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

Optimal result

Integrand size = 20, antiderivative size = 749 \[ \int x^3 \left (a+b \sec \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 i a b x^{7/2} \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,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} \]

[Out]

315/2*b^2*polylog(7,-exp(2*I*(c+d*x^(1/2))))/d^8+28*I*a*b*x^3*polylog(2,-I*exp(I*(c+d*x^(1/2))))/d^2+840*I*a*b
*x^2*polylog(4,I*exp(I*(c+d*x^(1/2))))/d^4+10080*I*a*b*x*polylog(6,-I*exp(I*(c+d*x^(1/2))))/d^6+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*b^2*x^(7/2)*tan(c+d*x^(1/2))/d-2*I*b^2*x^(7/2)/d-8*I*a*b*x^(7/2)*arctan(exp(I*(c+d*x^(1/
2))))/d-28*I*a*b*x^3*polylog(2,I*exp(I*(c+d*x^(1/2))))/d^2-840*I*a*b*x^2*polylog(4,-I*exp(I*(c+d*x^(1/2))))/d^
4+20160*I*a*b*polylog(8,I*exp(I*(c+d*x^(1/2))))/d^8+3360*a*b*x^(3/2)*polylog(5,-I*exp(I*(c+d*x^(1/2))))/d^5-33
60*a*b*x^(3/2)*polylog(5,I*exp(I*(c+d*x^(1/2))))/d^5-20160*a*b*polylog(7,-I*exp(I*(c+d*x^(1/2))))*x^(1/2)/d^7+
20160*a*b*polylog(7,I*exp(I*(c+d*x^(1/2))))*x^(1/2)/d^7-168*a*b*x^(5/2)*polylog(3,-I*exp(I*(c+d*x^(1/2))))/d^3
+168*a*b*x^(5/2)*polylog(3,I*exp(I*(c+d*x^(1/2))))/d^3-42*I*b^2*x^(5/2)*polylog(2,-exp(2*I*(c+d*x^(1/2))))/d^3
-20160*I*a*b*polylog(8,-I*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
0080*I*a*b*x*polylog(6,I*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+1/
4*a^2*x^4

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 749, 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 = {4289, 4275, 4266, 2611, 6744, 2320, 6724, 4269, 3800, 2221} \[ \int x^3 \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^4}{4}-\frac {8 i a b x^{7/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i 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} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {2 i b^2 x^{7/2}}{d} \]

[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 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 3800

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 4266

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 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 4289

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 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 \sec (c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {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) \text {Subst}\left (\int x^7 \sec (c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {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} \arctan \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) \text {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) \text {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 ) \text {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} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,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) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,-i 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,i 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 i a b x^{7/2} \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,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) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,-i 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,i 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 i a b x^{7/2} \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,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) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,-i 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,i 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 i a b x^{7/2} \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,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) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,-i 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,i 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 i a b x^{7/2} \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,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) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,-i 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,i 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 i a b x^{7/2} \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,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) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,-i 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,i 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 i a b x^{7/2} \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,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) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(7,-i 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,i 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 i a b x^{7/2} \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,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 ) \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 i a b x^{7/2} \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,i 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,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,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] (verified)

Time = 2.15 (sec) , antiderivative size = 739, normalized size of antiderivative = 0.99 \[ \int x^3 \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {-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 \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 )+112 i a b d^6 x^3 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )-112 i a b d^6 x^3 \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )-168 i b^2 d^5 x^{5/2} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )-672 a b d^5 x^{5/2} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )+672 a b d^5 x^{5/2} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )+420 b^2 d^4 x^2 \operatorname {PolyLog}\left (3,-e^{2 i \left (c+d \sqrt {x}\right )}\right )-3360 i a b d^4 x^2 \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )+3360 i a b d^4 x^2 \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )+840 i b^2 d^3 x^{3/2} \operatorname {PolyLog}\left (4,-e^{2 i \left (c+d \sqrt {x}\right )}\right )+13440 a b d^3 x^{3/2} \operatorname {PolyLog}\left (5,-i e^{i \left (c+d \sqrt {x}\right )}\right )-13440 a b d^3 x^{3/2} \operatorname {PolyLog}\left (5,i e^{i \left (c+d \sqrt {x}\right )}\right )-1260 b^2 d^2 x \operatorname {PolyLog}\left (5,-e^{2 i \left (c+d \sqrt {x}\right )}\right )+40320 i a b d^2 x \operatorname {PolyLog}\left (6,-i e^{i \left (c+d \sqrt {x}\right )}\right )-40320 i a b d^2 x \operatorname {PolyLog}\left (6,i e^{i \left (c+d \sqrt {x}\right )}\right )-1260 i b^2 d \sqrt {x} \operatorname {PolyLog}\left (6,-e^{2 i \left (c+d \sqrt {x}\right )}\right )-80640 a b d \sqrt {x} \operatorname {PolyLog}\left (7,-i e^{i \left (c+d \sqrt {x}\right )}\right )+80640 a b d \sqrt {x} \operatorname {PolyLog}\left (7,i e^{i \left (c+d \sqrt {x}\right )}\right )+630 b^2 \operatorname {PolyLog}\left (7,-e^{2 i \left (c+d \sqrt {x}\right )}\right )-80640 i a b \operatorname {PolyLog}\left (8,-i e^{i \left (c+d \sqrt {x}\right )}\right )+80640 i a b \operatorname {PolyLog}\left (8,i e^{i \left (c+d \sqrt {x}\right )}\right )+8 b^2 d^7 x^{7/2} \tan \left (c+d \sqrt {x}\right )}{4 d^8} \]

[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)

Maple [F]

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

[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)

Fricas [F]

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

[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)

Sympy [F]

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

[In]

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

[Out]

Integral(x**3*(a + b*sec(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. 6347 vs. \(2 (574) = 1148\).

Time = 0.69 (sec) , antiderivative size = 6347, normalized size of antiderivative = 8.47 \[ \int x^3 \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\text {Too large to display} \]

[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 - 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 + ((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) + (I*(d*sqrt(x) + c)^
7*a*b - 7*I*(d*sqrt(x) + c)^6*a*b*c + 21*I*(d*sqrt(x) + c)^5*a*b*c^2 - 35*I*(d*sqrt(x) + c)^4*a*b*c^3 + 35*I*(
d*sqrt(x) + c)^3*a*b*c^4 - 21*I*(d*sqrt(x) + c)^2*a*b*c^5 + 7*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 - 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 + ((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) + (I*(d*sqrt(x) + c)^7*a*b - 7*I*(d*sqrt(x) + c)^6*a*b*c +
21*I*(d*sqrt(x) + c)^5*a*b*c^2 - 35*I*(d*sqrt(x) + c)^4*a*b*c^3 + 35*I*(d*sqrt(x) + c)^3*a*b*c^4 - 21*I*(d*sq
rt(x) + c)^2*a*b*c^5 + 7*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) - 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 +
(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) -
(-80*I*(d*sqrt(x) + c)^6*b^2 + 288*I*(d*sqrt(x) + c)^5*b^2*c - 450*I*(d*sqrt(x) + c)^4*b^2*c^2 + 400*I*(d*sqrt
(x) + c)^3*b^2*c^3 - 225*I*(d*sqrt(x) + c)^2*b^2*c^4 + 90*I*(d*sqrt(x) + c)*b^2*c^5 - 15*I*b^2*c^6)*sin(2*d*sq
rt(x) + 2*c))*arctan2(sin(2*d*sqrt(x) + 2*c), cos(2*d*sqrt(x) + 2*c) + 1) + 60*((d*sqrt(x) + c)^7*b^2 - 7*(d*s
qrt(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) + 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*sqrt(x) + c)^2*b^2*c^3 + 15*(d*sqr
t(x) + c)*b^2*c^4 - 3*b^2*c^5 + (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*sqrt(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) + (16*
I*(d*sqrt(x) + c)^5*b^2 - 48*I*(d*sqrt(x) + c)^4*b^2*c + 60*I*(d*sqrt(x) + c)^3*b^2*c^2 - 40*I*(d*sqrt(x) + c)
^2*b^2*c^3 + 15*I*(d*sqrt(x) + c)*b^2*c^4 - 3*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 - 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 + ((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) + (I*(d*sqrt(x) + c)^6*a*b - 6*I*(d*sqr
t(x) + c)^5*a*b*c + 15*I*(d*sqrt(x) + c)^4*a*b*c^2 - 20*I*(d*sqrt(x) + c)^3*a*b*c^3 + 15*I*(d*sqrt(x) + c)^2*a
*b*c^4 - 6*I*(d*sqrt(x) + c)*a*b*c^5 + 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 - 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 + ((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) - (-I*(d*sqrt(x) + c)^6*a*b + 6*I*(d*sqrt(x) + c)
^5*a*b*c - 15*I*(d*sqrt(x) + c)^4*a*b*c^2 + 20*I*(d*sqrt(x) + c)^3*a*b*c^3 - 15*I*(d*sqrt(x) + c)^2*a*b*c^4 +
6*I*(d*sqrt(x) + c)*a*b*c^5 - I*a*b*c^6)*sin(2*d*sqrt(x) + 2*c))*dilog(-I*e^(I*d*sqrt(x) + I*c)) + 7*(80*I*(d*
sqrt(x) + c)^6*b^2 - 288*I*(d*sqrt(x) + c)^5*b^2*c + 450*I*(d*sqrt(x) + c)^4*b^2*c^2 - 400*I*(d*sqrt(x) + c)^3
*b^2*c^3 + 225*I*(d*sqrt(x) + c)^2*b^2*c^4 - 90*I*(d*sqrt(x) + c)*b^2*c^5 + 15*I*b^2*c^6 + (80*I*(d*sqrt(x) +
c)^6*b^2 - 288*I*(d*sqrt(x) + c)^5*b^2*c + 450*I*(d*sqrt(x) + c)^4*b^2*c^2 - 400*I*(d*sqrt(x) + c)^3*b^2*c^3 +
225*I*(d*sqrt(x) + c)^2*b^2*c^4 - 90*I*(d*sqrt(x) + c)*b^2*c^5 + 15*I*b^2*c^6)*cos(2*d*sqrt(x) + 2*c) - (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)*sin(2*d*sqrt(x) + 2*c))*log(co
s(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 - 7*I*(d*sqrt(x) + c)^6*a*b*c + 21*I*(d*sqrt(x) + c)^5*a*b*c^2 - 35*I*(d*sqrt(x) + c)^4*a*b*c^3 + 35*I*(d*sq
rt(x) + c)^3*a*b*c^4 - 21*I*(d*sqrt(x) + c)^2*a*b*c^5 + 7*I*(d*sqrt(x) + c)*a*b*c^6 + (I*(d*sqrt(x) + c)^7*a*b
- 7*I*(d*sqrt(x) + c)^6*a*b*c + 21*I*(d*sqrt(x) + c)^5*a*b*c^2 - 35*I*(d*sqrt(x) + c)^4*a*b*c^3 + 35*I*(d*sqr
t(x) + c)^3*a*b*c^4 - 21*I*(d*sqrt(x) + c)^2*a*b*c^5 + 7*I*(d*sqrt(x) + c)*a*b*c^6)*cos(2*d*sqrt(x) + 2*c) - (
(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 + 7*I*(d*sqrt(x) + c)^6*a*b*c - 21*I*(d*sqrt(x) + c)^5*a*b*c^2 + 35*I*(d*sqrt(x) + c)^4*a*b*c^3 - 35*I*(d*
sqrt(x) + c)^3*a*b*c^4 + 21*I*(d*sqrt(x) + c)^2*a*b*c^5 - 7*I*(d*sqrt(x) + c)*a*b*c^6 + (-I*(d*sqrt(x) + c)^7*
a*b + 7*I*(d*sqrt(x) + c)^6*a*b*c - 21*I*(d*sqrt(x) + c)^5*a*b*c^2 + 35*I*(d*sqrt(x) + c)^4*a*b*c^3 - 35*I*(d*
sqrt(x) + c)^3*a*b*c^4 + 21*I*(d*sqrt(x) + c)^2*a*b*c^5 - 7*I*(d*sqrt(x) + c)*a*b*c^6)*cos(2*d*sqrt(x) + 2*c)
+ ((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*sq
rt(x) + 2*c) + I*a*b*sin(2*d*sqrt(x) + 2*c) + a*b)*polylog(8, I*e^(I*d*sqrt(x) + I*c)) + 302400*(a*b*cos(2*d*s
qrt(x) + 2*c) + I*a*b*sin(2*d*sqrt(x) + 2*c) + a*b)*polylog(8, -I*e^(I*d*sqrt(x) + I*c)) + 12600*(I*b^2*cos(2*
d*sqrt(x) + 2*c) - b^2*sin(2*d*sqrt(x) + 2*c) + I*b^2)*polylog(7, -e^(2*I*d*sqrt(x) + 2*I*c)) + 302400*(I*(d*s
qrt(x) + c)*a*b - I*a*b*c + (I*(d*sqrt(x) + c)*a*b - I*a*b*c)*cos(2*d*sqrt(x) + 2*c) - ((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 + I*a*b*c
+ (-I*(d*sqrt(x) + c)*a*b + I*a*b*c)*cos(2*d*sqrt(x) + 2*c) + ((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)) + 5040*(5*(d*sqrt(x) + c)*b^2 - 3*b^2*c + (5*(d*sqrt(x) + c)*b^2 -
3*b^2*c)*cos(2*d*sqrt(x) + 2*c) + (5*I*(d*sqrt(x) + c)*b^2 - 3*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 - 2*(d*sqrt(x) + c)*a*b*c + a*b*c^2 + ((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) + (I*(d*sqrt(x) + c)^2*a*b - 2*I*(d*sqrt(x
) + c)*a*b*c + 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 - 2*(d*sqrt(x) + c)*a*b*c + a*b*c^2 + ((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) - (-I*(d*sqrt(x) + c)^2*a*b + 2*I*(d*sqrt(x) + c)*a*b*c - I*a*b*c^2)*sin(2*d*sqrt(x) + 2*c)
)*polylog(6, -I*e^(I*d*sqrt(x) + I*c)) + 630*(-40*I*(d*sqrt(x) + c)^2*b^2 + 48*I*(d*sqrt(x) + c)*b^2*c - 15*I*
b^2*c^2 + (-40*I*(d*sqrt(x) + c)^2*b^2 + 48*I*(d*sqrt(x) + c)*b^2*c - 15*I*b^2*c^2)*cos(2*d*sqrt(x) + 2*c) + (
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 + 3*I*(d*sqrt(x) + c)^2*a*b*c - 3*I*(d*sqrt(x) + c)*a*b*c
^2 + I*a*b*c^3 + (-I*(d*sqrt(x) + c)^3*a*b + 3*I*(d*sqrt(x) + c)^2*a*b*c - 3*I*(d*sqrt(x) + c)*a*b*c^2 + I*a*b
*c^3)*cos(2*d*sqrt(x) + 2*c) + ((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 - 3*I*
(d*sqrt(x) + c)^2*a*b*c + 3*I*(d*sqrt(x) + c)*a*b*c^2 - I*a*b*c^3 + (I*(d*sqrt(x) + c)^3*a*b - 3*I*(d*sqrt(x)
+ c)^2*a*b*c + 3*I*(d*sqrt(x) + c)*a*b*c^2 - I*a*b*c^3)*cos(2*d*sqrt(x) + 2*c) - ((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)) - 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 + (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) - (-40*I*(d*sqrt(x) + c)^3*b^2 + 72*I*(d*sqrt(x) + c)^2*b^2*c - 45*I*(d*sqrt(x) + c)*b^2*c^2
+ 10*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
- 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 + ((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)*c
os(2*d*sqrt(x) + 2*c) - (-I*(d*sqrt(x) + c)^4*a*b + 4*I*(d*sqrt(x) + c)^3*a*b*c - 6*I*(d*sqrt(x) + c)^2*a*b*c^
2 + 4*I*(d*sqrt(x) + c)*a*b*c^3 - I*a*b*c^4)*sin(2*d*sqrt(x) + 2*c))*polylog(4, I*e^(I*d*sqrt(x) + I*c)) + 126
00*((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 + ((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) + (I*(d*sqrt(x) + c)^4*a*b - 4*I*(d*sqrt(x) + c)^3*a*b*c + 6*I*
(d*sqrt(x) + c)^2*a*b*c^2 - 4*I*(d*sqrt(x) + c)*a*b*c^3 + I*a*b*c^4)*sin(2*d*sqrt(x) + 2*c))*polylog(4, -I*e^(
I*d*sqrt(x) + I*c)) + 105*(80*I*(d*sqrt(x) + c)^4*b^2 - 192*I*(d*sqrt(x) + c)^3*b^2*c + 180*I*(d*sqrt(x) + c)^
2*b^2*c^2 - 80*I*(d*sqrt(x) + c)*b^2*c^3 + 15*I*b^2*c^4 + (80*I*(d*sqrt(x) + c)^4*b^2 - 192*I*(d*sqrt(x) + c)^
3*b^2*c + 180*I*(d*sqrt(x) + c)^2*b^2*c^2 - 80*I*(d*sqrt(x) + c)*b^2*c^3 + 15*I*b^2*c^4)*cos(2*d*sqrt(x) + 2*c
) - (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 - 5*I*(d*sqrt(x) + c)^4*a*b*c + 10*I*(d*sqrt(x) + c)^3*a*b*c^2 - 10*I*(d*sqrt(x) + c)^2*a*b*c^3 + 5*
I*(d*sqrt(x) + c)*a*b*c^4 - I*a*b*c^5 + (I*(d*sqrt(x) + c)^5*a*b - 5*I*(d*sqrt(x) + c)^4*a*b*c + 10*I*(d*sqrt(
x) + c)^3*a*b*c^2 - 10*I*(d*sqrt(x) + c)^2*a*b*c^3 + 5*I*(d*sqrt(x) + c)*a*b*c^4 - I*a*b*c^5)*cos(2*d*sqrt(x)
+ 2*c) - ((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 + 5*I*(d*sqrt(x) + c)^4*a*b*c - 10*I*(d*sqrt(x) + c)^3*a*b*c^2 + 10*I*(d*sqr
t(x) + c)^2*a*b*c^3 - 5*I*(d*sqrt(x) + c)*a*b*c^4 + I*a*b*c^5 + (-I*(d*sqrt(x) + c)^5*a*b + 5*I*(d*sqrt(x) + c
)^4*a*b*c - 10*I*(d*sqrt(x) + c)^3*a*b*c^2 + 10*I*(d*sqrt(x) + c)^2*a*b*c^3 - 5*I*(d*sqrt(x) + c)*a*b*c^4 + I*
a*b*c^5)*cos(2*d*sqrt(x) + 2*c) + ((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 - 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))/(-30*I*cos(2*d*sqrt(x) + 2*c) + 30*sin(2*d*sqrt(x) + 2*c
) - 30*I))/d^8

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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