b*Csc[c + d*Sqrt[x]])^2*((2*d^7*x^(7/2))/E^((2*I)*c) + (7*I)*d^6*(1 - E^((-2*I)*c))*x^3*Log[1 - E^((
-I)*(c + d*Sqrt[x]))] + (7*I)*d^6*(1 - E^((-2*I)*c))*x^3*Log[1 + E^((-I)*(c + d*Sqrt[x]))] - 42*d^5*(1 - E^((-
2*I)*c))*x^(5/2)*PolyLog[2, -E^((-I)*(c + d*Sqrt[x]))] - 42*d^5*(1 - E^((-2*I)*c))*x^(5/2)*PolyLog[2, E^((-I)*
(c + d*Sqrt[x]))] + (210*I)*d^4*(1 - E^((-2*I)*c))*x^2*PolyLog[3, -E^((-I)*(c + d*Sqrt[x]))] + (210*I)*d^4*(1
- E^((-2*I)*c))*x^2*PolyLog[3, E^((-I)*(c + d*Sqrt[x]))] + 840*d^3*(1 - E^((-2*I)*c))*x^(3/2)*PolyLog[4, -E^((
-I)*(c + d*Sqrt[x]))] + 840*d^3*(1 - E^((-2*I)*c))*x^(3/2)*PolyLog[4, E^((-I)*(c + d*Sqrt[x]))] - (2520*I)*d^2
*(1 - E^((-2*I)*c))*x*PolyLog[5, -E^((-I)*(c + d*Sqrt[x]))] - (2520*I)*d^2*(1 - E^((-2*I)*c))*x*PolyLog[5, E^(
(-I)*(c + d*Sqrt[x]))] - 5040*d*(1 - E^((-2*I)*c))*Sqrt[x]*PolyLog[6, -E^((-I)*(c + d*Sqrt[x]))] - 5040*d*(1 -
E^((-2*I)*c))*Sqrt[x]*PolyLog[6, E^((-I)*(c + d*Sqrt[x]))] + (5040*I)*(1 - E^((-2*I)*c))*PolyLog[7, -E^((-I)*
(c + d*Sqrt[x]))] + (5040*I)*(1 - E^((-2*I)*c))*PolyLog[7, E^((-I)*(c + d*Sqrt[x]))])*Sin[c + d*Sqrt[x]]^2)/(d
^8*(b + a*Sin[c + d*Sqrt[x]])^2) + (4*a*b*(a + b*Csc[c + d*Sqrt[x]])^2*(d^7*x^(7/2)*Log[1 - E^(I*(c + d*Sqrt[x
]))] - d^7*x^(7/2)*Log[1 + E^(I*(c + d*Sqrt[x]))] + (7*I)*d^6*x^3*PolyLog[2, -E^(I*(c + d*Sqrt[x]))] - (7*I)*d
^6*x^3*PolyLog[2, E^(I*(c + d*Sqrt[x]))] - 42*d^5*x^(5/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))] + 42*d^5*x^(5/2)*
PolyLog[3, E^(I*(c + d*Sqrt[x]))] - (210*I)*d^4*x^2*PolyLog[4, -E^(I*(c + d*Sqrt[x]))] + (210*I)*d^4*x^2*PolyL
og[4, E^(I*(c + d*Sqrt[x]))] + 840*d^3*x^(3/2)*PolyLog[5, -E^(I*(c + d*Sqrt[x]))] - 840*d^3*x^(3/2)*PolyLog[5,
E^(I*(c + d*Sqrt[x]))] + (2520*I)*d^2*x*PolyLog[6, -E^(I*(c + d*Sqrt[x]))] - (2520*I)*d^2*x*PolyLog[6, E^(I*(
c + d*Sqrt[x]))] - 5040*d*Sqrt[x]*PolyLog[7, -E^(I*(c + d*Sqrt[x]))] + 5040*d*Sqrt[x]*PolyLog[7, E^(I*(c + d*S
qrt[x]))] - (5040*I)*PolyLog[8, -E^(I*(c + d*Sqrt[x]))] + (5040*I)*PolyLog[8, E^(I*(c + d*Sqrt[x]))])*Sin[c +
d*Sqrt[x]]^2)/(d^8*(b + a*Sin[c + d*Sqrt[x]])^2) + (b^2*x^(7/2)*Csc[c/2]*Csc[c/2 + (d*Sqrt[x])/2]*(a + b*Csc[c
+ d*Sqrt[x]])^2*Sin[c + d*Sqrt[x]]^2*Sin[(d*Sqrt[x])/2])/(d*(b + a*Sin[c + d*Sqrt[x]])^2) + (b^2*x^(7/2)*(a +
b*Csc[c + d*Sqrt[x]])^2*Sec[c/2]*Sec[c/2 + (d*Sqrt[x])/2]*Sin[c + d*Sqrt[x]]^2*Sin[(d*Sqrt[x])/2])/(d*(b + a*
Sin[c + d*Sqrt[x]])^2)

Maple [F]

\[\int x^{3} \left (a +b \csc \left (c +d \sqrt {x}\right )\right )^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{3} \,d x } \]

[In]

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

[Out]

integral(b^2*x^3*csc(d*sqrt(x) + c)^2 + 2*a*b*x^3*csc(d*sqrt(x) + c) + a^2*x^3, x)

Sympy [F]

\[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^{3} \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]

[In]

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

[Out]

Integral(x**3*(a + b*csc(c + d*sqrt(x)))**2, x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 6462 vs. \(2 (550) = 1100\).

Time = 0.67 (sec) , antiderivative size = 6462, normalized size of antiderivative = 9.30 \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*((d*sqrt(x) + c)^8*a^2 - 8*(d*sqrt(x) + c)^7*a^2*c + 28*(d*sqrt(x) + c)^6*a^2*c^2 - 56*(d*sqrt(x) + c)^5*a
^2*c^3 + 70*(d*sqrt(x) + c)^4*a^2*c^4 - 56*(d*sqrt(x) + c)^3*a^2*c^5 + 28*(d*sqrt(x) + c)^2*a^2*c^6 - 8*(d*sqr
t(x) + c)*a^2*c^7 + 16*a*b*c^7*log(cot(d*sqrt(x) + c) + csc(d*sqrt(x) + c)) + 8*(4*b^2*c^7 + 2*(2*(d*sqrt(x) +
c)^7*a*b - 7*b^2*c^6 - 7*(2*a*b*c + b^2)*(d*sqrt(x) + c)^6 + 42*(a*b*c^2 + b^2*c)*(d*sqrt(x) + c)^5 - 35*(2*a
*b*c^3 + 3*b^2*c^2)*(d*sqrt(x) + c)^4 + 70*(a*b*c^4 + 2*b^2*c^3)*(d*sqrt(x) + c)^3 - 21*(2*a*b*c^5 + 5*b^2*c^4
)*(d*sqrt(x) + c)^2 + 14*(a*b*c^6 + 3*b^2*c^5)*(d*sqrt(x) + c) - (2*(d*sqrt(x) + c)^7*a*b - 7*b^2*c^6 - 7*(2*a
*b*c + b^2)*(d*sqrt(x) + c)^6 + 42*(a*b*c^2 + b^2*c)*(d*sqrt(x) + c)^5 - 35*(2*a*b*c^3 + 3*b^2*c^2)*(d*sqrt(x)
+ c)^4 + 70*(a*b*c^4 + 2*b^2*c^3)*(d*sqrt(x) + c)^3 - 21*(2*a*b*c^5 + 5*b^2*c^4)*(d*sqrt(x) + c)^2 + 14*(a*b*
c^6 + 3*b^2*c^5)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + (-2*I*(d*sqrt(x) + c)^7*a*b + 7*I*b^2*c^6 + 7*(2*I*
a*b*c + I*b^2)*(d*sqrt(x) + c)^6 + 42*(-I*a*b*c^2 - I*b^2*c)*(d*sqrt(x) + c)^5 + 35*(2*I*a*b*c^3 + 3*I*b^2*c^2
)*(d*sqrt(x) + c)^4 + 70*(-I*a*b*c^4 - 2*I*b^2*c^3)*(d*sqrt(x) + c)^3 + 21*(2*I*a*b*c^5 + 5*I*b^2*c^4)*(d*sqrt
(x) + c)^2 + 14*(-I*a*b*c^6 - 3*I*b^2*c^5)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*arctan2(sin(d*sqrt(x) + c)
, cos(d*sqrt(x) + c) + 1) + 14*(b^2*c^6*cos(2*d*sqrt(x) + 2*c) + I*b^2*c^6*sin(2*d*sqrt(x) + 2*c) - b^2*c^6)*a
rctan2(sin(d*sqrt(x) + c), cos(d*sqrt(x) + c) - 1) + 2*(2*(d*sqrt(x) + c)^7*a*b - 7*(2*a*b*c - b^2)*(d*sqrt(x)
+ c)^6 + 42*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c)^5 - 35*(2*a*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + c)^4 + 70*(a*b*c^4
- 2*b^2*c^3)*(d*sqrt(x) + c)^3 - 21*(2*a*b*c^5 - 5*b^2*c^4)*(d*sqrt(x) + c)^2 + 14*(a*b*c^6 - 3*b^2*c^5)*(d*sq
rt(x) + c) - (2*(d*sqrt(x) + c)^7*a*b - 7*(2*a*b*c - b^2)*(d*sqrt(x) + c)^6 + 42*(a*b*c^2 - b^2*c)*(d*sqrt(x)
+ c)^5 - 35*(2*a*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + c)^4 + 70*(a*b*c^4 - 2*b^2*c^3)*(d*sqrt(x) + c)^3 - 21*(2*a*b
*c^5 - 5*b^2*c^4)*(d*sqrt(x) + c)^2 + 14*(a*b*c^6 - 3*b^2*c^5)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + (-2*I
*(d*sqrt(x) + c)^7*a*b + 7*(2*I*a*b*c - I*b^2)*(d*sqrt(x) + c)^6 + 42*(-I*a*b*c^2 + I*b^2*c)*(d*sqrt(x) + c)^5
+ 35*(2*I*a*b*c^3 - 3*I*b^2*c^2)*(d*sqrt(x) + c)^4 + 70*(-I*a*b*c^4 + 2*I*b^2*c^3)*(d*sqrt(x) + c)^3 + 21*(2*
I*a*b*c^5 - 5*I*b^2*c^4)*(d*sqrt(x) + c)^2 + 14*(-I*a*b*c^6 + 3*I*b^2*c^5)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) +
2*c))*arctan2(sin(d*sqrt(x) + c), -cos(d*sqrt(x) + c) + 1) - 4*((d*sqrt(x) + c)^7*b^2 - 7*(d*sqrt(x) + c)^6*b^
2*c + 21*(d*sqrt(x) + c)^5*b^2*c^2 - 35*(d*sqrt(x) + c)^4*b^2*c^3 + 35*(d*sqrt(x) + c)^3*b^2*c^4 - 21*(d*sqrt(
x) + c)^2*b^2*c^5 + 7*(d*sqrt(x) + c)*b^2*c^6)*cos(2*d*sqrt(x) + 2*c) - 28*((d*sqrt(x) + c)^6*a*b + a*b*c^6 +
3*b^2*c^5 - 3*(2*a*b*c + b^2)*(d*sqrt(x) + c)^5 + 15*(a*b*c^2 + b^2*c)*(d*sqrt(x) + c)^4 - 10*(2*a*b*c^3 + 3*b
^2*c^2)*(d*sqrt(x) + c)^3 + 15*(a*b*c^4 + 2*b^2*c^3)*(d*sqrt(x) + c)^2 - 3*(2*a*b*c^5 + 5*b^2*c^4)*(d*sqrt(x)
+ c) - ((d*sqrt(x) + c)^6*a*b + a*b*c^6 + 3*b^2*c^5 - 3*(2*a*b*c + b^2)*(d*sqrt(x) + c)^5 + 15*(a*b*c^2 + b^2*
c)*(d*sqrt(x) + c)^4 - 10*(2*a*b*c^3 + 3*b^2*c^2)*(d*sqrt(x) + c)^3 + 15*(a*b*c^4 + 2*b^2*c^3)*(d*sqrt(x) + c)
^2 - 3*(2*a*b*c^5 + 5*b^2*c^4)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) - (I*(d*sqrt(x) + c)^6*a*b + I*a*b*c^6
+ 3*I*b^2*c^5 + 3*(-2*I*a*b*c - I*b^2)*(d*sqrt(x) + c)^5 + 15*(I*a*b*c^2 + I*b^2*c)*(d*sqrt(x) + c)^4 + 10*(-2
*I*a*b*c^3 - 3*I*b^2*c^2)*(d*sqrt(x) + c)^3 + 15*(I*a*b*c^4 + 2*I*b^2*c^3)*(d*sqrt(x) + c)^2 + 3*(-2*I*a*b*c^5
- 5*I*b^2*c^4)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*dilog(-e^(I*d*sqrt(x) + I*c)) + 28*((d*sqrt(x) + c)^6
*a*b + a*b*c^6 - 3*b^2*c^5 - 3*(2*a*b*c - b^2)*(d*sqrt(x) + c)^5 + 15*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c)^4 - 10
*(2*a*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + c)^3 + 15*(a*b*c^4 - 2*b^2*c^3)*(d*sqrt(x) + c)^2 - 3*(2*a*b*c^5 - 5*b^2
*c^4)*(d*sqrt(x) + c) - ((d*sqrt(x) + c)^6*a*b + a*b*c^6 - 3*b^2*c^5 - 3*(2*a*b*c - b^2)*(d*sqrt(x) + c)^5 + 1
5*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c)^4 - 10*(2*a*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + c)^3 + 15*(a*b*c^4 - 2*b^2*c^3
)*(d*sqrt(x) + c)^2 - 3*(2*a*b*c^5 - 5*b^2*c^4)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + (-I*(d*sqrt(x) + c)^
6*a*b - I*a*b*c^6 + 3*I*b^2*c^5 + 3*(2*I*a*b*c - I*b^2)*(d*sqrt(x) + c)^5 + 15*(-I*a*b*c^2 + I*b^2*c)*(d*sqrt(
x) + c)^4 + 10*(2*I*a*b*c^3 - 3*I*b^2*c^2)*(d*sqrt(x) + c)^3 + 15*(-I*a*b*c^4 + 2*I*b^2*c^3)*(d*sqrt(x) + c)^2
+ 3*(2*I*a*b*c^5 - 5*I*b^2*c^4)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*dilog(e^(I*d*sqrt(x) + I*c)) - (2*I*
(d*sqrt(x) + c)^7*a*b - 7*I*b^2*c^6 - 7*(2*I*a*b*c + I*b^2)*(d*sqrt(x) + c)^6 - 42*(-I*a*b*c^2 - I*b^2*c)*(d*s
qrt(x) + c)^5 - 35*(2*I*a*b*c^3 + 3*I*b^2*c^2)*(d*sqrt(x) + c)^4 - 70*(-I*a*b*c^4 - 2*I*b^2*c^3)*(d*sqrt(x) +
c)^3 - 21*(2*I*a*b*c^5 + 5*I*b^2*c^4)*(d*sqrt(x) + c)^2 - 14*(-I*a*b*c^6 - 3*I*b^2*c^5)*(d*sqrt(x) + c) + (-2*
I*(d*sqrt(x) + c)^7*a*b + 7*I*b^2*c^6 - 7*(-2*I*a*b*c - I*b^2)*(d*sqrt(x) + c)^6 - 42*(I*a*b*c^2 + I*b^2*c)*(d
*sqrt(x) + c)^5 - 35*(-2*I*a*b*c^3 - 3*I*b^2*c^2)*(d*sqrt(x) + c)^4 - 70*(I*a*b*c^4 + 2*I*b^2*c^3)*(d*sqrt(x)
+ c)^3 - 21*(-2*I*a*b*c^5 - 5*I*b^2*c^4)*(d*sqrt(x) + c)^2 - 14*(I*a*b*c^6 + 3*I*b^2*c^5)*(d*sqrt(x) + c))*cos
(2*d*sqrt(x) + 2*c) + (2*(d*sqrt(x) + c)^7*a*b - 7*b^2*c^6 - 7*(2*a*b*c + b^2)*(d*sqrt(x) + c)^6 + 42*(a*b*c^2
+ b^2*c)*(d*sqrt(x) + c)^5 - 35*(2*a*b*c^3 + 3*b^2*c^2)*(d*sqrt(x) + c)^4 + 70*(a*b*c^4 + 2*b^2*c^3)*(d*sqrt(
x) + c)^3 - 21*(2*a*b*c^5 + 5*b^2*c^4)*(d*sqrt(x) + c)^2 + 14*(a*b*c^6 + 3*b^2*c^5)*(d*sqrt(x) + c))*sin(2*d*s
qrt(x) + 2*c))*log(cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 + 2*cos(d*sqrt(x) + c) + 1) - (-2*I*(d*sqrt(x)
+ c)^7*a*b - 7*I*b^2*c^6 - 7*(-2*I*a*b*c + I*b^2)*(d*sqrt(x) + c)^6 - 42*(I*a*b*c^2 - I*b^2*c)*(d*sqrt(x) + c)
^5 - 35*(-2*I*a*b*c^3 + 3*I*b^2*c^2)*(d*sqrt(x) + c)^4 - 70*(I*a*b*c^4 - 2*I*b^2*c^3)*(d*sqrt(x) + c)^3 - 21*(
-2*I*a*b*c^5 + 5*I*b^2*c^4)*(d*sqrt(x) + c)^2 - 14*(I*a*b*c^6 - 3*I*b^2*c^5)*(d*sqrt(x) + c) + (2*I*(d*sqrt(x)
+ c)^7*a*b + 7*I*b^2*c^6 - 7*(2*I*a*b*c - I*b^2)*(d*sqrt(x) + c)^6 - 42*(-I*a*b*c^2 + I*b^2*c)*(d*sqrt(x) + c
)^5 - 35*(2*I*a*b*c^3 - 3*I*b^2*c^2)*(d*sqrt(x) + c)^4 - 70*(-I*a*b*c^4 + 2*I*b^2*c^3)*(d*sqrt(x) + c)^3 - 21*
(2*I*a*b*c^5 - 5*I*b^2*c^4)*(d*sqrt(x) + c)^2 - 14*(-I*a*b*c^6 + 3*I*b^2*c^5)*(d*sqrt(x) + c))*cos(2*d*sqrt(x)
+ 2*c) - (2*(d*sqrt(x) + c)^7*a*b + 7*b^2*c^6 - 7*(2*a*b*c - b^2)*(d*sqrt(x) + c)^6 + 42*(a*b*c^2 - b^2*c)*(d
*sqrt(x) + c)^5 - 35*(2*a*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + c)^4 + 70*(a*b*c^4 - 2*b^2*c^3)*(d*sqrt(x) + c)^3 -
21*(2*a*b*c^5 - 5*b^2*c^4)*(d*sqrt(x) + c)^2 + 14*(a*b*c^6 - 3*b^2*c^5)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c
))*log(cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 - 2*cos(d*sqrt(x) + c) + 1) - 20160*(a*b*cos(2*d*sqrt(x) +
2*c) + I*a*b*sin(2*d*sqrt(x) + 2*c) - a*b)*polylog(8, -e^(I*d*sqrt(x) + I*c)) + 20160*(a*b*cos(2*d*sqrt(x) + 2
*c) + I*a*b*sin(2*d*sqrt(x) + 2*c) - a*b)*polylog(8, e^(I*d*sqrt(x) + I*c)) + 10080*(-2*I*(d*sqrt(x) + c)*a*b
+ 2*I*a*b*c + I*b^2 + (2*I*(d*sqrt(x) + c)*a*b - 2*I*a*b*c - I*b^2)*cos(2*d*sqrt(x) + 2*c) - (2*(d*sqrt(x) + c
)*a*b - 2*a*b*c - b^2)*sin(2*d*sqrt(x) + 2*c))*polylog(7, -e^(I*d*sqrt(x) + I*c)) + 10080*(2*I*(d*sqrt(x) + c)
*a*b - 2*I*a*b*c + I*b^2 + (-2*I*(d*sqrt(x) + c)*a*b + 2*I*a*b*c - I*b^2)*cos(2*d*sqrt(x) + 2*c) + (2*(d*sqrt(
x) + c)*a*b - 2*a*b*c + b^2)*sin(2*d*sqrt(x) + 2*c))*polylog(7, e^(I*d*sqrt(x) + I*c)) - 10080*((d*sqrt(x) + c
)^2*a*b + a*b*c^2 + b^2*c - (2*a*b*c + b^2)*(d*sqrt(x) + c) - ((d*sqrt(x) + c)^2*a*b + a*b*c^2 + b^2*c - (2*a*
b*c + b^2)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) - (I*(d*sqrt(x) + c)^2*a*b + I*a*b*c^2 + I*b^2*c + (-2*I*a*
b*c - I*b^2)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*polylog(6, -e^(I*d*sqrt(x) + I*c)) + 10080*((d*sqrt(x) +
c)^2*a*b + a*b*c^2 - b^2*c - (2*a*b*c - b^2)*(d*sqrt(x) + c) - ((d*sqrt(x) + c)^2*a*b + a*b*c^2 - b^2*c - (2*
a*b*c - b^2)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + (-I*(d*sqrt(x) + c)^2*a*b - I*a*b*c^2 + I*b^2*c + (2*I*
a*b*c - I*b^2)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*polylog(6, e^(I*d*sqrt(x) + I*c)) + 1680*(2*I*(d*sqrt(
x) + c)^3*a*b - 2*I*a*b*c^3 - 3*I*b^2*c^2 + 3*(-2*I*a*b*c - I*b^2)*(d*sqrt(x) + c)^2 + 6*(I*a*b*c^2 + I*b^2*c)
*(d*sqrt(x) + c) + (-2*I*(d*sqrt(x) + c)^3*a*b + 2*I*a*b*c^3 + 3*I*b^2*c^2 + 3*(2*I*a*b*c + I*b^2)*(d*sqrt(x)
+ c)^2 + 6*(-I*a*b*c^2 - I*b^2*c)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + (2*(d*sqrt(x) + c)^3*a*b - 2*a*b*c
^3 - 3*b^2*c^2 - 3*(2*a*b*c + b^2)*(d*sqrt(x) + c)^2 + 6*(a*b*c^2 + b^2*c)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) +
2*c))*polylog(5, -e^(I*d*sqrt(x) + I*c)) + 1680*(-2*I*(d*sqrt(x) + c)^3*a*b + 2*I*a*b*c^3 - 3*I*b^2*c^2 + 3*(2
*I*a*b*c - I*b^2)*(d*sqrt(x) + c)^2 + 6*(-I*a*b*c^2 + I*b^2*c)*(d*sqrt(x) + c) + (2*I*(d*sqrt(x) + c)^3*a*b -
2*I*a*b*c^3 + 3*I*b^2*c^2 + 3*(-2*I*a*b*c + I*b^2)*(d*sqrt(x) + c)^2 + 6*(I*a*b*c^2 - I*b^2*c)*(d*sqrt(x) + c)
)*cos(2*d*sqrt(x) + 2*c) - (2*(d*sqrt(x) + c)^3*a*b - 2*a*b*c^3 + 3*b^2*c^2 - 3*(2*a*b*c - b^2)*(d*sqrt(x) + c
)^2 + 6*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*polylog(5, e^(I*d*sqrt(x) + I*c)) + 840*((d
*sqrt(x) + c)^4*a*b + a*b*c^4 + 2*b^2*c^3 - 2*(2*a*b*c + b^2)*(d*sqrt(x) + c)^3 + 6*(a*b*c^2 + b^2*c)*(d*sqrt(
x) + c)^2 - 2*(2*a*b*c^3 + 3*b^2*c^2)*(d*sqrt(x) + c) - ((d*sqrt(x) + c)^4*a*b + a*b*c^4 + 2*b^2*c^3 - 2*(2*a*
b*c + b^2)*(d*sqrt(x) + c)^3 + 6*(a*b*c^2 + b^2*c)*(d*sqrt(x) + c)^2 - 2*(2*a*b*c^3 + 3*b^2*c^2)*(d*sqrt(x) +
c))*cos(2*d*sqrt(x) + 2*c) + (-I*(d*sqrt(x) + c)^4*a*b - I*a*b*c^4 - 2*I*b^2*c^3 + 2*(2*I*a*b*c + I*b^2)*(d*sq
rt(x) + c)^3 + 6*(-I*a*b*c^2 - I*b^2*c)*(d*sqrt(x) + c)^2 + 2*(2*I*a*b*c^3 + 3*I*b^2*c^2)*(d*sqrt(x) + c))*sin
(2*d*sqrt(x) + 2*c))*polylog(4, -e^(I*d*sqrt(x) + I*c)) - 840*((d*sqrt(x) + c)^4*a*b + a*b*c^4 - 2*b^2*c^3 - 2
*(2*a*b*c - b^2)*(d*sqrt(x) + c)^3 + 6*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c)^2 - 2*(2*a*b*c^3 - 3*b^2*c^2)*(d*sqrt
(x) + c) - ((d*sqrt(x) + c)^4*a*b + a*b*c^4 - 2*b^2*c^3 - 2*(2*a*b*c - b^2)*(d*sqrt(x) + c)^3 + 6*(a*b*c^2 - b
^2*c)*(d*sqrt(x) + c)^2 - 2*(2*a*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) - (I*(d*sqrt(x) +
c)^4*a*b + I*a*b*c^4 - 2*I*b^2*c^3 + 2*(-2*I*a*b*c + I*b^2)*(d*sqrt(x) + c)^3 + 6*(I*a*b*c^2 - I*b^2*c)*(d*sqr
t(x) + c)^2 + 2*(-2*I*a*b*c^3 + 3*I*b^2*c^2)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*polylog(4, e^(I*d*sqrt(x
) + I*c)) + 84*(-2*I*(d*sqrt(x) + c)^5*a*b + 2*I*a*b*c^5 + 5*I*b^2*c^4 + 5*(2*I*a*b*c + I*b^2)*(d*sqrt(x) + c)
^4 + 20*(-I*a*b*c^2 - I*b^2*c)*(d*sqrt(x) + c)^3 + 10*(2*I*a*b*c^3 + 3*I*b^2*c^2)*(d*sqrt(x) + c)^2 + 10*(-I*a
*b*c^4 - 2*I*b^2*c^3)*(d*sqrt(x) + c) + (2*I*(d*sqrt(x) + c)^5*a*b - 2*I*a*b*c^5 - 5*I*b^2*c^4 + 5*(-2*I*a*b*c
- I*b^2)*(d*sqrt(x) + c)^4 + 20*(I*a*b*c^2 + I*b^2*c)*(d*sqrt(x) + c)^3 + 10*(-2*I*a*b*c^3 - 3*I*b^2*c^2)*(d*
sqrt(x) + c)^2 + 10*(I*a*b*c^4 + 2*I*b^2*c^3)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) - (2*(d*sqrt(x) + c)^5*a
*b - 2*a*b*c^5 - 5*b^2*c^4 - 5*(2*a*b*c + b^2)*(d*sqrt(x) + c)^4 + 20*(a*b*c^2 + b^2*c)*(d*sqrt(x) + c)^3 - 10
*(2*a*b*c^3 + 3*b^2*c^2)*(d*sqrt(x) + c)^2 + 10*(a*b*c^4 + 2*b^2*c^3)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))
*polylog(3, -e^(I*d*sqrt(x) + I*c)) + 84*(2*I*(d*sqrt(x) + c)^5*a*b - 2*I*a*b*c^5 + 5*I*b^2*c^4 + 5*(-2*I*a*b*
c + I*b^2)*(d*sqrt(x) + c)^4 + 20*(I*a*b*c^2 - I*b^2*c)*(d*sqrt(x) + c)^3 + 10*(-2*I*a*b*c^3 + 3*I*b^2*c^2)*(d
*sqrt(x) + c)^2 + 10*(I*a*b*c^4 - 2*I*b^2*c^3)*(d*sqrt(x) + c) + (-2*I*(d*sqrt(x) + c)^5*a*b + 2*I*a*b*c^5 - 5
*I*b^2*c^4 + 5*(2*I*a*b*c - I*b^2)*(d*sqrt(x) + c)^4 + 20*(-I*a*b*c^2 + I*b^2*c)*(d*sqrt(x) + c)^3 + 10*(2*I*a
*b*c^3 - 3*I*b^2*c^2)*(d*sqrt(x) + c)^2 + 10*(-I*a*b*c^4 + 2*I*b^2*c^3)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c
) + (2*(d*sqrt(x) + c)^5*a*b - 2*a*b*c^5 + 5*b^2*c^4 - 5*(2*a*b*c - b^2)*(d*sqrt(x) + c)^4 + 20*(a*b*c^2 - b^2
*c)*(d*sqrt(x) + c)^3 - 10*(2*a*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + c)^2 + 10*(a*b*c^4 - 2*b^2*c^3)*(d*sqrt(x) + c
))*sin(2*d*sqrt(x) + 2*c))*polylog(3, e^(I*d*sqrt(x) + I*c)) + 4*(-I*(d*sqrt(x) + c)^7*b^2 + 7*I*(d*sqrt(x) +
c)^6*b^2*c - 21*I*(d*sqrt(x) + c)^5*b^2*c^2 + 35*I*(d*sqrt(x) + c)^4*b^2*c^3 - 35*I*(d*sqrt(x) + c)^3*b^2*c^4
+ 21*I*(d*sqrt(x) + c)^2*b^2*c^5 - 7*I*(d*sqrt(x) + c)*b^2*c^6)*sin(2*d*sqrt(x) + 2*c))/(-2*I*cos(2*d*sqrt(x)
+ 2*c) + 2*sin(2*d*sqrt(x) + 2*c) + 2*I))/d^8

Giac [F]

\[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{3} \,d x } \]

[In]

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

[Out]

integrate((b*csc(d*sqrt(x) + c) + a)^2*x^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^3\,{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]

[In]

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

[Out]

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