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

Optimal. Leaf size=749 \[ \text{result too large to display} \]

[Out]

((-2*I)*b^2*x^(7/2))/d + (a^2*x^4)/4 - ((8*I)*a*b*x^(7/2)*ArcTan[E^(I*(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, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((28*I)*a*b
*x^3*PolyLog[2, I*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, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (168*a*b*x^(5/2)*PolyLog[3, I*E^(I*(c + d*S
qrt[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, (-I)*E
^(I*(c + d*Sqrt[x]))])/d^4 + ((840*I)*a*b*x^2*PolyLog[4, I*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, (-I)*E^(I*(c + d*Sqrt[x]))])/d^5 -
(3360*a*b*x^(3/2)*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))])/d^5 - (315*b^2*x*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))]
)/d^6 + ((10080*I)*a*b*x*PolyLog[6, (-I)*E^(I*(c + d*Sqrt[x]))])/d^6 - ((10080*I)*a*b*x*PolyLog[6, I*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]*Pol
yLog[7, (-I)*E^(I*(c + d*Sqrt[x]))])/d^7 + (20160*a*b*Sqrt[x]*PolyLog[7, I*E^(I*(c + d*Sqrt[x]))])/d^7 + (315*
b^2*PolyLog[7, -E^((2*I)*(c + d*Sqrt[x]))])/(2*d^8) - ((20160*I)*a*b*PolyLog[8, (-I)*E^(I*(c + d*Sqrt[x]))])/d
^8 + ((20160*I)*a*b*PolyLog[8, I*E^(I*(c + d*Sqrt[x]))])/d^8 + (2*b^2*x^(7/2)*Tan[c + d*Sqrt[x]])/d

________________________________________________________________________________________

Rubi [A]  time = 0.860258, antiderivative size = 749, 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 = {4204, 4190, 4181, 2531, 6609, 2282, 6589, 4184, 3719, 2190} \[ \frac{28 i a b x^3 \text{PolyLog}\left (2,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{PolyLog}\left (2,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{168 a b x^{5/2} \text{PolyLog}\left (3,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{PolyLog}\left (3,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{840 i a b x^2 \text{PolyLog}\left (4,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{PolyLog}\left (4,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{3360 a b x^{3/2} \text{PolyLog}\left (5,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{PolyLog}\left (5,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{10080 i a b x \text{PolyLog}\left (6,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 i a b x \text{PolyLog}\left (6,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{20160 a b \sqrt{x} \text{PolyLog}\left (7,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{PolyLog}\left (7,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}-\frac{20160 i a b \text{PolyLog}\left (8,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}+\frac{20160 i a b \text{PolyLog}\left (8,i 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 i a b x^{7/2} \tan ^{-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} \tan \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*Sec[c + d*Sqrt[x]])^2,x]

[Out]

((-2*I)*b^2*x^(7/2))/d + (a^2*x^4)/4 - ((8*I)*a*b*x^(7/2)*ArcTan[E^(I*(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, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((28*I)*a*b
*x^3*PolyLog[2, I*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, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (168*a*b*x^(5/2)*PolyLog[3, I*E^(I*(c + d*S
qrt[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, (-I)*E
^(I*(c + d*Sqrt[x]))])/d^4 + ((840*I)*a*b*x^2*PolyLog[4, I*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, (-I)*E^(I*(c + d*Sqrt[x]))])/d^5 -
(3360*a*b*x^(3/2)*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))])/d^5 - (315*b^2*x*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))]
)/d^6 + ((10080*I)*a*b*x*PolyLog[6, (-I)*E^(I*(c + d*Sqrt[x]))])/d^6 - ((10080*I)*a*b*x*PolyLog[6, I*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]*Pol
yLog[7, (-I)*E^(I*(c + d*Sqrt[x]))])/d^7 + (20160*a*b*Sqrt[x]*PolyLog[7, I*E^(I*(c + d*Sqrt[x]))])/d^7 + (315*
b^2*PolyLog[7, -E^((2*I)*(c + d*Sqrt[x]))])/(2*d^8) - ((20160*I)*a*b*PolyLog[8, (-I)*E^(I*(c + d*Sqrt[x]))])/d
^8 + ((20160*I)*a*b*PolyLog[8, I*E^(I*(c + d*Sqrt[x]))])/d^8 + (2*b^2*x^(7/2)*Tan[c + d*Sqrt[x]])/d

Rule 4204

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sec[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 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && 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 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
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 \sec \left (c+d \sqrt{x}\right )\right )^2 \, dx &=2 \operatorname{Subst}\left (\int x^7 (a+b \sec (c+d x))^2 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^2 x^7+2 a b x^7 \sec (c+d x)+b^2 x^7 \sec ^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 \sec (c+d x) \, dx,x,\sqrt{x}\right )+\left (2 b^2\right ) \operatorname{Subst}\left (\int x^7 \sec ^2(c+d x) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^4}{4}-\frac{8 i a b x^{7/2} \tan ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{2 b^2 x^{7/2} \tan \left (c+d \sqrt{x}\right )}{d}-\frac{(28 a b) \operatorname{Subst}\left (\int x^6 \log \left (1-i 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+i e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d}-\frac{\left (14 b^2\right ) \operatorname{Subst}\left (\int x^6 \tan (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 i a b x^{7/2} \tan ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{28 i a b x^3 \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}+\frac{2 b^2 x^{7/2} \tan \left (c+d \sqrt{x}\right )}{d}-\frac{(168 i a b) \operatorname{Subst}\left (\int x^5 \text{Li}_2\left (-i 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 (i 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 i a b x^{7/2} \tan ^{-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{28 i a b x^3 \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{168 a b x^{5/2} \text{Li}_3\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{2 b^2 x^{7/2} \tan \left (c+d \sqrt{x}\right )}{d}+\frac{(840 a b) \operatorname{Subst}\left (\int x^4 \text{Li}_3\left (-i 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 (i 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 i a b x^{7/2} \tan ^{-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{28 i a b x^3 \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{840 i a b x^2 \text{Li}_4\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{2 b^2 x^{7/2} \tan \left (c+d \sqrt{x}\right )}{d}+\frac{(3360 i a b) \operatorname{Subst}\left (\int x^3 \text{Li}_4\left (-i 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 (i 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 i a b x^{7/2} \tan ^{-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{28 i a b x^3 \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{2 b^2 x^{7/2} \tan \left (c+d \sqrt{x}\right )}{d}-\frac{(10080 a b) \operatorname{Subst}\left (\int x^2 \text{Li}_5\left (-i 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 (i 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 i a b x^{7/2} \tan ^{-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{28 i a b x^3 \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{10080 i a b x \text{Li}_6\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 i a b x \text{Li}_6\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}+\frac{2 b^2 x^{7/2} \tan \left (c+d \sqrt{x}\right )}{d}-\frac{(20160 i a b) \operatorname{Subst}\left (\int x \text{Li}_6\left (-i 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 (i 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 i a b x^{7/2} \tan ^{-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{28 i a b x^3 \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 i a b x \text{Li}_6\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{20160 a b \sqrt{x} \text{Li}_7\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{2 b^2 x^{7/2} \tan \left (c+d \sqrt{x}\right )}{d}+\frac{(20160 a b) \operatorname{Subst}\left (\int \text{Li}_7\left (-i 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 (i 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 i a b x^{7/2} \tan ^{-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{28 i a b x^3 \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 i a b x \text{Li}_6\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{2 b^2 x^{7/2} \tan \left (c+d \sqrt{x}\right )}{d}-\frac{(20160 i a b) \operatorname{Subst}\left (\int \frac{\text{Li}_7(-i 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(i 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 i a b x^{7/2} \tan ^{-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{28 i a b x^3 \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 i a b x \text{Li}_6\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}-\frac{20160 i a b \text{Li}_8\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}+\frac{20160 i a b \text{Li}_8\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}+\frac{2 b^2 x^{7/2} \tan \left (c+d \sqrt{x}\right )}{d}+\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 i a b x^{7/2} \tan ^{-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{28 i a b x^3 \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 i a b x \text{Li}_6\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left (i 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 (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}+\frac{20160 i a b \text{Li}_8\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^8}+\frac{2 b^2 x^{7/2} \tan \left (c+d \sqrt{x}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 2.06019, size = 739, normalized size = 0.99 \[ \frac{112 i a b d^6 x^3 \text{PolyLog}\left (2,-i e^{i \left (c+d \sqrt{x}\right )}\right )-112 i a b d^6 x^3 \text{PolyLog}\left (2,i e^{i \left (c+d \sqrt{x}\right )}\right )-672 a b d^5 x^{5/2} \text{PolyLog}\left (3,-i e^{i \left (c+d \sqrt{x}\right )}\right )+672 a b d^5 x^{5/2} \text{PolyLog}\left (3,i e^{i \left (c+d \sqrt{x}\right )}\right )-3360 i a b d^4 x^2 \text{PolyLog}\left (4,-i e^{i \left (c+d \sqrt{x}\right )}\right )+3360 i a b d^4 x^2 \text{PolyLog}\left (4,i e^{i \left (c+d \sqrt{x}\right )}\right )+13440 a b d^3 x^{3/2} \text{PolyLog}\left (5,-i e^{i \left (c+d \sqrt{x}\right )}\right )-13440 a b d^3 x^{3/2} \text{PolyLog}\left (5,i e^{i \left (c+d \sqrt{x}\right )}\right )+40320 i a b d^2 x \text{PolyLog}\left (6,-i e^{i \left (c+d \sqrt{x}\right )}\right )-40320 i a b d^2 x \text{PolyLog}\left (6,i e^{i \left (c+d \sqrt{x}\right )}\right )-80640 a b d \sqrt{x} \text{PolyLog}\left (7,-i e^{i \left (c+d \sqrt{x}\right )}\right )+80640 a b d \sqrt{x} \text{PolyLog}\left (7,i e^{i \left (c+d \sqrt{x}\right )}\right )-80640 i a b \text{PolyLog}\left (8,-i e^{i \left (c+d \sqrt{x}\right )}\right )+80640 i a b \text{PolyLog}\left (8,i e^{i \left (c+d \sqrt{x}\right )}\right )-168 i b^2 d^5 x^{5/2} \text{PolyLog}\left (2,-e^{2 i \left (c+d \sqrt{x}\right )}\right )+420 b^2 d^4 x^2 \text{PolyLog}\left (3,-e^{2 i \left (c+d \sqrt{x}\right )}\right )+840 i b^2 d^3 x^{3/2} \text{PolyLog}\left (4,-e^{2 i \left (c+d \sqrt{x}\right )}\right )-1260 b^2 d^2 x \text{PolyLog}\left (5,-e^{2 i \left (c+d \sqrt{x}\right )}\right )-1260 i b^2 d \sqrt{x} \text{PolyLog}\left (6,-e^{2 i \left (c+d \sqrt{x}\right )}\right )+630 b^2 \text{PolyLog}\left (7,-e^{2 i \left (c+d \sqrt{x}\right )}\right )+a^2 d^8 x^4-32 i a b d^7 x^{7/2} \tan ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )+56 b^2 d^6 x^3 \log \left (1+e^{2 i \left (c+d \sqrt{x}\right )}\right )+8 b^2 d^7 x^{7/2} \tan \left (c+d \sqrt{x}\right )-8 i b^2 d^7 x^{7/2}}{4 d^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*I)*b^2*d^7*x^(7/2) + a^2*d^8*x^4 - (32*I)*a*b*d^7*x^(7/2)*ArcTan[E^(I*(c + d*Sqrt[x]))] + 56*b^2*d^6*x^3*
Log[1 + E^((2*I)*(c + d*Sqrt[x]))] + (112*I)*a*b*d^6*x^3*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))] - (112*I)*a*b*
d^6*x^3*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))] - (168*I)*b^2*d^5*x^(5/2)*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))] -
672*a*b*d^5*x^(5/2)*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))] + 672*a*b*d^5*x^(5/2)*PolyLog[3, I*E^(I*(c + d*Sqr
t[x]))] + 420*b^2*d^4*x^2*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))] - (3360*I)*a*b*d^4*x^2*PolyLog[4, (-I)*E^(I*(
c + d*Sqrt[x]))] + (3360*I)*a*b*d^4*x^2*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))] + (840*I)*b^2*d^3*x^(3/2)*PolyLog[
4, -E^((2*I)*(c + d*Sqrt[x]))] + 13440*a*b*d^3*x^(3/2)*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))] - 13440*a*b*d^3*
x^(3/2)*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))] - 1260*b^2*d^2*x*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))] + (40320*I
)*a*b*d^2*x*PolyLog[6, (-I)*E^(I*(c + d*Sqrt[x]))] - (40320*I)*a*b*d^2*x*PolyLog[6, I*E^(I*(c + d*Sqrt[x]))] -
(1260*I)*b^2*d*Sqrt[x]*PolyLog[6, -E^((2*I)*(c + d*Sqrt[x]))] - 80640*a*b*d*Sqrt[x]*PolyLog[7, (-I)*E^(I*(c +
d*Sqrt[x]))] + 80640*a*b*d*Sqrt[x]*PolyLog[7, I*E^(I*(c + d*Sqrt[x]))] + 630*b^2*PolyLog[7, -E^((2*I)*(c + d*
Sqrt[x]))] - (80640*I)*a*b*PolyLog[8, (-I)*E^(I*(c + d*Sqrt[x]))] + (80640*I)*a*b*PolyLog[8, I*E^(I*(c + d*Sqr
t[x]))] + 8*b^2*d^7*x^(7/2)*Tan[c + d*Sqrt[x]])/(4*d^8)

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\sec \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*sec(c+d*x^(1/2)))^2,x)

[Out]

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

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Maxima [B]  time = 4.26139, size = 8608, normalized size = 11.49 \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*sec(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(sec(d*sqrt(x) + c) + tan(d*sqrt(x) + c)) - 8*(60*b^2*c^7 + (60*(d*sqrt(x) +
c)^7*a*b - 420*(d*sqrt(x) + c)^6*a*b*c + 1260*(d*sqrt(x) + c)^5*a*b*c^2 - 2100*(d*sqrt(x) + c)^4*a*b*c^3 + 21
00*(d*sqrt(x) + c)^3*a*b*c^4 - 1260*(d*sqrt(x) + c)^2*a*b*c^5 + 420*(d*sqrt(x) + c)*a*b*c^6 + 60*((d*sqrt(x) +
c)^7*a*b - 7*(d*sqrt(x) + c)^6*a*b*c + 21*(d*sqrt(x) + c)^5*a*b*c^2 - 35*(d*sqrt(x) + c)^4*a*b*c^3 + 35*(d*sq
rt(x) + c)^3*a*b*c^4 - 21*(d*sqrt(x) + c)^2*a*b*c^5 + 7*(d*sqrt(x) + c)*a*b*c^6)*cos(2*d*sqrt(x) + 2*c) - (-60
*I*(d*sqrt(x) + c)^7*a*b + 420*I*(d*sqrt(x) + c)^6*a*b*c - 1260*I*(d*sqrt(x) + c)^5*a*b*c^2 + 2100*I*(d*sqrt(x
) + c)^4*a*b*c^3 - 2100*I*(d*sqrt(x) + c)^3*a*b*c^4 + 1260*I*(d*sqrt(x) + c)^2*a*b*c^5 - 420*I*(d*sqrt(x) + c)
*a*b*c^6)*sin(2*d*sqrt(x) + 2*c))*arctan2(cos(d*sqrt(x) + c), sin(d*sqrt(x) + c) + 1) + (60*(d*sqrt(x) + c)^7*
a*b - 420*(d*sqrt(x) + c)^6*a*b*c + 1260*(d*sqrt(x) + c)^5*a*b*c^2 - 2100*(d*sqrt(x) + c)^4*a*b*c^3 + 2100*(d*
sqrt(x) + c)^3*a*b*c^4 - 1260*(d*sqrt(x) + c)^2*a*b*c^5 + 420*(d*sqrt(x) + c)*a*b*c^6 + 60*((d*sqrt(x) + c)^7*
a*b - 7*(d*sqrt(x) + c)^6*a*b*c + 21*(d*sqrt(x) + c)^5*a*b*c^2 - 35*(d*sqrt(x) + c)^4*a*b*c^3 + 35*(d*sqrt(x)
+ c)^3*a*b*c^4 - 21*(d*sqrt(x) + c)^2*a*b*c^5 + 7*(d*sqrt(x) + c)*a*b*c^6)*cos(2*d*sqrt(x) + 2*c) - (-60*I*(d*
sqrt(x) + c)^7*a*b + 420*I*(d*sqrt(x) + c)^6*a*b*c - 1260*I*(d*sqrt(x) + c)^5*a*b*c^2 + 2100*I*(d*sqrt(x) + c)
^4*a*b*c^3 - 2100*I*(d*sqrt(x) + c)^3*a*b*c^4 + 1260*I*(d*sqrt(x) + c)^2*a*b*c^5 - 420*I*(d*sqrt(x) + c)*a*b*c
^6)*sin(2*d*sqrt(x) + 2*c))*arctan2(cos(d*sqrt(x) + c), -sin(d*sqrt(x) + c) + 1) - (1120*(d*sqrt(x) + c)^6*b^2
- 4032*(d*sqrt(x) + c)^5*b^2*c + 6300*(d*sqrt(x) + c)^4*b^2*c^2 - 5600*(d*sqrt(x) + c)^3*b^2*c^3 + 3150*(d*sq
rt(x) + c)^2*b^2*c^4 - 1260*(d*sqrt(x) + c)*b^2*c^5 + 210*b^2*c^6 + 14*(80*(d*sqrt(x) + c)^6*b^2 - 288*(d*sqrt
(x) + c)^5*b^2*c + 450*(d*sqrt(x) + c)^4*b^2*c^2 - 400*(d*sqrt(x) + c)^3*b^2*c^3 + 225*(d*sqrt(x) + c)^2*b^2*c
^4 - 90*(d*sqrt(x) + c)*b^2*c^5 + 15*b^2*c^6)*cos(2*d*sqrt(x) + 2*c) + (1120*I*(d*sqrt(x) + c)^6*b^2 - 4032*I*
(d*sqrt(x) + c)^5*b^2*c + 6300*I*(d*sqrt(x) + c)^4*b^2*c^2 - 5600*I*(d*sqrt(x) + c)^3*b^2*c^3 + 3150*I*(d*sqrt
(x) + c)^2*b^2*c^4 - 1260*I*(d*sqrt(x) + c)*b^2*c^5 + 210*I*b^2*c^6)*sin(2*d*sqrt(x) + 2*c))*arctan2(sin(2*d*s
qrt(x) + 2*c), cos(2*d*sqrt(x) + 2*c) + 1) + 60*((d*sqrt(x) + c)^7*b^2 - 7*(d*sqrt(x) + c)^6*b^2*c + 21*(d*sqr
t(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) + (3360*(d*sqrt(x) + c)^5*b^2 - 10080*(d*sqrt(x) + c)^4
*b^2*c + 12600*(d*sqrt(x) + c)^3*b^2*c^2 - 8400*(d*sqrt(x) + c)^2*b^2*c^3 + 3150*(d*sqrt(x) + c)*b^2*c^4 - 630
*b^2*c^5 + 210*(16*(d*sqrt(x) + c)^5*b^2 - 48*(d*sqrt(x) + c)^4*b^2*c + 60*(d*sqrt(x) + c)^3*b^2*c^2 - 40*(d*s
qrt(x) + c)^2*b^2*c^3 + 15*(d*sqrt(x) + c)*b^2*c^4 - 3*b^2*c^5)*cos(2*d*sqrt(x) + 2*c) - (-3360*I*(d*sqrt(x) +
c)^5*b^2 + 10080*I*(d*sqrt(x) + c)^4*b^2*c - 12600*I*(d*sqrt(x) + c)^3*b^2*c^2 + 8400*I*(d*sqrt(x) + c)^2*b^2
*c^3 - 3150*I*(d*sqrt(x) + c)*b^2*c^4 + 630*I*b^2*c^5)*sin(2*d*sqrt(x) + 2*c))*dilog(-e^(2*I*d*sqrt(x) + 2*I*c
)) + (420*(d*sqrt(x) + c)^6*a*b - 2520*(d*sqrt(x) + c)^5*a*b*c + 6300*(d*sqrt(x) + c)^4*a*b*c^2 - 8400*(d*sqrt
(x) + c)^3*a*b*c^3 + 6300*(d*sqrt(x) + c)^2*a*b*c^4 - 2520*(d*sqrt(x) + c)*a*b*c^5 + 420*a*b*c^6 + 420*((d*sqr
t(x) + c)^6*a*b - 6*(d*sqrt(x) + c)^5*a*b*c + 15*(d*sqrt(x) + c)^4*a*b*c^2 - 20*(d*sqrt(x) + c)^3*a*b*c^3 + 15
*(d*sqrt(x) + c)^2*a*b*c^4 - 6*(d*sqrt(x) + c)*a*b*c^5 + a*b*c^6)*cos(2*d*sqrt(x) + 2*c) - (-420*I*(d*sqrt(x)
+ c)^6*a*b + 2520*I*(d*sqrt(x) + c)^5*a*b*c - 6300*I*(d*sqrt(x) + c)^4*a*b*c^2 + 8400*I*(d*sqrt(x) + c)^3*a*b*
c^3 - 6300*I*(d*sqrt(x) + c)^2*a*b*c^4 + 2520*I*(d*sqrt(x) + c)*a*b*c^5 - 420*I*a*b*c^6)*sin(2*d*sqrt(x) + 2*c
))*dilog(I*e^(I*d*sqrt(x) + I*c)) - (420*(d*sqrt(x) + c)^6*a*b - 2520*(d*sqrt(x) + c)^5*a*b*c + 6300*(d*sqrt(x
) + c)^4*a*b*c^2 - 8400*(d*sqrt(x) + c)^3*a*b*c^3 + 6300*(d*sqrt(x) + c)^2*a*b*c^4 - 2520*(d*sqrt(x) + c)*a*b*
c^5 + 420*a*b*c^6 + 420*((d*sqrt(x) + c)^6*a*b - 6*(d*sqrt(x) + c)^5*a*b*c + 15*(d*sqrt(x) + c)^4*a*b*c^2 - 20
*(d*sqrt(x) + c)^3*a*b*c^3 + 15*(d*sqrt(x) + c)^2*a*b*c^4 - 6*(d*sqrt(x) + c)*a*b*c^5 + a*b*c^6)*cos(2*d*sqrt(
x) + 2*c) + (420*I*(d*sqrt(x) + c)^6*a*b - 2520*I*(d*sqrt(x) + c)^5*a*b*c + 6300*I*(d*sqrt(x) + c)^4*a*b*c^2 -
8400*I*(d*sqrt(x) + c)^3*a*b*c^3 + 6300*I*(d*sqrt(x) + c)^2*a*b*c^4 - 2520*I*(d*sqrt(x) + c)*a*b*c^5 + 420*I*
a*b*c^6)*sin(2*d*sqrt(x) + 2*c))*dilog(-I*e^(I*d*sqrt(x) + I*c)) - (-560*I*(d*sqrt(x) + c)^6*b^2 + 2016*I*(d*s
qrt(x) + c)^5*b^2*c - 3150*I*(d*sqrt(x) + c)^4*b^2*c^2 + 2800*I*(d*sqrt(x) + c)^3*b^2*c^3 - 1575*I*(d*sqrt(x)
+ c)^2*b^2*c^4 + 630*I*(d*sqrt(x) + c)*b^2*c^5 - 105*I*b^2*c^6 + (-560*I*(d*sqrt(x) + c)^6*b^2 + 2016*I*(d*sqr
t(x) + c)^5*b^2*c - 3150*I*(d*sqrt(x) + c)^4*b^2*c^2 + 2800*I*(d*sqrt(x) + c)^3*b^2*c^3 - 1575*I*(d*sqrt(x) +
c)^2*b^2*c^4 + 630*I*(d*sqrt(x) + c)*b^2*c^5 - 105*I*b^2*c^6)*cos(2*d*sqrt(x) + 2*c) + 7*(80*(d*sqrt(x) + c)^6
*b^2 - 288*(d*sqrt(x) + c)^5*b^2*c + 450*(d*sqrt(x) + c)^4*b^2*c^2 - 400*(d*sqrt(x) + c)^3*b^2*c^3 + 225*(d*sq
rt(x) + c)^2*b^2*c^4 - 90*(d*sqrt(x) + c)*b^2*c^5 + 15*b^2*c^6)*sin(2*d*sqrt(x) + 2*c))*log(cos(2*d*sqrt(x) +
2*c)^2 + sin(2*d*sqrt(x) + 2*c)^2 + 2*cos(2*d*sqrt(x) + 2*c) + 1) - (-30*I*(d*sqrt(x) + c)^7*a*b + 210*I*(d*sq
rt(x) + c)^6*a*b*c - 630*I*(d*sqrt(x) + c)^5*a*b*c^2 + 1050*I*(d*sqrt(x) + c)^4*a*b*c^3 - 1050*I*(d*sqrt(x) +
c)^3*a*b*c^4 + 630*I*(d*sqrt(x) + c)^2*a*b*c^5 - 210*I*(d*sqrt(x) + c)*a*b*c^6 + (-30*I*(d*sqrt(x) + c)^7*a*b
+ 210*I*(d*sqrt(x) + c)^6*a*b*c - 630*I*(d*sqrt(x) + c)^5*a*b*c^2 + 1050*I*(d*sqrt(x) + c)^4*a*b*c^3 - 1050*I*
(d*sqrt(x) + c)^3*a*b*c^4 + 630*I*(d*sqrt(x) + c)^2*a*b*c^5 - 210*I*(d*sqrt(x) + c)*a*b*c^6)*cos(2*d*sqrt(x) +
2*c) + 30*((d*sqrt(x) + c)^7*a*b - 7*(d*sqrt(x) + c)^6*a*b*c + 21*(d*sqrt(x) + c)^5*a*b*c^2 - 35*(d*sqrt(x) +
c)^4*a*b*c^3 + 35*(d*sqrt(x) + c)^3*a*b*c^4 - 21*(d*sqrt(x) + c)^2*a*b*c^5 + 7*(d*sqrt(x) + c)*a*b*c^6)*sin(2
*d*sqrt(x) + 2*c))*log(cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 + 2*sin(d*sqrt(x) + c) + 1) - (30*I*(d*sqrt
(x) + c)^7*a*b - 210*I*(d*sqrt(x) + c)^6*a*b*c + 630*I*(d*sqrt(x) + c)^5*a*b*c^2 - 1050*I*(d*sqrt(x) + c)^4*a*
b*c^3 + 1050*I*(d*sqrt(x) + c)^3*a*b*c^4 - 630*I*(d*sqrt(x) + c)^2*a*b*c^5 + 210*I*(d*sqrt(x) + c)*a*b*c^6 + (
30*I*(d*sqrt(x) + c)^7*a*b - 210*I*(d*sqrt(x) + c)^6*a*b*c + 630*I*(d*sqrt(x) + c)^5*a*b*c^2 - 1050*I*(d*sqrt(
x) + c)^4*a*b*c^3 + 1050*I*(d*sqrt(x) + c)^3*a*b*c^4 - 630*I*(d*sqrt(x) + c)^2*a*b*c^5 + 210*I*(d*sqrt(x) + c)
*a*b*c^6)*cos(2*d*sqrt(x) + 2*c) - 30*((d*sqrt(x) + c)^7*a*b - 7*(d*sqrt(x) + c)^6*a*b*c + 21*(d*sqrt(x) + c)^
5*a*b*c^2 - 35*(d*sqrt(x) + c)^4*a*b*c^3 + 35*(d*sqrt(x) + c)^3*a*b*c^4 - 21*(d*sqrt(x) + c)^2*a*b*c^5 + 7*(d*
sqrt(x) + c)*a*b*c^6)*sin(2*d*sqrt(x) + 2*c))*log(cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 - 2*sin(d*sqrt(x
) + c) + 1) - (302400*a*b*cos(2*d*sqrt(x) + 2*c) + 302400*I*a*b*sin(2*d*sqrt(x) + 2*c) + 302400*a*b)*polylog(8
, I*e^(I*d*sqrt(x) + I*c)) + (302400*a*b*cos(2*d*sqrt(x) + 2*c) + 302400*I*a*b*sin(2*d*sqrt(x) + 2*c) + 302400
*a*b)*polylog(8, -I*e^(I*d*sqrt(x) + I*c)) - (-12600*I*b^2*cos(2*d*sqrt(x) + 2*c) + 12600*b^2*sin(2*d*sqrt(x)
+ 2*c) - 12600*I*b^2)*polylog(7, -e^(2*I*d*sqrt(x) + 2*I*c)) - (-302400*I*(d*sqrt(x) + c)*a*b + 302400*I*a*b*c
+ (-302400*I*(d*sqrt(x) + c)*a*b + 302400*I*a*b*c)*cos(2*d*sqrt(x) + 2*c) + 302400*((d*sqrt(x) + c)*a*b - a*b
*c)*sin(2*d*sqrt(x) + 2*c))*polylog(7, I*e^(I*d*sqrt(x) + I*c)) - (302400*I*(d*sqrt(x) + c)*a*b - 302400*I*a*b
*c + (302400*I*(d*sqrt(x) + c)*a*b - 302400*I*a*b*c)*cos(2*d*sqrt(x) + 2*c) - 302400*((d*sqrt(x) + c)*a*b - a*
b*c)*sin(2*d*sqrt(x) + 2*c))*polylog(7, -I*e^(I*d*sqrt(x) + I*c)) + (25200*(d*sqrt(x) + c)*b^2 - 15120*b^2*c +
5040*(5*(d*sqrt(x) + c)*b^2 - 3*b^2*c)*cos(2*d*sqrt(x) + 2*c) - (-25200*I*(d*sqrt(x) + c)*b^2 + 15120*I*b^2*c
)*sin(2*d*sqrt(x) + 2*c))*polylog(6, -e^(2*I*d*sqrt(x) + 2*I*c)) + (151200*(d*sqrt(x) + c)^2*a*b - 302400*(d*s
qrt(x) + c)*a*b*c + 151200*a*b*c^2 + 151200*((d*sqrt(x) + c)^2*a*b - 2*(d*sqrt(x) + c)*a*b*c + a*b*c^2)*cos(2*
d*sqrt(x) + 2*c) - (-151200*I*(d*sqrt(x) + c)^2*a*b + 302400*I*(d*sqrt(x) + c)*a*b*c - 151200*I*a*b*c^2)*sin(2
*d*sqrt(x) + 2*c))*polylog(6, I*e^(I*d*sqrt(x) + I*c)) - (151200*(d*sqrt(x) + c)^2*a*b - 302400*(d*sqrt(x) + c
)*a*b*c + 151200*a*b*c^2 + 151200*((d*sqrt(x) + c)^2*a*b - 2*(d*sqrt(x) + c)*a*b*c + a*b*c^2)*cos(2*d*sqrt(x)
+ 2*c) + (151200*I*(d*sqrt(x) + c)^2*a*b - 302400*I*(d*sqrt(x) + c)*a*b*c + 151200*I*a*b*c^2)*sin(2*d*sqrt(x)
+ 2*c))*polylog(6, -I*e^(I*d*sqrt(x) + I*c)) - (25200*I*(d*sqrt(x) + c)^2*b^2 - 30240*I*(d*sqrt(x) + c)*b^2*c
+ 9450*I*b^2*c^2 + (25200*I*(d*sqrt(x) + c)^2*b^2 - 30240*I*(d*sqrt(x) + c)*b^2*c + 9450*I*b^2*c^2)*cos(2*d*sq
rt(x) + 2*c) - 630*(40*(d*sqrt(x) + c)^2*b^2 - 48*(d*sqrt(x) + c)*b^2*c + 15*b^2*c^2)*sin(2*d*sqrt(x) + 2*c))*
polylog(5, -e^(2*I*d*sqrt(x) + 2*I*c)) - (50400*I*(d*sqrt(x) + c)^3*a*b - 151200*I*(d*sqrt(x) + c)^2*a*b*c + 1
51200*I*(d*sqrt(x) + c)*a*b*c^2 - 50400*I*a*b*c^3 + (50400*I*(d*sqrt(x) + c)^3*a*b - 151200*I*(d*sqrt(x) + c)^
2*a*b*c + 151200*I*(d*sqrt(x) + c)*a*b*c^2 - 50400*I*a*b*c^3)*cos(2*d*sqrt(x) + 2*c) - 50400*((d*sqrt(x) + c)^
3*a*b - 3*(d*sqrt(x) + c)^2*a*b*c + 3*(d*sqrt(x) + c)*a*b*c^2 - a*b*c^3)*sin(2*d*sqrt(x) + 2*c))*polylog(5, I*
e^(I*d*sqrt(x) + I*c)) - (-50400*I*(d*sqrt(x) + c)^3*a*b + 151200*I*(d*sqrt(x) + c)^2*a*b*c - 151200*I*(d*sqrt
(x) + c)*a*b*c^2 + 50400*I*a*b*c^3 + (-50400*I*(d*sqrt(x) + c)^3*a*b + 151200*I*(d*sqrt(x) + c)^2*a*b*c - 1512
00*I*(d*sqrt(x) + c)*a*b*c^2 + 50400*I*a*b*c^3)*cos(2*d*sqrt(x) + 2*c) + 50400*((d*sqrt(x) + c)^3*a*b - 3*(d*s
qrt(x) + c)^2*a*b*c + 3*(d*sqrt(x) + c)*a*b*c^2 - a*b*c^3)*sin(2*d*sqrt(x) + 2*c))*polylog(5, -I*e^(I*d*sqrt(x
) + I*c)) - (16800*(d*sqrt(x) + c)^3*b^2 - 30240*(d*sqrt(x) + c)^2*b^2*c + 18900*(d*sqrt(x) + c)*b^2*c^2 - 420
0*b^2*c^3 + 420*(40*(d*sqrt(x) + c)^3*b^2 - 72*(d*sqrt(x) + c)^2*b^2*c + 45*(d*sqrt(x) + c)*b^2*c^2 - 10*b^2*c
^3)*cos(2*d*sqrt(x) + 2*c) + (16800*I*(d*sqrt(x) + c)^3*b^2 - 30240*I*(d*sqrt(x) + c)^2*b^2*c + 18900*I*(d*sqr
t(x) + c)*b^2*c^2 - 4200*I*b^2*c^3)*sin(2*d*sqrt(x) + 2*c))*polylog(4, -e^(2*I*d*sqrt(x) + 2*I*c)) - (12600*(d
*sqrt(x) + c)^4*a*b - 50400*(d*sqrt(x) + c)^3*a*b*c + 75600*(d*sqrt(x) + c)^2*a*b*c^2 - 50400*(d*sqrt(x) + c)*
a*b*c^3 + 12600*a*b*c^4 + 12600*((d*sqrt(x) + c)^4*a*b - 4*(d*sqrt(x) + c)^3*a*b*c + 6*(d*sqrt(x) + c)^2*a*b*c
^2 - 4*(d*sqrt(x) + c)*a*b*c^3 + a*b*c^4)*cos(2*d*sqrt(x) + 2*c) + (12600*I*(d*sqrt(x) + c)^4*a*b - 50400*I*(d
*sqrt(x) + c)^3*a*b*c + 75600*I*(d*sqrt(x) + c)^2*a*b*c^2 - 50400*I*(d*sqrt(x) + c)*a*b*c^3 + 12600*I*a*b*c^4)
*sin(2*d*sqrt(x) + 2*c))*polylog(4, I*e^(I*d*sqrt(x) + I*c)) + (12600*(d*sqrt(x) + c)^4*a*b - 50400*(d*sqrt(x)
+ c)^3*a*b*c + 75600*(d*sqrt(x) + c)^2*a*b*c^2 - 50400*(d*sqrt(x) + c)*a*b*c^3 + 12600*a*b*c^4 + 12600*((d*sq
rt(x) + c)^4*a*b - 4*(d*sqrt(x) + c)^3*a*b*c + 6*(d*sqrt(x) + c)^2*a*b*c^2 - 4*(d*sqrt(x) + c)*a*b*c^3 + a*b*c
^4)*cos(2*d*sqrt(x) + 2*c) - (-12600*I*(d*sqrt(x) + c)^4*a*b + 50400*I*(d*sqrt(x) + c)^3*a*b*c - 75600*I*(d*sq
rt(x) + c)^2*a*b*c^2 + 50400*I*(d*sqrt(x) + c)*a*b*c^3 - 12600*I*a*b*c^4)*sin(2*d*sqrt(x) + 2*c))*polylog(4, -
I*e^(I*d*sqrt(x) + I*c)) - (-8400*I*(d*sqrt(x) + c)^4*b^2 + 20160*I*(d*sqrt(x) + c)^3*b^2*c - 18900*I*(d*sqrt(
x) + c)^2*b^2*c^2 + 8400*I*(d*sqrt(x) + c)*b^2*c^3 - 1575*I*b^2*c^4 + (-8400*I*(d*sqrt(x) + c)^4*b^2 + 20160*I
*(d*sqrt(x) + c)^3*b^2*c - 18900*I*(d*sqrt(x) + c)^2*b^2*c^2 + 8400*I*(d*sqrt(x) + c)*b^2*c^3 - 1575*I*b^2*c^4
)*cos(2*d*sqrt(x) + 2*c) + 105*(80*(d*sqrt(x) + c)^4*b^2 - 192*(d*sqrt(x) + c)^3*b^2*c + 180*(d*sqrt(x) + c)^2
*b^2*c^2 - 80*(d*sqrt(x) + c)*b^2*c^3 + 15*b^2*c^4)*sin(2*d*sqrt(x) + 2*c))*polylog(3, -e^(2*I*d*sqrt(x) + 2*I
*c)) - (-2520*I*(d*sqrt(x) + c)^5*a*b + 12600*I*(d*sqrt(x) + c)^4*a*b*c - 25200*I*(d*sqrt(x) + c)^3*a*b*c^2 +
25200*I*(d*sqrt(x) + c)^2*a*b*c^3 - 12600*I*(d*sqrt(x) + c)*a*b*c^4 + 2520*I*a*b*c^5 + (-2520*I*(d*sqrt(x) + c
)^5*a*b + 12600*I*(d*sqrt(x) + c)^4*a*b*c - 25200*I*(d*sqrt(x) + c)^3*a*b*c^2 + 25200*I*(d*sqrt(x) + c)^2*a*b*
c^3 - 12600*I*(d*sqrt(x) + c)*a*b*c^4 + 2520*I*a*b*c^5)*cos(2*d*sqrt(x) + 2*c) + 2520*((d*sqrt(x) + c)^5*a*b -
5*(d*sqrt(x) + c)^4*a*b*c + 10*(d*sqrt(x) + c)^3*a*b*c^2 - 10*(d*sqrt(x) + c)^2*a*b*c^3 + 5*(d*sqrt(x) + c)*a
*b*c^4 - a*b*c^5)*sin(2*d*sqrt(x) + 2*c))*polylog(3, I*e^(I*d*sqrt(x) + I*c)) - (2520*I*(d*sqrt(x) + c)^5*a*b
- 12600*I*(d*sqrt(x) + c)^4*a*b*c + 25200*I*(d*sqrt(x) + c)^3*a*b*c^2 - 25200*I*(d*sqrt(x) + c)^2*a*b*c^3 + 12
600*I*(d*sqrt(x) + c)*a*b*c^4 - 2520*I*a*b*c^5 + (2520*I*(d*sqrt(x) + c)^5*a*b - 12600*I*(d*sqrt(x) + c)^4*a*b
*c + 25200*I*(d*sqrt(x) + c)^3*a*b*c^2 - 25200*I*(d*sqrt(x) + c)^2*a*b*c^3 + 12600*I*(d*sqrt(x) + c)*a*b*c^4 -
2520*I*a*b*c^5)*cos(2*d*sqrt(x) + 2*c) - 2520*((d*sqrt(x) + c)^5*a*b - 5*(d*sqrt(x) + c)^4*a*b*c + 10*(d*sqrt
(x) + c)^3*a*b*c^2 - 10*(d*sqrt(x) + c)^2*a*b*c^3 + 5*(d*sqrt(x) + c)*a*b*c^4 - a*b*c^5)*sin(2*d*sqrt(x) + 2*c
))*polylog(3, -I*e^(I*d*sqrt(x) + I*c)) - (-60*I*(d*sqrt(x) + c)^7*b^2 + 420*I*(d*sqrt(x) + c)^6*b^2*c - 1260*
I*(d*sqrt(x) + c)^5*b^2*c^2 + 2100*I*(d*sqrt(x) + c)^4*b^2*c^3 - 2100*I*(d*sqrt(x) + c)^3*b^2*c^4 + 1260*I*(d*
sqrt(x) + c)^2*b^2*c^5 - 420*I*(d*sqrt(x) + c)*b^2*c^6)*sin(2*d*sqrt(x) + 2*c))/(-30*I*cos(2*d*sqrt(x) + 2*c)
+ 30*sin(2*d*sqrt(x) + 2*c) - 30*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} \sec \left (d \sqrt{x} + c\right )^{2} + 2 \, a b x^{3} \sec \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*sec(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^3*sec(d*sqrt(x) + c)^2 + 2*a*b*x^3*sec(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*sec(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 \sec \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*sec(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

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