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

Optimal. Leaf size=695 \[ \text{result too large to display} \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.820226, antiderivative size = 695, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4205, 4190, 4183, 2531, 6609, 2282, 6589, 4184, 3717, 2190} \[ \frac{28 i a b x^3 \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{168 a b x^{5/2} \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{840 i a b x^2 \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{3360 a b x^{3/2} \text{PolyLog}\left (5,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{PolyLog}\left (5,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{10080 i a b x \text{PolyLog}\left (6,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 i a b x \text{PolyLog}\left (6,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{20160 a b \sqrt{x} \text{PolyLog}\left (7,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{PolyLog}\left (7,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}-\frac{20160 i a b \text{PolyLog}\left (8,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}+\frac{20160 i a b \text{PolyLog}\left (8,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}-\frac{42 i b^2 x^{5/2} \text{PolyLog}\left (2,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{105 b^2 x^2 \text{PolyLog}\left (3,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{210 i b^2 x^{3/2} \text{PolyLog}\left (4,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{315 b^2 x \text{PolyLog}\left (5,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{315 i b^2 \sqrt{x} \text{PolyLog}\left (6,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{315 b^2 \text{PolyLog}\left (7,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{2 d^8}+\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\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} \]

Antiderivative was successfully verified.

[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 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 4190

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 4183

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 2531

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

Rule 6609

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

Rule 2282

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

Rule 6589

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

Rule 4184

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 3717

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 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]

Rubi steps

\begin{align*} \int x^3 \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2 \, dx &=2 \operatorname{Subst}\left (\int x^7 (a+b \csc (c+d x))^2 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{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) \operatorname{Subst}\left (\int x^7 \csc (c+d x) \, dx,x,\sqrt{x}\right )+\left (2 b^2\right ) \operatorname{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} \tanh ^{-1}\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) \operatorname{Subst}\left (\int x^6 \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{(28 a b) \operatorname{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 ) \operatorname{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} \tanh ^{-1}\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 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{(168 i a b) \operatorname{Subst}\left (\int x^5 \text{Li}_2\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}+\frac{(168 i a b) \operatorname{Subst}\left (\int x^5 \text{Li}_2\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}-\frac{\left (28 i b^2\right ) \operatorname{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} \tanh ^{-1}\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 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{(840 a b) \operatorname{Subst}\left (\int x^4 \text{Li}_3\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}-\frac{(840 a b) \operatorname{Subst}\left (\int x^4 \text{Li}_3\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}-\frac{\left (84 b^2\right ) \operatorname{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} \tanh ^{-1}\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 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{42 i b^2 x^{5/2} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{840 i a b x^2 \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{(3360 i a b) \operatorname{Subst}\left (\int x^3 \text{Li}_4\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^4}-\frac{(3360 i a b) \operatorname{Subst}\left (\int x^3 \text{Li}_4\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^4}+\frac{\left (210 i b^2\right ) \operatorname{Subst}\left (\int x^4 \text{Li}_2\left (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} \tanh ^{-1}\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 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{42 i b^2 x^{5/2} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{105 b^2 x^2 \text{Li}_3\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{840 i a b x^2 \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{(10080 a b) \operatorname{Subst}\left (\int x^2 \text{Li}_5\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^5}+\frac{(10080 a b) \operatorname{Subst}\left (\int x^2 \text{Li}_5\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^5}-\frac{\left (420 b^2\right ) \operatorname{Subst}\left (\int x^3 \text{Li}_3\left (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} \tanh ^{-1}\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 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{42 i b^2 x^{5/2} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{105 b^2 x^2 \text{Li}_3\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{840 i a b x^2 \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{210 i b^2 x^{3/2} \text{Li}_4\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{10080 i a b x \text{Li}_6\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 i a b x \text{Li}_6\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{(20160 i a b) \operatorname{Subst}\left (\int x \text{Li}_6\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^6}+\frac{(20160 i a b) \operatorname{Subst}\left (\int x \text{Li}_6\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^6}-\frac{\left (630 i b^2\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_4\left (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} \tanh ^{-1}\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 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{42 i b^2 x^{5/2} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{105 b^2 x^2 \text{Li}_3\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{840 i a b x^2 \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{210 i b^2 x^{3/2} \text{Li}_4\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{315 b^2 x \text{Li}_5\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^6}+\frac{10080 i a b x \text{Li}_6\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 i a b x \text{Li}_6\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{20160 a b \sqrt{x} \text{Li}_7\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{(20160 a b) \operatorname{Subst}\left (\int \text{Li}_7\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^7}-\frac{(20160 a b) \operatorname{Subst}\left (\int \text{Li}_7\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^7}+\frac{\left (630 b^2\right ) \operatorname{Subst}\left (\int x \text{Li}_5\left (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} \tanh ^{-1}\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 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{42 i b^2 x^{5/2} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{105 b^2 x^2 \text{Li}_3\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{840 i a b x^2 \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{210 i b^2 x^{3/2} \text{Li}_4\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{315 b^2 x \text{Li}_5\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^6}+\frac{10080 i a b x \text{Li}_6\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 i a b x \text{Li}_6\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{315 i b^2 \sqrt{x} \text{Li}_6\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^7}-\frac{20160 a b \sqrt{x} \text{Li}_7\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}-\frac{(20160 i a b) \operatorname{Subst}\left (\int \frac{\text{Li}_7(-x)}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}+\frac{(20160 i a b) \operatorname{Subst}\left (\int \frac{\text{Li}_7(x)}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}+\frac{\left (315 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_6\left (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} \tanh ^{-1}\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 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{42 i b^2 x^{5/2} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{105 b^2 x^2 \text{Li}_3\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{840 i a b x^2 \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{210 i b^2 x^{3/2} \text{Li}_4\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{315 b^2 x \text{Li}_5\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^6}+\frac{10080 i a b x \text{Li}_6\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 i a b x \text{Li}_6\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{315 i b^2 \sqrt{x} \text{Li}_6\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^7}-\frac{20160 a b \sqrt{x} \text{Li}_7\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}-\frac{20160 i a b \text{Li}_8\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}+\frac{20160 i a b \text{Li}_8\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}+\frac{\left (315 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_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} \tanh ^{-1}\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 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{42 i b^2 x^{5/2} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{105 b^2 x^2 \text{Li}_3\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{840 i a b x^2 \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{210 i b^2 x^{3/2} \text{Li}_4\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{315 b^2 x \text{Li}_5\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^6}+\frac{10080 i a b x \text{Li}_6\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 i a b x \text{Li}_6\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{315 i b^2 \sqrt{x} \text{Li}_6\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^7}-\frac{20160 a b \sqrt{x} \text{Li}_7\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{315 b^2 \text{Li}_7\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{2 d^8}-\frac{20160 i a b \text{Li}_8\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}+\frac{20160 i a b \text{Li}_8\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}\\ \end{align*}

Mathematica [A]  time = 20.8497, size = 954, normalized size = 1.37 \[ \frac{a^2 \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2 \sin ^2\left (c+d \sqrt{x}\right ) x^4}{4 \left (b+a \sin \left (c+d \sqrt{x}\right )\right )^2}+\frac{b^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 ) x^{7/2}}{d \left (b+a \sin \left (c+d \sqrt{x}\right )\right )^2}+\frac{b^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 ) x^{7/2}}{d \left (b+a \sin \left (c+d \sqrt{x}\right )\right )^2}-\frac{i b \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2 \left (\frac{8 b e^{2 i c} x^{7/2} d^7}{-1+e^{2 i c}}+8 i a x^{7/2} \log \left (1-e^{i \left (c+d \sqrt{x}\right )}\right ) d^7-8 i a x^{7/2} \log \left (1+e^{i \left (c+d \sqrt{x}\right )}\right ) d^7+28 i b x^3 \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right ) d^6-56 a x^3 \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right ) d^6+56 a x^3 \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right ) d^6+84 b x^{5/2} \text{PolyLog}\left (2,e^{2 i \left (c+d \sqrt{x}\right )}\right ) d^5-336 i a x^{5/2} \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right ) d^5+336 i a x^{5/2} \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right ) d^5+210 i b x^2 \text{PolyLog}\left (3,e^{2 i \left (c+d \sqrt{x}\right )}\right ) d^4+1680 a x^2 \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right ) d^4-1680 a x^2 \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right ) d^4-420 b x^{3/2} \text{PolyLog}\left (4,e^{2 i \left (c+d \sqrt{x}\right )}\right ) d^3+6720 i a x^{3/2} \text{PolyLog}\left (5,-e^{i \left (c+d \sqrt{x}\right )}\right ) d^3-6720 i a x^{3/2} \text{PolyLog}\left (5,e^{i \left (c+d \sqrt{x}\right )}\right ) d^3-630 i b x \text{PolyLog}\left (5,e^{2 i \left (c+d \sqrt{x}\right )}\right ) d^2-20160 a x \text{PolyLog}\left (6,-e^{i \left (c+d \sqrt{x}\right )}\right ) d^2+20160 a x \text{PolyLog}\left (6,e^{i \left (c+d \sqrt{x}\right )}\right ) d^2+630 b \sqrt{x} \text{PolyLog}\left (6,e^{2 i \left (c+d \sqrt{x}\right )}\right ) d-40320 i a \sqrt{x} \text{PolyLog}\left (7,-e^{i \left (c+d \sqrt{x}\right )}\right ) d+40320 i a \sqrt{x} \text{PolyLog}\left (7,e^{i \left (c+d \sqrt{x}\right )}\right ) d+315 i b \text{PolyLog}\left (7,e^{2 i \left (c+d \sqrt{x}\right )}\right )+40320 a \text{PolyLog}\left (8,-e^{i \left (c+d \sqrt{x}\right )}\right )-40320 a \text{PolyLog}\left (8,e^{i \left (c+d \sqrt{x}\right )}\right )\right ) \sin ^2\left (c+d \sqrt{x}\right )}{2 d^8 \left (b+a \sin \left (c+d \sqrt{x}\right )\right )^2} \]

Antiderivative was successfully verified.

[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) - ((I/2)*b*(a + b
*Csc[c + d*Sqrt[x]])^2*((8*b*d^7*E^((2*I)*c)*x^(7/2))/(-1 + E^((2*I)*c)) + (8*I)*a*d^7*x^(7/2)*Log[1 - E^(I*(c
+ d*Sqrt[x]))] - (8*I)*a*d^7*x^(7/2)*Log[1 + E^(I*(c + d*Sqrt[x]))] + (28*I)*b*d^6*x^3*Log[1 - E^((2*I)*(c +
d*Sqrt[x]))] - 56*a*d^6*x^3*PolyLog[2, -E^(I*(c + d*Sqrt[x]))] + 56*a*d^6*x^3*PolyLog[2, E^(I*(c + d*Sqrt[x]))
] + 84*b*d^5*x^(5/2)*PolyLog[2, E^((2*I)*(c + d*Sqrt[x]))] - (336*I)*a*d^5*x^(5/2)*PolyLog[3, -E^(I*(c + d*Sqr
t[x]))] + (336*I)*a*d^5*x^(5/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))] + (210*I)*b*d^4*x^2*PolyLog[3, E^((2*I)*(c +
d*Sqrt[x]))] + 1680*a*d^4*x^2*PolyLog[4, -E^(I*(c + d*Sqrt[x]))] - 1680*a*d^4*x^2*PolyLog[4, E^(I*(c + d*Sqrt
[x]))] - 420*b*d^3*x^(3/2)*PolyLog[4, E^((2*I)*(c + d*Sqrt[x]))] + (6720*I)*a*d^3*x^(3/2)*PolyLog[5, -E^(I*(c
+ d*Sqrt[x]))] - (6720*I)*a*d^3*x^(3/2)*PolyLog[5, E^(I*(c + d*Sqrt[x]))] - (630*I)*b*d^2*x*PolyLog[5, E^((2*I
)*(c + d*Sqrt[x]))] - 20160*a*d^2*x*PolyLog[6, -E^(I*(c + d*Sqrt[x]))] + 20160*a*d^2*x*PolyLog[6, E^(I*(c + d*
Sqrt[x]))] + 630*b*d*Sqrt[x]*PolyLog[6, E^((2*I)*(c + d*Sqrt[x]))] - (40320*I)*a*d*Sqrt[x]*PolyLog[7, -E^(I*(c
+ d*Sqrt[x]))] + (40320*I)*a*d*Sqrt[x]*PolyLog[7, E^(I*(c + d*Sqrt[x]))] + (315*I)*b*PolyLog[7, E^((2*I)*(c +
d*Sqrt[x]))] + 40320*a*PolyLog[8, -E^(I*(c + d*Sqrt[x]))] - 40320*a*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)

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Maple [F]  time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \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^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)

________________________________________________________________________________________

Maxima [B]  time = 3.53755, size = 8640, normalized size = 12.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 + (4*(d*sqrt(x) + c
)^7*a*b - 14*b^2*c^6 - 14*(2*a*b*c + b^2)*(d*sqrt(x) + c)^6 + 84*(a*b*c^2 + b^2*c)*(d*sqrt(x) + c)^5 - 70*(2*a
*b*c^3 + 3*b^2*c^2)*(d*sqrt(x) + c)^4 + 140*(a*b*c^4 + 2*b^2*c^3)*(d*sqrt(x) + c)^3 - 42*(2*a*b*c^5 + 5*b^2*c^
4)*(d*sqrt(x) + c)^2 + 28*(a*b*c^6 + 3*b^2*c^5)*(d*sqrt(x) + c) - 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))*cos(2*d*sqrt(x) + 2*c) - (4*I*(d*sqrt(x) + c)^7*a*b - 14*I*b^2*c^6 + (-28
*I*a*b*c - 14*I*b^2)*(d*sqrt(x) + c)^6 + (84*I*a*b*c^2 + 84*I*b^2*c)*(d*sqrt(x) + c)^5 + (-140*I*a*b*c^3 - 210
*I*b^2*c^2)*(d*sqrt(x) + c)^4 + (140*I*a*b*c^4 + 280*I*b^2*c^3)*(d*sqrt(x) + c)^3 + (-84*I*a*b*c^5 - 210*I*b^2
*c^4)*(d*sqrt(x) + c)^2 + (28*I*a*b*c^6 + 84*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)*arctan2(sin(d*sqrt(x) + c), cos(d*sqrt(x) + c) - 1) + (4*(d*sqrt(x) + c)^7*a*b - 14*(2*a*b*c - b^2
)*(d*sqrt(x) + c)^6 + 84*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c)^5 - 70*(2*a*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + c)^4 +
140*(a*b*c^4 - 2*b^2*c^3)*(d*sqrt(x) + c)^3 - 42*(2*a*b*c^5 - 5*b^2*c^4)*(d*sqrt(x) + c)^2 + 28*(a*b*c^6 - 3*b
^2*c^5)*(d*sqrt(x) + c) - 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*sqrt(x) + c))*cos(2*d*sqrt(x)
+ 2*c) - (4*I*(d*sqrt(x) + c)^7*a*b + (-28*I*a*b*c + 14*I*b^2)*(d*sqrt(x) + c)^6 + (84*I*a*b*c^2 - 84*I*b^2*c
)*(d*sqrt(x) + c)^5 + (-140*I*a*b*c^3 + 210*I*b^2*c^2)*(d*sqrt(x) + c)^4 + (140*I*a*b*c^4 - 280*I*b^2*c^3)*(d*
sqrt(x) + c)^3 + (-84*I*a*b*c^5 + 210*I*b^2*c^4)*(d*sqrt(x) + c)^2 + (28*I*a*b*c^6 - 84*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 + 28*a*b*c^6 + 84*b^2*c^5 - 84*(2*a*b*c + b^2)*(d*sqrt(x) + c)^5 + 420*(a*b*c^2 + b^2*c)*(d*sqrt(x
) + c)^4 - 280*(2*a*b*c^3 + 3*b^2*c^2)*(d*sqrt(x) + c)^3 + 420*(a*b*c^4 + 2*b^2*c^3)*(d*sqrt(x) + c)^2 - 84*(2
*a*b*c^5 + 5*b^2*c^4)*(d*sqrt(x) + 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))*cos(2*d*sqrt(x) + 2*c) + (
-28*I*(d*sqrt(x) + c)^6*a*b - 28*I*a*b*c^6 - 84*I*b^2*c^5 + (168*I*a*b*c + 84*I*b^2)*(d*sqrt(x) + c)^5 + (-420
*I*a*b*c^2 - 420*I*b^2*c)*(d*sqrt(x) + c)^4 + (560*I*a*b*c^3 + 840*I*b^2*c^2)*(d*sqrt(x) + c)^3 + (-420*I*a*b*
c^4 - 840*I*b^2*c^3)*(d*sqrt(x) + c)^2 + (168*I*a*b*c^5 + 420*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 + 28*a*b*c^6 - 84*b^2*c^5 - 84*(2*a*b*c - b^2)*(
d*sqrt(x) + c)^5 + 420*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c)^4 - 280*(2*a*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + c)^3 + 4
20*(a*b*c^4 - 2*b^2*c^3)*(d*sqrt(x) + c)^2 - 84*(2*a*b*c^5 - 5*b^2*c^4)*(d*sqrt(x) + 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 - 1
0*(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) - (28*I*(d*sqrt(x) + c)^6*a*b + 28*I*a*b*c^6 - 84*I*b^2*c^5 + (
-168*I*a*b*c + 84*I*b^2)*(d*sqrt(x) + c)^5 + (420*I*a*b*c^2 - 420*I*b^2*c)*(d*sqrt(x) + c)^4 + (-560*I*a*b*c^3
+ 840*I*b^2*c^2)*(d*sqrt(x) + c)^3 + (420*I*a*b*c^4 - 840*I*b^2*c^3)*(d*sqrt(x) + c)^2 + (-168*I*a*b*c^5 + 42
0*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 + (-14*I*a*b*c - 7*I*b^2)*(d*sqrt(x) + c)^6 + (42*I*a*b*c^2 + 42*I*b^2*c)*(d*sqrt(x) + c)^5 +
(-70*I*a*b*c^3 - 105*I*b^2*c^2)*(d*sqrt(x) + c)^4 + (70*I*a*b*c^4 + 140*I*b^2*c^3)*(d*sqrt(x) + c)^3 + (-42*I*
a*b*c^5 - 105*I*b^2*c^4)*(d*sqrt(x) + c)^2 + (14*I*a*b*c^6 + 42*I*b^2*c^5)*(d*sqrt(x) + c) + (-2*I*(d*sqrt(x)
+ c)^7*a*b + 7*I*b^2*c^6 + (14*I*a*b*c + 7*I*b^2)*(d*sqrt(x) + c)^6 + (-42*I*a*b*c^2 - 42*I*b^2*c)*(d*sqrt(x)
+ c)^5 + (70*I*a*b*c^3 + 105*I*b^2*c^2)*(d*sqrt(x) + c)^4 + (-70*I*a*b*c^4 - 140*I*b^2*c^3)*(d*sqrt(x) + c)^3
+ (42*I*a*b*c^5 + 105*I*b^2*c^4)*(d*sqrt(x) + c)^2 + (-14*I*a*b*c^6 - 42*I*b^2*c^5)*(d*sqrt(x) + c))*cos(2*d*s
qrt(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) - (-2*I*(d*sqrt(x) + c)^7
*a*b - 7*I*b^2*c^6 + (14*I*a*b*c - 7*I*b^2)*(d*sqrt(x) + c)^6 + (-42*I*a*b*c^2 + 42*I*b^2*c)*(d*sqrt(x) + c)^5
+ (70*I*a*b*c^3 - 105*I*b^2*c^2)*(d*sqrt(x) + c)^4 + (-70*I*a*b*c^4 + 140*I*b^2*c^3)*(d*sqrt(x) + c)^3 + (42*
I*a*b*c^5 - 105*I*b^2*c^4)*(d*sqrt(x) + c)^2 + (-14*I*a*b*c^6 + 42*I*b^2*c^5)*(d*sqrt(x) + c) + (2*I*(d*sqrt(x
) + c)^7*a*b + 7*I*b^2*c^6 + (-14*I*a*b*c + 7*I*b^2)*(d*sqrt(x) + c)^6 + (42*I*a*b*c^2 - 42*I*b^2*c)*(d*sqrt(x
) + c)^5 + (-70*I*a*b*c^3 + 105*I*b^2*c^2)*(d*sqrt(x) + c)^4 + (70*I*a*b*c^4 - 140*I*b^2*c^3)*(d*sqrt(x) + c)^
3 + (-42*I*a*b*c^5 + 105*I*b^2*c^4)*(d*sqrt(x) + c)^2 + (14*I*a*b*c^6 - 42*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*sq
rt(x) + 2*c) + 20160*I*a*b*sin(2*d*sqrt(x) + 2*c) - 20160*a*b)*polylog(8, -e^(I*d*sqrt(x) + I*c)) + (20160*a*b
*cos(2*d*sqrt(x) + 2*c) + 20160*I*a*b*sin(2*d*sqrt(x) + 2*c) - 20160*a*b)*polylog(8, e^(I*d*sqrt(x) + I*c)) -
(20160*I*(d*sqrt(x) + c)*a*b - 20160*I*a*b*c - 10080*I*b^2 + (-20160*I*(d*sqrt(x) + c)*a*b + 20160*I*a*b*c + 1
0080*I*b^2)*cos(2*d*sqrt(x) + 2*c) + 10080*(2*(d*sqrt(x) + c)*a*b - 2*a*b*c - b^2)*sin(2*d*sqrt(x) + 2*c))*pol
ylog(7, -e^(I*d*sqrt(x) + I*c)) - (-20160*I*(d*sqrt(x) + c)*a*b + 20160*I*a*b*c - 10080*I*b^2 + (20160*I*(d*sq
rt(x) + c)*a*b - 20160*I*a*b*c + 10080*I*b^2)*cos(2*d*sqrt(x) + 2*c) - 10080*(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 + 10080*a*b*c^
2 + 10080*b^2*c - 10080*(2*a*b*c + b^2)*(d*sqrt(x) + 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))*cos(2*d*sqrt(x) + 2*c) + (-10080*I*(d*sqrt(x) + c)^2*a*b - 10080*I*a*b*c^2 - 100
80*I*b^2*c + (20160*I*a*b*c + 10080*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 + 10080*a*b*c^2 - 10080*b^2*c - 10080*(2*a*b*c - b^2)*(d*sqrt(x) + 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))*cos(2*d*sqrt(x) + 2*c) - (
10080*I*(d*sqrt(x) + c)^2*a*b + 10080*I*a*b*c^2 - 10080*I*b^2*c + (-20160*I*a*b*c + 10080*I*b^2)*(d*sqrt(x) +
c))*sin(2*d*sqrt(x) + 2*c))*polylog(6, e^(I*d*sqrt(x) + I*c)) - (-3360*I*(d*sqrt(x) + c)^3*a*b + 3360*I*a*b*c^
3 + 5040*I*b^2*c^2 + (10080*I*a*b*c + 5040*I*b^2)*(d*sqrt(x) + c)^2 + (-10080*I*a*b*c^2 - 10080*I*b^2*c)*(d*sq
rt(x) + c) + (3360*I*(d*sqrt(x) + c)^3*a*b - 3360*I*a*b*c^3 - 5040*I*b^2*c^2 + (-10080*I*a*b*c - 5040*I*b^2)*(
d*sqrt(x) + c)^2 + (10080*I*a*b*c^2 + 10080*I*b^2*c)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) - 1680*(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)) - (3360*I*(d*sqrt(x) + c)^3*a*b - 3360*I*a*b*
c^3 + 5040*I*b^2*c^2 + (-10080*I*a*b*c + 5040*I*b^2)*(d*sqrt(x) + c)^2 + (10080*I*a*b*c^2 - 10080*I*b^2*c)*(d*
sqrt(x) + c) + (-3360*I*(d*sqrt(x) + c)^3*a*b + 3360*I*a*b*c^3 - 5040*I*b^2*c^2 + (10080*I*a*b*c - 5040*I*b^2)
*(d*sqrt(x) + c)^2 + (-10080*I*a*b*c^2 + 10080*I*b^2*c)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + 1680*(2*(d*s
qrt(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 + 840*a*b*c^4
+ 1680*b^2*c^3 - 1680*(2*a*b*c + b^2)*(d*sqrt(x) + c)^3 + 5040*(a*b*c^2 + b^2*c)*(d*sqrt(x) + c)^2 - 1680*(2*a
*b*c^3 + 3*b^2*c^2)*(d*sqrt(x) + 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))*cos(2*d*sq
rt(x) + 2*c) - (840*I*(d*sqrt(x) + c)^4*a*b + 840*I*a*b*c^4 + 1680*I*b^2*c^3 + (-3360*I*a*b*c - 1680*I*b^2)*(d
*sqrt(x) + c)^3 + (5040*I*a*b*c^2 + 5040*I*b^2*c)*(d*sqrt(x) + c)^2 + (-3360*I*a*b*c^3 - 5040*I*b^2*c^2)*(d*sq
rt(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 + 840*a*b*
c^4 - 1680*b^2*c^3 - 1680*(2*a*b*c - b^2)*(d*sqrt(x) + c)^3 + 5040*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c)^2 - 1680*
(2*a*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + 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))*cos(2*
d*sqrt(x) + 2*c) + (-840*I*(d*sqrt(x) + c)^4*a*b - 840*I*a*b*c^4 + 1680*I*b^2*c^3 + (3360*I*a*b*c - 1680*I*b^2
)*(d*sqrt(x) + c)^3 + (-5040*I*a*b*c^2 + 5040*I*b^2*c)*(d*sqrt(x) + c)^2 + (3360*I*a*b*c^3 - 5040*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)) - (168*I*(d*sqrt(x) + c)^5*a*b - 168
*I*a*b*c^5 - 420*I*b^2*c^4 + (-840*I*a*b*c - 420*I*b^2)*(d*sqrt(x) + c)^4 + (1680*I*a*b*c^2 + 1680*I*b^2*c)*(d
*sqrt(x) + c)^3 + (-1680*I*a*b*c^3 - 2520*I*b^2*c^2)*(d*sqrt(x) + c)^2 + (840*I*a*b*c^4 + 1680*I*b^2*c^3)*(d*s
qrt(x) + c) + (-168*I*(d*sqrt(x) + c)^5*a*b + 168*I*a*b*c^5 + 420*I*b^2*c^4 + (840*I*a*b*c + 420*I*b^2)*(d*sqr
t(x) + c)^4 + (-1680*I*a*b*c^2 - 1680*I*b^2*c)*(d*sqrt(x) + c)^3 + (1680*I*a*b*c^3 + 2520*I*b^2*c^2)*(d*sqrt(x
) + c)^2 + (-840*I*a*b*c^4 - 1680*I*b^2*c^3)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + 84*(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)) - (-168*I*(d*sqrt(x) + c)^5*a*b + 168*I*a*b*c^5 - 420*I*b^2*c^4 + (840*I
*a*b*c - 420*I*b^2)*(d*sqrt(x) + c)^4 + (-1680*I*a*b*c^2 + 1680*I*b^2*c)*(d*sqrt(x) + c)^3 + (1680*I*a*b*c^3 -
2520*I*b^2*c^2)*(d*sqrt(x) + c)^2 + (-840*I*a*b*c^4 + 1680*I*b^2*c^3)*(d*sqrt(x) + c) + (168*I*(d*sqrt(x) + c
)^5*a*b - 168*I*a*b*c^5 + 420*I*b^2*c^4 + (-840*I*a*b*c + 420*I*b^2)*(d*sqrt(x) + c)^4 + (1680*I*a*b*c^2 - 168
0*I*b^2*c)*(d*sqrt(x) + c)^3 + (-1680*I*a*b*c^3 + 2520*I*b^2*c^2)*(d*sqrt(x) + c)^2 + (840*I*a*b*c^4 - 1680*I*
b^2*c^3)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) - 84*(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 - 28*I*(d*sqrt(x) + c)^6*b^2*c + 84*I*(d*sqrt(x) + c)^5*b^2*c^2 - 140*I*(d*sqrt(
x) + c)^4*b^2*c^3 + 140*I*(d*sqrt(x) + c)^3*b^2*c^4 - 84*I*(d*sqrt(x) + c)^2*b^2*c^5 + 28*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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x^{3} \csc \left (d \sqrt{x} + c\right )^{2} + 2 \, a b x^{3} \csc \left (d \sqrt{x} + c\right ) + a^{2} x^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \csc \left (d \sqrt{x} + c\right ) + a\right )}^{2} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)