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

Optimal. Leaf size=749 \[ \frac {a^2 x^4}{4}-\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 {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 {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 {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 {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 {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 {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 {8 i a b x^{7/2} \tan ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {315 b^2 \text {Li}_7\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {315 i b^2 \sqrt {x} \text {Li}_6\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {315 b^2 x \text {Li}_5\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\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 {105 b^2 x^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\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 {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} \]

[Out]

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*polyl
og(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+315/2*b^2*polylog(7,-exp(
2*I*(c+d*x^(1/2))))/d^8+1/4*a^2*x^4-8*I*a*b*x^(7/2)*arctan(exp(I*(c+d*x^(1/2))))/d-28*I*a*b*x^3*polylog(2,I*ex
p(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+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^6-10080*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+20160*I*a*b*polylog(8,I*exp(I*(c+d*x^(1/2))))/d^8-168*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+3360*a*b*x^(3/2)*polylog(5,-I*exp(I*(c+d*x^(1/2)
)))/d^5-3360*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^7-42*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

________________________________________________________________________________________

Rubi [A]  time = 0.86, antiderivative size = 749, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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]

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 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 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 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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 2.20, size = 739, normalized size = 0.99 \[ \frac {a^2 d^8 x^4-32 i a b d^7 x^{7/2} \tan ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )+112 i a b d^6 x^3 \text {Li}_2\left (-i e^{i \left (c+d \sqrt {x}\right )}\right )-112 i a b d^6 x^3 \text {Li}_2\left (i e^{i \left (c+d \sqrt {x}\right )}\right )-672 a b d^5 x^{5/2} \text {Li}_3\left (-i e^{i \left (c+d \sqrt {x}\right )}\right )+672 a b d^5 x^{5/2} \text {Li}_3\left (i e^{i \left (c+d \sqrt {x}\right )}\right )-3360 i a b d^4 x^2 \text {Li}_4\left (-i e^{i \left (c+d \sqrt {x}\right )}\right )+3360 i a b d^4 x^2 \text {Li}_4\left (i e^{i \left (c+d \sqrt {x}\right )}\right )+13440 a b d^3 x^{3/2} \text {Li}_5\left (-i e^{i \left (c+d \sqrt {x}\right )}\right )-13440 a b d^3 x^{3/2} \text {Li}_5\left (i e^{i \left (c+d \sqrt {x}\right )}\right )+40320 i a b d^2 x \text {Li}_6\left (-i e^{i \left (c+d \sqrt {x}\right )}\right )-40320 i a b d^2 x \text {Li}_6\left (i e^{i \left (c+d \sqrt {x}\right )}\right )-80640 a b d \sqrt {x} \text {Li}_7\left (-i e^{i \left (c+d \sqrt {x}\right )}\right )+80640 a b d \sqrt {x} \text {Li}_7\left (i e^{i \left (c+d \sqrt {x}\right )}\right )-80640 i a b \text {Li}_8\left (-i e^{i \left (c+d \sqrt {x}\right )}\right )+80640 i a b \text {Li}_8\left (i e^{i \left (c+d \sqrt {x}\right )}\right )+8 b^2 d^7 x^{7/2} \tan \left (c+d \sqrt {x}\right )+56 b^2 d^6 x^3 \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )-168 i b^2 d^5 x^{5/2} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )+420 b^2 d^4 x^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )+840 i b^2 d^3 x^{3/2} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )-1260 b^2 d^2 x \text {Li}_5\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )-1260 i b^2 d \sqrt {x} \text {Li}_6\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )+630 b^2 \text {Li}_7\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )-8 i b^2 d^7 x^{7/2}}{4 d^8} \]

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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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{3} \sec \left (d \sqrt {x} + c\right )^{2} + 2 \, a b x^{3} \sec \left (d \sqrt {x} + c\right ) + a^{2} x^{3}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{3}\,{d x} \]

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)

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maple [F]  time = 1.38, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a +b \sec \left (c +d \sqrt {x}\right )\right )^{2}\, dx \]

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 = 1.62, size = 6373, normalized size = 8.51 \[ \text {result too large to display} \]

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) + I*a*b*sin(2*d*sqrt(x) + 2*c) + a*b)*polylog(8, I*e^(I*d*sqr
t(x) + I*c)) + 302400*(a*b*cos(2*d*sqrt(x) + 2*c) + I*a*b*sin(2*d*sqrt(x) + 2*c) + a*b)*polylog(8, -I*e^(I*d*s
qrt(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)*polylo
g(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))*po
lylog(6, -e^(2*I*d*sqrt(x) + 2*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)) - (151200*(d*sqrt(x) + c)^2*a*b - 302400*(d*sqrt(x) + c)*a*b*c + 151200*a*b*c^2 + 1
51200*((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*sqrt(x) + 2*c) - 630*(40*(d*sq
rt(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 + 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 + 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)) - (-5
0400*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 - 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*s
qrt(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 - 4200*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*sqrt(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 + 12
600*((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 + 7560
0*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))*pol
ylog(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*sqr
t(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)) - (
-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 - 1
8900*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) + 1
05*(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*sq
rt(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 + 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*sqr
t(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*s
qrt(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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a + b \sec {\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]

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]

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

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