\(\int x^2 (a+b \sec (c+d \sqrt {x}))^2 \, dx\) [37]

Optimal result

Integrand size = 20, antiderivative size = 551 \[ \int x^2 \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {2 i b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 i a b x^{5/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 b^2 x^2 \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {30 b^2 x \operatorname {PolyLog}\left (3,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {240 i a b x \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {240 i a b x \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (4,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {15 b^2 \operatorname {PolyLog}\left (5,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {480 i a b \operatorname {PolyLog}\left (6,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {480 i a b \operatorname {PolyLog}\left (6,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {2 b^2 x^{5/2} \tan \left (c+d \sqrt {x}\right )}{d} \]

[Out]

20*I*a*b*x^2*polylog(2,-I*exp(I*(c+d*x^(1/2))))/d^2+1/3*a^2*x^3-20*I*b^2*x^(3/2)*polylog(2,-exp(2*I*(c+d*x^(1/
2))))/d^3+10*b^2*x^2*ln(1+exp(2*I*(c+d*x^(1/2))))/d^2+240*I*a*b*x*polylog(4,I*exp(I*(c+d*x^(1/2))))/d^4-2*I*b^
2*x^(5/2)/d-8*I*a*b*x^(5/2)*arctan(exp(I*(c+d*x^(1/2))))/d-80*a*b*x^(3/2)*polylog(3,-I*exp(I*(c+d*x^(1/2))))/d
^3+80*a*b*x^(3/2)*polylog(3,I*exp(I*(c+d*x^(1/2))))/d^3+30*b^2*x*polylog(3,-exp(2*I*(c+d*x^(1/2))))/d^4-20*I*a
*b*x^2*polylog(2,I*exp(I*(c+d*x^(1/2))))/d^2+30*I*b^2*polylog(4,-exp(2*I*(c+d*x^(1/2))))*x^(1/2)/d^5-15*b^2*po
lylog(5,-exp(2*I*(c+d*x^(1/2))))/d^6-480*I*a*b*polylog(6,I*exp(I*(c+d*x^(1/2))))/d^6-240*I*a*b*x*polylog(4,-I*
exp(I*(c+d*x^(1/2))))/d^4+480*I*a*b*polylog(6,-I*exp(I*(c+d*x^(1/2))))/d^6+480*a*b*polylog(5,-I*exp(I*(c+d*x^(
1/2))))*x^(1/2)/d^5-480*a*b*polylog(5,I*exp(I*(c+d*x^(1/2))))*x^(1/2)/d^5+2*b^2*x^(5/2)*tan(c+d*x^(1/2))/d

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.00, number of steps used = 24, 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} \[ \int x^2 \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^3}{3}-\frac {8 i a b x^{5/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {480 i a b \operatorname {PolyLog}\left (6,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {480 i a b \operatorname {PolyLog}\left (6,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {240 i a b x \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {240 i a b x \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {15 b^2 \operatorname {PolyLog}\left (5,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (4,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {30 b^2 x \operatorname {PolyLog}\left (3,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {10 b^2 x^2 \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {2 b^2 x^{5/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {2 i b^2 x^{5/2}}{d} \]

[In]

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

[Out]

((-2*I)*b^2*x^(5/2))/d + (a^2*x^3)/3 - ((8*I)*a*b*x^(5/2)*ArcTan[E^(I*(c + d*Sqrt[x]))])/d + (10*b^2*x^2*Log[1
+ E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((20*I)*a*b*x^2*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((20*I)*a*b
*x^2*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - ((20*I)*b^2*x^(3/2)*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^
3 - (80*a*b*x^(3/2)*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (80*a*b*x^(3/2)*PolyLog[3, I*E^(I*(c + d*Sqr
t[x]))])/d^3 + (30*b^2*x*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))])/d^4 - ((240*I)*a*b*x*PolyLog[4, (-I)*E^(I*(c
+ d*Sqrt[x]))])/d^4 + ((240*I)*a*b*x*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))])/d^4 + ((30*I)*b^2*Sqrt[x]*PolyLog[4,
-E^((2*I)*(c + d*Sqrt[x]))])/d^5 + (480*a*b*Sqrt[x]*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))])/d^5 - (480*a*b*Sq
rt[x]*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))])/d^5 - (15*b^2*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))])/d^6 + ((480*I
)*a*b*PolyLog[6, (-I)*E^(I*(c + d*Sqrt[x]))])/d^6 - ((480*I)*a*b*PolyLog[6, I*E^(I*(c + d*Sqrt[x]))])/d^6 + (2
*b^2*x^(5/2)*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^5 (a+b \sec (c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^5+2 a b x^5 \sec (c+d x)+b^2 x^5 \sec ^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^3}{3}+(4 a b) \text {Subst}\left (\int x^5 \sec (c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^5 \sec ^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^3}{3}-\frac {8 i a b x^{5/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {2 b^2 x^{5/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {(20 a b) \text {Subst}\left (\int x^4 \log \left (1-i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(20 a b) \text {Subst}\left (\int x^4 \log \left (1+i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}-\frac {\left (10 b^2\right ) \text {Subst}\left (\int x^4 \tan (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 i a b x^{5/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {2 b^2 x^{5/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {(80 i a b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(80 i a b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {\left (20 i b^2\right ) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} x^4}{1+e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 i a b x^{5/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 b^2 x^2 \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {2 b^2 x^{5/2} \tan \left (c+d \sqrt {x}\right )}{d}+\frac {(240 a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {(240 a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (40 b^2\right ) \text {Subst}\left (\int x^3 \log \left (1+e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = -\frac {2 i b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 i a b x^{5/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 b^2 x^2 \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {240 i a b x \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {240 i a b x \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {2 b^2 x^{5/2} \tan \left (c+d \sqrt {x}\right )}{d}+\frac {(480 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(480 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}+\frac {\left (60 i b^2\right ) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = -\frac {2 i b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 i a b x^{5/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 b^2 x^2 \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {30 b^2 x \operatorname {PolyLog}\left (3,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {240 i a b x \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {240 i a b x \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {2 b^2 x^{5/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {(480 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (5,-i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5}+\frac {(480 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (5,i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5}-\frac {\left (60 b^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4} \\ & = -\frac {2 i b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 i a b x^{5/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 b^2 x^2 \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {30 b^2 x \operatorname {PolyLog}\left (3,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {240 i a b x \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {240 i a b x \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (4,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {2 b^2 x^{5/2} \tan \left (c+d \sqrt {x}\right )}{d}+\frac {(480 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(5,-i x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {(480 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(5,i x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {\left (30 i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (4,-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5} \\ & = -\frac {2 i b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 i a b x^{5/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 b^2 x^2 \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {30 b^2 x \operatorname {PolyLog}\left (3,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {240 i a b x \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {240 i a b x \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (4,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {480 i a b \operatorname {PolyLog}\left (6,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {480 i a b \operatorname {PolyLog}\left (6,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {2 b^2 x^{5/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,-x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6} \\ & = -\frac {2 i b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 i a b x^{5/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 b^2 x^2 \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {30 b^2 x \operatorname {PolyLog}\left (3,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {240 i a b x \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {240 i a b x \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (4,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {15 b^2 \operatorname {PolyLog}\left (5,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {480 i a b \operatorname {PolyLog}\left (6,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {480 i a b \operatorname {PolyLog}\left (6,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {2 b^2 x^{5/2} \tan \left (c+d \sqrt {x}\right )}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 543, normalized size of antiderivative = 0.99 \[ \int x^2 \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {-6 i b^2 d^5 x^{5/2}+a^2 d^6 x^3-24 i a b d^5 x^{5/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )+30 b^2 d^4 x^2 \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )+60 i a b d^4 x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )-60 i a b d^4 x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )-60 i b^2 d^3 x^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )-240 a b d^3 x^{3/2} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )+240 a b d^3 x^{3/2} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )+90 b^2 d^2 x \operatorname {PolyLog}\left (3,-e^{2 i \left (c+d \sqrt {x}\right )}\right )-720 i a b d^2 x \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )+720 i a b d^2 x \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )+90 i b^2 d \sqrt {x} \operatorname {PolyLog}\left (4,-e^{2 i \left (c+d \sqrt {x}\right )}\right )+1440 a b d \sqrt {x} \operatorname {PolyLog}\left (5,-i e^{i \left (c+d \sqrt {x}\right )}\right )-1440 a b d \sqrt {x} \operatorname {PolyLog}\left (5,i e^{i \left (c+d \sqrt {x}\right )}\right )-45 b^2 \operatorname {PolyLog}\left (5,-e^{2 i \left (c+d \sqrt {x}\right )}\right )+1440 i a b \operatorname {PolyLog}\left (6,-i e^{i \left (c+d \sqrt {x}\right )}\right )-1440 i a b \operatorname {PolyLog}\left (6,i e^{i \left (c+d \sqrt {x}\right )}\right )+6 b^2 d^5 x^{5/2} \tan \left (c+d \sqrt {x}\right )}{3 d^6} \]

[In]

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

[Out]

((-6*I)*b^2*d^5*x^(5/2) + a^2*d^6*x^3 - (24*I)*a*b*d^5*x^(5/2)*ArcTan[E^(I*(c + d*Sqrt[x]))] + 30*b^2*d^4*x^2*
Log[1 + E^((2*I)*(c + d*Sqrt[x]))] + (60*I)*a*b*d^4*x^2*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))] - (60*I)*a*b*d^
4*x^2*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))] - (60*I)*b^2*d^3*x^(3/2)*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))] - 24
0*a*b*d^3*x^(3/2)*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))] + 240*a*b*d^3*x^(3/2)*PolyLog[3, I*E^(I*(c + d*Sqrt[x
]))] + 90*b^2*d^2*x*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))] - (720*I)*a*b*d^2*x*PolyLog[4, (-I)*E^(I*(c + d*Sqr
t[x]))] + (720*I)*a*b*d^2*x*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))] + (90*I)*b^2*d*Sqrt[x]*PolyLog[4, -E^((2*I)*(c
+ d*Sqrt[x]))] + 1440*a*b*d*Sqrt[x]*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))] - 1440*a*b*d*Sqrt[x]*PolyLog[5, I*
E^(I*(c + d*Sqrt[x]))] - 45*b^2*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))] + (1440*I)*a*b*PolyLog[6, (-I)*E^(I*(c
+ d*Sqrt[x]))] - (1440*I)*a*b*PolyLog[6, I*E^(I*(c + d*Sqrt[x]))] + 6*b^2*d^5*x^(5/2)*Tan[c + d*Sqrt[x]])/(3*d
^6)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x**2*(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. 3879 vs. \(2 (422) = 844\).

Time = 0.61 (sec) , antiderivative size = 3879, normalized size of antiderivative = 7.04 \[ \int x^2 \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3*((d*sqrt(x) + c)^6*a^2 - 6*(d*sqrt(x) + c)^5*a^2*c + 15*(d*sqrt(x) + c)^4*a^2*c^2 - 20*(d*sqrt(x) + c)^3*a
^2*c^3 + 15*(d*sqrt(x) + c)^2*a^2*c^4 - 6*(d*sqrt(x) + c)*a^2*c^5 - 12*a*b*c^5*log(sec(d*sqrt(x) + c) + tan(d*
sqrt(x) + c)) - 6*(12*b^2*c^5 + 12*((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 + ((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)*cos(2*d*sq
rt(x) + 2*c) + (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)*sin(2*d*sqrt(x) + 2*c))*arctan2(cos(d*sqrt(x) + c), s
in(d*sqrt(x) + c) + 1) + 12*((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 + ((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)*cos(2*d*sqrt(x) +
2*c) + (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)*sin(2*d*sqrt(x) + 2*c))*arctan2(cos(d*sqrt(x) + c), -sin(d*s
qrt(x) + c) + 1) - 10*(6*(d*sqrt(x) + c)^4*b^2 - 16*(d*sqrt(x) + c)^3*b^2*c + 18*(d*sqrt(x) + c)^2*b^2*c^2 - 1
2*(d*sqrt(x) + c)*b^2*c^3 + 3*b^2*c^4 + (6*(d*sqrt(x) + c)^4*b^2 - 16*(d*sqrt(x) + c)^3*b^2*c + 18*(d*sqrt(x)
+ c)^2*b^2*c^2 - 12*(d*sqrt(x) + c)*b^2*c^3 + 3*b^2*c^4)*cos(2*d*sqrt(x) + 2*c) - (-6*I*(d*sqrt(x) + c)^4*b^2
+ 16*I*(d*sqrt(x) + c)^3*b^2*c - 18*I*(d*sqrt(x) + c)^2*b^2*c^2 + 12*I*(d*sqrt(x) + c)*b^2*c^3 - 3*I*b^2*c^4)*
sin(2*d*sqrt(x) + 2*c))*arctan2(sin(2*d*sqrt(x) + 2*c), cos(2*d*sqrt(x) + 2*c) + 1) + 12*((d*sqrt(x) + c)^5*b^
2 - 5*(d*sqrt(x) + c)^4*b^2*c + 10*(d*sqrt(x) + c)^3*b^2*c^2 - 10*(d*sqrt(x) + c)^2*b^2*c^3 + 5*(d*sqrt(x) + c
)*b^2*c^4)*cos(2*d*sqrt(x) + 2*c) + 60*(2*(d*sqrt(x) + c)^3*b^2 - 4*(d*sqrt(x) + c)^2*b^2*c + 3*(d*sqrt(x) + c
)*b^2*c^2 - b^2*c^3 + (2*(d*sqrt(x) + c)^3*b^2 - 4*(d*sqrt(x) + c)^2*b^2*c + 3*(d*sqrt(x) + c)*b^2*c^2 - b^2*c
^3)*cos(2*d*sqrt(x) + 2*c) + (2*I*(d*sqrt(x) + c)^3*b^2 - 4*I*(d*sqrt(x) + c)^2*b^2*c + 3*I*(d*sqrt(x) + c)*b^
2*c^2 - I*b^2*c^3)*sin(2*d*sqrt(x) + 2*c))*dilog(-e^(2*I*d*sqrt(x) + 2*I*c)) + 60*((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))*dilog(I*e^(I*d*sqrt(x) + I*c)) - 60*((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))*dilog(-I*e^(I*d*sqrt(x) + I*c))
+ 5*(6*I*(d*sqrt(x) + c)^4*b^2 - 16*I*(d*sqrt(x) + c)^3*b^2*c + 18*I*(d*sqrt(x) + c)^2*b^2*c^2 - 12*I*(d*sqrt
(x) + c)*b^2*c^3 + 3*I*b^2*c^4 + (6*I*(d*sqrt(x) + c)^4*b^2 - 16*I*(d*sqrt(x) + c)^3*b^2*c + 18*I*(d*sqrt(x) +
c)^2*b^2*c^2 - 12*I*(d*sqrt(x) + c)*b^2*c^3 + 3*I*b^2*c^4)*cos(2*d*sqrt(x) + 2*c) - (6*(d*sqrt(x) + c)^4*b^2
- 16*(d*sqrt(x) + c)^3*b^2*c + 18*(d*sqrt(x) + c)^2*b^2*c^2 - 12*(d*sqrt(x) + c)*b^2*c^3 + 3*b^2*c^4)*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) + 6*(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*(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)*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)*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) + 6*(-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*(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)
*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)*sin(2*d*sqrt(x) + 2*c))*log(cos(d*sqrt(x) + c)^2 + s
in(d*sqrt(x) + c)^2 - 2*sin(d*sqrt(x) + c) + 1) + 1440*(a*b*cos(2*d*sqrt(x) + 2*c) + I*a*b*sin(2*d*sqrt(x) + 2
*c) + a*b)*polylog(6, I*e^(I*d*sqrt(x) + I*c)) - 1440*(a*b*cos(2*d*sqrt(x) + 2*c) + I*a*b*sin(2*d*sqrt(x) + 2*
c) + a*b)*polylog(6, -I*e^(I*d*sqrt(x) + I*c)) + 90*(-I*b^2*cos(2*d*sqrt(x) + 2*c) + b^2*sin(2*d*sqrt(x) + 2*c
) - I*b^2)*polylog(5, -e^(2*I*d*sqrt(x) + 2*I*c)) + 1440*(-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(5, I*
e^(I*d*sqrt(x) + I*c)) + 1440*(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*sqr
t(x) + 2*c) - ((d*sqrt(x) + c)*a*b - a*b*c)*sin(2*d*sqrt(x) + 2*c))*polylog(5, -I*e^(I*d*sqrt(x) + I*c)) - 60*
(3*(d*sqrt(x) + c)*b^2 - 2*b^2*c + (3*(d*sqrt(x) + c)*b^2 - 2*b^2*c)*cos(2*d*sqrt(x) + 2*c) - (-3*I*(d*sqrt(x)
+ c)*b^2 + 2*I*b^2*c)*sin(2*d*sqrt(x) + 2*c))*polylog(4, -e^(2*I*d*sqrt(x) + 2*I*c)) - 720*((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))*p
olylog(4, I*e^(I*d*sqrt(x) + I*c)) + 720*((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(4, -I*e^(I*d*sqrt(x) + I*c)) + 30*(6*I*(d*sqr
t(x) + c)^2*b^2 - 8*I*(d*sqrt(x) + c)*b^2*c + 3*I*b^2*c^2 + (6*I*(d*sqrt(x) + c)^2*b^2 - 8*I*(d*sqrt(x) + c)*b
^2*c + 3*I*b^2*c^2)*cos(2*d*sqrt(x) + 2*c) - (6*(d*sqrt(x) + c)^2*b^2 - 8*(d*sqrt(x) + c)*b^2*c + 3*b^2*c^2)*s
in(2*d*sqrt(x) + 2*c))*polylog(3, -e^(2*I*d*sqrt(x) + 2*I*c)) + 240*(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(3, I*e^(I*d*sqrt(x) + I*c))
+ 240*(-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*sq
rt(x) + 2*c))*polylog(3, -I*e^(I*d*sqrt(x) + I*c)) + 12*(I*(d*sqrt(x) + c)^5*b^2 - 5*I*(d*sqrt(x) + c)^4*b^2*c
+ 10*I*(d*sqrt(x) + c)^3*b^2*c^2 - 10*I*(d*sqrt(x) + c)^2*b^2*c^3 + 5*I*(d*sqrt(x) + c)*b^2*c^4)*sin(2*d*sqrt
(x) + 2*c))/(-6*I*cos(2*d*sqrt(x) + 2*c) + 6*sin(2*d*sqrt(x) + 2*c) - 6*I))/d^6

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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