\(\int \frac {x^2}{(a+b \tan (c+d \sqrt {x}))^2} \, dx\) [42]

Optimal result

Integrand size = 20, antiderivative size = 1147 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {4 i b^2 x^{5/2}}{\left (a^2+b^2\right )^2 d}+\frac {4 b^2 x^{5/2}}{(a+i b) (i a+b)^2 d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {10 b^2 x^2 \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {10 b x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {10 b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {30 b^2 x \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac {20 b x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac {30 b x \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {30 b^2 x \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac {15 b^2 \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}-\frac {30 b \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}+\frac {15 b \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^6}-\frac {15 b^2 \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6} \]

[Out]

-20*I*b^2*x^(3/2)*polylog(3,-(a-I*b)*exp(2*I*(c+d*x^(1/2)))/(a+I*b))/(a^2+b^2)^2/d^3+4*b^2*x^(5/2)/(a+I*b)/(I*
a+b)^2/d/(I*a-b+(I*a+b)*exp(2*I*(c+d*x^(1/2))))+1/3*x^3/(a-I*b)^2+4/3*b*x^3/(I*a-b)/(a-I*b)^2-4/3*b^2*x^3/(a^2
+b^2)^2+10*b^2*x^2*ln(1+(a-I*b)*exp(2*I*(c+d*x^(1/2)))/(a+I*b))/(a^2+b^2)^2/d^2+4*b*x^(5/2)*ln(1+(a-I*b)*exp(2
*I*(c+d*x^(1/2)))/(a+I*b))/(a-I*b)^2/(a+I*b)/d-4*I*b^2*x^(5/2)/(a^2+b^2)^2/d+30*I*b^2*polylog(4,-(a-I*b)*exp(2
*I*(c+d*x^(1/2)))/(a+I*b))*x^(1/2)/(a^2+b^2)^2/d^5+10*b*x^2*polylog(2,-(a-I*b)*exp(2*I*(c+d*x^(1/2)))/(a+I*b))
/(I*a-b)/(a-I*b)^2/d^2-10*b^2*x^2*polylog(2,-(a-I*b)*exp(2*I*(c+d*x^(1/2)))/(a+I*b))/(a^2+b^2)^2/d^2+30*b^2*x*
polylog(3,-(a-I*b)*exp(2*I*(c+d*x^(1/2)))/(a+I*b))/(a^2+b^2)^2/d^4+20*b*x^(3/2)*polylog(3,-(a-I*b)*exp(2*I*(c+
d*x^(1/2)))/(a+I*b))/(a-I*b)^2/(a+I*b)/d^3+30*I*b^2*polylog(5,-(a-I*b)*exp(2*I*(c+d*x^(1/2)))/(a+I*b))*x^(1/2)
/(a^2+b^2)^2/d^5-30*b*x*polylog(4,-(a-I*b)*exp(2*I*(c+d*x^(1/2)))/(a+I*b))/(I*a-b)/(a-I*b)^2/d^4+30*b^2*x*poly
log(4,-(a-I*b)*exp(2*I*(c+d*x^(1/2)))/(a+I*b))/(a^2+b^2)^2/d^4-15*b^2*polylog(5,-(a-I*b)*exp(2*I*(c+d*x^(1/2))
)/(a+I*b))/(a^2+b^2)^2/d^6+15*b*polylog(6,-(a-I*b)*exp(2*I*(c+d*x^(1/2)))/(a+I*b))/(I*a-b)/(a-I*b)^2/d^6-15*b^
2*polylog(6,-(a-I*b)*exp(2*I*(c+d*x^(1/2)))/(a+I*b))/(a^2+b^2)^2/d^6-4*I*b^2*x^(5/2)*ln(1+(a-I*b)*exp(2*I*(c+d
*x^(1/2)))/(a+I*b))/(a^2+b^2)^2/d-30*b*polylog(5,-(a-I*b)*exp(2*I*(c+d*x^(1/2)))/(a+I*b))*x^(1/2)/(a-I*b)^2/(a
+I*b)/d^5-20*I*b^2*x^(3/2)*polylog(2,-(a-I*b)*exp(2*I*(c+d*x^(1/2)))/(a+I*b))/(a^2+b^2)^2/d^3

Rubi [A] (verified)

Time = 2.72 (sec) , antiderivative size = 1147, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3832, 3815, 2216, 2215, 2221, 2611, 6744, 2320, 6724, 2222} \[ \int \frac {x^2}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {4 b x^3}{3 (i a-b) (a-i b)^2}+\frac {x^3}{3 (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {4 b \log \left (\frac {e^{2 i \left (c+d \sqrt {x}\right )} (a-i b)}{a+i b}+1\right ) x^{5/2}}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 \log \left (\frac {e^{2 i \left (c+d \sqrt {x}\right )} (a-i b)}{a+i b}+1\right ) x^{5/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 i b^2 x^{5/2}}{\left (a^2+b^2\right )^2 d}+\frac {4 b^2 x^{5/2}}{(a+i b) (i a+b)^2 d \left (i a+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}-b\right )}+\frac {10 b^2 \log \left (\frac {e^{2 i \left (c+d \sqrt {x}\right )} (a-i b)}{a+i b}+1\right ) x^2}{\left (a^2+b^2\right )^2 d^2}+\frac {10 b \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right ) x^2}{(i a-b) (a-i b)^2 d^2}-\frac {10 b^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right ) x^2}{\left (a^2+b^2\right )^2 d^2}-\frac {20 i b^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right ) x^{3/2}}{\left (a^2+b^2\right )^2 d^3}+\frac {20 b \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right ) x^{3/2}}{(a-i b)^2 (a+i b) d^3}-\frac {20 i b^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right ) x^{3/2}}{\left (a^2+b^2\right )^2 d^3}+\frac {30 b^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right ) x}{\left (a^2+b^2\right )^2 d^4}-\frac {30 b \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right ) x}{(i a-b) (a-i b)^2 d^4}+\frac {30 b^2 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right ) x}{\left (a^2+b^2\right )^2 d^4}+\frac {30 i b^2 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right ) \sqrt {x}}{\left (a^2+b^2\right )^2 d^5}-\frac {30 b \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right ) \sqrt {x}}{(a-i b)^2 (a+i b) d^5}+\frac {30 i b^2 \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right ) \sqrt {x}}{\left (a^2+b^2\right )^2 d^5}-\frac {15 b^2 \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}+\frac {15 b \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^6}-\frac {15 b^2 \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6} \]

[In]

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

[Out]

((-4*I)*b^2*x^(5/2))/((a^2 + b^2)^2*d) + (4*b^2*x^(5/2))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^((2*I
)*(c + d*Sqrt[x])))) + x^3/(3*(a - I*b)^2) + (4*b*x^3)/(3*(I*a - b)*(a - I*b)^2) - (4*b^2*x^3)/(3*(a^2 + b^2)^
2) + (10*b^2*x^2*Log[1 + ((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b)])/((a^2 + b^2)^2*d^2) + (4*b*x^(5/2)*
Log[1 + ((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b)])/((a - I*b)^2*(a + I*b)*d) - ((4*I)*b^2*x^(5/2)*Log[1
+ ((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b)])/((a^2 + b^2)^2*d) - ((20*I)*b^2*x^(3/2)*PolyLog[2, -(((a
- I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (10*b*x^2*PolyLog[2, -(((a - I*b)*E^((2*I
)*(c + d*Sqrt[x])))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^2) - (10*b^2*x^2*PolyLog[2, -(((a - I*b)*E^((2*I)*(c
+ d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^2) + (30*b^2*x*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x]))
)/(a + I*b))])/((a^2 + b^2)^2*d^4) + (20*b*x^(3/2)*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b
))])/((a - I*b)^2*(a + I*b)*d^3) - ((20*I)*b^2*x^(3/2)*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a +
I*b))])/((a^2 + b^2)^2*d^3) + ((30*I)*b^2*Sqrt[x]*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b
))])/((a^2 + b^2)^2*d^5) - (30*b*x*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((I*a - b)*
(a - I*b)^2*d^4) + (30*b^2*x*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^
4) - (15*b^2*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^6) - (30*b*Sqrt[
x]*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^5) + ((30*I)*b^2*S
qrt[x]*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^5) + (15*b*PolyLog[6,
-(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^6) - (15*b^2*PolyLog[6, -(((a -
I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^6)

Rule 2215

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

Rule 2216

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

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 2222

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

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 3815

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(
c + d*x)^m, (1/(a - I*b) - 2*I*(b/(a^2 + b^2 + (a - I*b)^2*E^(2*I*(e + f*x)))))^(-n), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]

Rule 3832

Int[(x_)^(m_.)*((a_.) + (b_.)*Tan[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Tan[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 \frac {x^5}{(a+b \tan (c+d x))^2} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {x^5}{(a-i b)^2}-\frac {4 b^2 x^5}{(i a+b)^2 \left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )^2}+\frac {4 b x^5}{(a-i b)^2 \left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {x^3}{3 (a-i b)^2}+\frac {(8 b) \text {Subst}\left (\int \frac {x^5}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt {x}\right )}{(a-i b)^2}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {x^5}{\left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )^2} \, dx,x,\sqrt {x}\right )}{(i a+b)^2} \\ & = \frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}+\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {x^5}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt {x}\right )}{(i a-b) (a-i b)^2}-\frac {(8 b) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^5}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt {x}\right )}{a^2+b^2}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^5}{\left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )^2} \, dx,x,\sqrt {x}\right )}{a^2+b^2} \\ & = -\frac {4 b^2 x^{5/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {4 b x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^5}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt {x}\right )}{(a+i b)^2 (i a+b)}-\frac {(20 b) \text {Subst}\left (\int x^4 \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{(a-i b)^2 (a+i b) d}+\frac {\left (20 b^2\right ) \text {Subst}\left (\int \frac {x^4}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt {x}\right )}{(a-i b)^2 (a+i b) d} \\ & = -\frac {4 i b^2 x^{5/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 b^2 x^{5/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {4 b x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac {10 b x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {(40 b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (2,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {\left (20 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^4}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt {x}\right )}{(a-i b) (a+i b)^2 d}+\frac {\left (20 i b^2\right ) \text {Subst}\left (\int x^4 \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right )^2 d} \\ & = -\frac {4 i b^2 x^{5/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 b^2 x^{5/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {10 b^2 x^2 \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac {10 b x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {10 b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {20 b x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {(60 b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (3,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {\left (40 b^2\right ) \text {Subst}\left (\int x^3 \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (40 b^2\right ) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (2,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right )^2 d^2} \\ & = -\frac {4 i b^2 x^{5/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 b^2 x^{5/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {10 b^2 x^2 \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {10 b x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {10 b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {20 b x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {30 b x \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {(60 b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (4,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {\left (60 i b^2\right ) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (2,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {\left (60 i b^2\right ) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (3,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right )^2 d^3} \\ & = -\frac {4 i b^2 x^{5/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 b^2 x^{5/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {10 b^2 x^2 \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {10 b x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {10 b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {30 b^2 x \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac {20 b x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {30 b x \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {30 b^2 x \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac {30 b \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac {(30 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (5,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{(a-i b)^2 (a+i b) d^5}-\frac {\left (60 b^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (3,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac {\left (60 b^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (4,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right )^2 d^4} \\ & = -\frac {4 i b^2 x^{5/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 b^2 x^{5/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {10 b^2 x^2 \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {10 b x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {10 b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {30 b^2 x \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac {20 b x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac {30 b x \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {30 b^2 x \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac {30 b \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}+\frac {(15 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (5,-\frac {(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{(i a-b) (a-i b)^2 d^6}-\frac {\left (30 i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (4,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac {\left (30 i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (5,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right )^2 d^5} \\ & = -\frac {4 i b^2 x^{5/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 b^2 x^{5/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {10 b^2 x^2 \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {10 b x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {10 b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {30 b^2 x \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac {20 b x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac {30 b x \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {30 b^2 x \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac {30 b \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}+\frac {15 b \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^6}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (4,-\frac {(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{\left (a^2+b^2\right )^2 d^6}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (5,-\frac {(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{\left (a^2+b^2\right )^2 d^6} \\ & = -\frac {4 i b^2 x^{5/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 b^2 x^{5/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {10 b^2 x^2 \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {10 b x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {10 b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {30 b^2 x \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac {20 b x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {20 i b^2 x^{3/2} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac {30 b x \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {30 b^2 x \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac {15 b^2 \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}-\frac {30 b \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac {30 i b^2 \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}+\frac {15 b \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^6}-\frac {15 b^2 \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.26 (sec) , antiderivative size = 848, normalized size of antiderivative = 0.74 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {-\frac {i b \left (12 (a+i b) b (i a+b) d^5 x^{5/2}+4 a (a+i b) (i a+b) d^6 x^3+30 (a-i b) b d^4 \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) x^2 \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )+12 a (a-i b) d^5 \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) x^{5/2} \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )+15 (a-i b) b \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) \left (4 i d^3 x^{3/2} \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )+6 d^2 x \operatorname {PolyLog}\left (3,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )-6 i d \sqrt {x} \operatorname {PolyLog}\left (4,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )-3 \operatorname {PolyLog}\left (5,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )\right )+15 a (a-i b) \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) \left (2 i d^4 x^2 \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )+4 d^3 x^{3/2} \operatorname {PolyLog}\left (3,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )-6 i d^2 x \operatorname {PolyLog}\left (4,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )-6 d \sqrt {x} \operatorname {PolyLog}\left (5,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )+3 i \operatorname {PolyLog}\left (6,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )\right )\right )}{d^6 \left (b-b e^{2 i c}-i a \left (1+e^{2 i c}\right )\right )}+\frac {(a-i b)^2 (a+i b) x^3 (a \cos (c)-b \sin (c))}{a \cos (c)+b \sin (c)}+\frac {6 (a-i b)^2 (a+i b) b^2 x^{5/2} \sin \left (d \sqrt {x}\right )}{d (a \cos (c)+b \sin (c)) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )}}{3 (a-i b)^3 (a+i b)^2} \]

[In]

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

[Out]

(((-I)*b*(12*(a + I*b)*b*(I*a + b)*d^5*x^(5/2) + 4*a*(a + I*b)*(I*a + b)*d^6*x^3 + 30*(a - I*b)*b*d^4*((-I)*b*
(-1 + E^((2*I)*c)) + a*(1 + E^((2*I)*c)))*x^2*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))] + 12*a*
(a - I*b)*d^5*((-I)*b*(-1 + E^((2*I)*c)) + a*(1 + E^((2*I)*c)))*x^(5/2)*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*
(c + d*Sqrt[x])))] + 15*(a - I*b)*b*((-I)*b*(-1 + E^((2*I)*c)) + a*(1 + E^((2*I)*c)))*((4*I)*d^3*x^(3/2)*PolyL
og[2, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))] + 6*d^2*x*PolyLog[3, (-a - I*b)/((a - I*b)*E^((2*I)*(c
+ d*Sqrt[x])))] - (6*I)*d*Sqrt[x]*PolyLog[4, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))] - 3*PolyLog[5,
(-a - I*b)/((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))]) + 15*a*(a - I*b)*((-I)*b*(-1 + E^((2*I)*c)) + a*(1 + E^((2
*I)*c)))*((2*I)*d^4*x^2*PolyLog[2, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))] + 4*d^3*x^(3/2)*PolyLog[3
, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))] - (6*I)*d^2*x*PolyLog[4, (-a - I*b)/((a - I*b)*E^((2*I)*(c
+ d*Sqrt[x])))] - 6*d*Sqrt[x]*PolyLog[5, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))] + (3*I)*PolyLog[6,
(-a - I*b)/((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))])))/(d^6*(b - b*E^((2*I)*c) - I*a*(1 + E^((2*I)*c)))) + ((a
- I*b)^2*(a + I*b)*x^3*(a*Cos[c] - b*Sin[c]))/(a*Cos[c] + b*Sin[c]) + (6*(a - I*b)^2*(a + I*b)*b^2*x^(5/2)*Sin
[d*Sqrt[x]])/(d*(a*Cos[c] + b*Sin[c])*(a*Cos[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]])))/(3*(a - I*b)^3*(a + I*b)
^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x**2/(a + b*tan(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. 4345 vs. \(2 (928) = 1856\).

Time = 1.39 (sec) , antiderivative size = 4345, normalized size of antiderivative = 3.79 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/15*(30*(2*a*b*log(b*tan(d*sqrt(x) + c) + a)/(a^4 + 2*a^2*b^2 + b^4) - a*b*log(tan(d*sqrt(x) + c)^2 + 1)/(a^
4 + 2*a^2*b^2 + b^4) + (a^2 - b^2)*(d*sqrt(x) + c)/(a^4 + 2*a^2*b^2 + b^4) - b/(a^3 + a*b^2 + (a^2*b + b^3)*ta
n(d*sqrt(x) + c)))*c^5 - (5*(a^3 - I*a^2*b + a*b^2 - I*b^3)*(d*sqrt(x) + c)^6 - 30*(a^3 - I*a^2*b + a*b^2 - I*
b^3)*(d*sqrt(x) + c)^5*c + 75*(a^3 - I*a^2*b + a*b^2 - I*b^3)*(d*sqrt(x) + c)^4*c^2 - 100*(a^3 - I*a^2*b + a*b
^2 - I*b^3)*(d*sqrt(x) + c)^3*c^3 + 75*(a^3 - I*a^2*b + a*b^2 - I*b^3)*(d*sqrt(x) + c)^2*c^4 - 150*((-I*a*b^2
- b^3)*c^4*cos(2*d*sqrt(x) + 2*c) + (a*b^2 - I*b^3)*c^4*sin(2*d*sqrt(x) + 2*c) + (-I*a*b^2 + b^3)*c^4)*arctan2
(-b*cos(2*d*sqrt(x) + 2*c) + a*sin(2*d*sqrt(x) + 2*c) + b, a*cos(2*d*sqrt(x) + 2*c) + b*sin(2*d*sqrt(x) + 2*c)
+ a) - 4*(48*(I*a^2*b - a*b^2)*(d*sqrt(x) + c)^5 + 75*(I*a*b^2 - b^3 + 2*(-I*a^2*b + a*b^2)*c)*(d*sqrt(x) + c
)^4 + 200*((I*a^2*b - a*b^2)*c^2 + (-I*a*b^2 + b^3)*c)*(d*sqrt(x) + c)^3 + 75*(2*(-I*a^2*b + a*b^2)*c^3 + 3*(I
*a*b^2 - b^3)*c^2)*(d*sqrt(x) + c)^2 + 75*((I*a^2*b - a*b^2)*c^4 + 2*(-I*a*b^2 + b^3)*c^3)*(d*sqrt(x) + c) + (
48*(I*a^2*b + a*b^2)*(d*sqrt(x) + c)^5 + 75*(I*a*b^2 + b^3 + 2*(-I*a^2*b - a*b^2)*c)*(d*sqrt(x) + c)^4 + 200*(
(I*a^2*b + a*b^2)*c^2 + (-I*a*b^2 - b^3)*c)*(d*sqrt(x) + c)^3 + 75*(2*(-I*a^2*b - a*b^2)*c^3 + 3*(I*a*b^2 + b^
3)*c^2)*(d*sqrt(x) + c)^2 + 75*((I*a^2*b + a*b^2)*c^4 + 2*(-I*a*b^2 - b^3)*c^3)*(d*sqrt(x) + c))*cos(2*d*sqrt(
x) + 2*c) - (48*(a^2*b - I*a*b^2)*(d*sqrt(x) + c)^5 + 75*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*sqrt(x) +
c)^4 + 200*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c)*(d*sqrt(x) + c)^3 - 75*(2*(a^2*b - I*a*b^2)*c^3 - 3*(a*
b^2 - I*b^3)*c^2)*(d*sqrt(x) + c)^2 + 75*((a^2*b - I*a*b^2)*c^4 - 2*(a*b^2 - I*b^3)*c^3)*(d*sqrt(x) + c))*sin(
2*d*sqrt(x) + 2*c))*arctan2((2*a*b*cos(2*d*sqrt(x) + 2*c) - (a^2 - b^2)*sin(2*d*sqrt(x) + 2*c))/(a^2 + b^2), (
2*a*b*sin(2*d*sqrt(x) + 2*c) + a^2 + b^2 + (a^2 - b^2)*cos(2*d*sqrt(x) + 2*c))/(a^2 + b^2)) + 5*((a^3 - 3*I*a^
2*b - 3*a*b^2 + I*b^3)*(d*sqrt(x) + c)^6 - 6*(2*I*a*b^2 + 2*b^3 + (a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c)*(d*sq
rt(x) + c)^5 - 60*(I*a*b^2 + b^3)*(d*sqrt(x) + c)*c^4 + 15*((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c^2 - 4*(-I*a*
b^2 - b^3)*c)*(d*sqrt(x) + c)^4 - 20*((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c^3 + 6*(I*a*b^2 + b^3)*c^2)*(d*sqrt
(x) + c)^3 + 15*((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c^4 - 8*(-I*a*b^2 - b^3)*c^3)*(d*sqrt(x) + c)^2)*cos(2*d*
sqrt(x) + 2*c) - 30*(16*(I*a^2*b - a*b^2)*(d*sqrt(x) + c)^4 + 5*(I*a^2*b - a*b^2)*c^4 + 20*(I*a*b^2 - b^3 + 2*
(-I*a^2*b + a*b^2)*c)*(d*sqrt(x) + c)^3 + 10*(-I*a*b^2 + b^3)*c^3 + 40*((I*a^2*b - a*b^2)*c^2 + (-I*a*b^2 + b^
3)*c)*(d*sqrt(x) + c)^2 + 10*(2*(-I*a^2*b + a*b^2)*c^3 + 3*(I*a*b^2 - b^3)*c^2)*(d*sqrt(x) + c) + (16*(I*a^2*b
+ a*b^2)*(d*sqrt(x) + c)^4 + 5*(I*a^2*b + a*b^2)*c^4 + 20*(I*a*b^2 + b^3 + 2*(-I*a^2*b - a*b^2)*c)*(d*sqrt(x)
+ c)^3 + 10*(-I*a*b^2 - b^3)*c^3 + 40*((I*a^2*b + a*b^2)*c^2 + (-I*a*b^2 - b^3)*c)*(d*sqrt(x) + c)^2 + 10*(2*
(-I*a^2*b - a*b^2)*c^3 + 3*(I*a*b^2 + b^3)*c^2)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) - (16*(a^2*b - I*a*b^2
)*(d*sqrt(x) + c)^4 + 5*(a^2*b - I*a*b^2)*c^4 + 20*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*sqrt(x) + c)^3 -
10*(a*b^2 - I*b^3)*c^3 + 40*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c)*(d*sqrt(x) + c)^2 - 10*(2*(a^2*b - I*
a*b^2)*c^3 - 3*(a*b^2 - I*b^3)*c^2)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*dilog((I*a + b)*e^(2*I*d*sqrt(x)
+ 2*I*c)/(-I*a + b)) + 75*((a*b^2 - I*b^3)*c^4*cos(2*d*sqrt(x) + 2*c) - (-I*a*b^2 - b^3)*c^4*sin(2*d*sqrt(x) +
2*c) + (a*b^2 + I*b^3)*c^4)*log((a^2 + b^2)*cos(2*d*sqrt(x) + 2*c)^2 + 4*a*b*sin(2*d*sqrt(x) + 2*c) + (a^2 +
b^2)*sin(2*d*sqrt(x) + 2*c)^2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2*d*sqrt(x) + 2*c)) + 2*(48*(a^2*b + I*a*b^2)*(d
*sqrt(x) + c)^5 + 75*(a*b^2 + I*b^3 - 2*(a^2*b + I*a*b^2)*c)*(d*sqrt(x) + c)^4 + 200*((a^2*b + I*a*b^2)*c^2 -
(a*b^2 + I*b^3)*c)*(d*sqrt(x) + c)^3 - 75*(2*(a^2*b + I*a*b^2)*c^3 - 3*(a*b^2 + I*b^3)*c^2)*(d*sqrt(x) + c)^2
+ 75*((a^2*b + I*a*b^2)*c^4 - 2*(a*b^2 + I*b^3)*c^3)*(d*sqrt(x) + c) + (48*(a^2*b - I*a*b^2)*(d*sqrt(x) + c)^5
+ 75*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*sqrt(x) + c)^4 + 200*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)
*c)*(d*sqrt(x) + c)^3 - 75*(2*(a^2*b - I*a*b^2)*c^3 - 3*(a*b^2 - I*b^3)*c^2)*(d*sqrt(x) + c)^2 + 75*((a^2*b -
I*a*b^2)*c^4 - 2*(a*b^2 - I*b^3)*c^3)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) - (48*(-I*a^2*b - a*b^2)*(d*sqrt
(x) + c)^5 + 75*(-I*a*b^2 - b^3 + 2*(I*a^2*b + a*b^2)*c)*(d*sqrt(x) + c)^4 + 200*((-I*a^2*b - a*b^2)*c^2 + (I*
a*b^2 + b^3)*c)*(d*sqrt(x) + c)^3 + 75*(2*(I*a^2*b + a*b^2)*c^3 + 3*(-I*a*b^2 - b^3)*c^2)*(d*sqrt(x) + c)^2 +
75*((-I*a^2*b - a*b^2)*c^4 + 2*(I*a*b^2 + b^3)*c^3)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*log(((a^2 + b^2)*
cos(2*d*sqrt(x) + 2*c)^2 + 4*a*b*sin(2*d*sqrt(x) + 2*c) + (a^2 + b^2)*sin(2*d*sqrt(x) + 2*c)^2 + a^2 + b^2 + 2
*(a^2 - b^2)*cos(2*d*sqrt(x) + 2*c))/(a^2 + b^2)) - 720*(I*a^2*b - a*b^2 + (I*a^2*b + a*b^2)*cos(2*d*sqrt(x) +
2*c) - (a^2*b - I*a*b^2)*sin(2*d*sqrt(x) + 2*c))*polylog(6, (I*a + b)*e^(2*I*d*sqrt(x) + 2*I*c)/(-I*a + b)) -
90*(5*a*b^2 + 5*I*b^3 + 16*(a^2*b + I*a*b^2)*(d*sqrt(x) + c) - 10*(a^2*b + I*a*b^2)*c + (5*a*b^2 - 5*I*b^3 +
16*(a^2*b - I*a*b^2)*(d*sqrt(x) + c) - 10*(a^2*b - I*a*b^2)*c)*cos(2*d*sqrt(x) + 2*c) + (5*I*a*b^2 + 5*b^3 + 1
6*(I*a^2*b + a*b^2)*(d*sqrt(x) + c) + 10*(-I*a^2*b - a*b^2)*c)*sin(2*d*sqrt(x) + 2*c))*polylog(5, (I*a + b)*e^
(2*I*d*sqrt(x) + 2*I*c)/(-I*a + b)) - 60*(24*(-I*a^2*b + a*b^2)*(d*sqrt(x) + c)^2 + 10*(-I*a^2*b + a*b^2)*c^2
+ 15*(-I*a*b^2 + b^3 + 2*(I*a^2*b - a*b^2)*c)*(d*sqrt(x) + c) + 10*(I*a*b^2 - b^3)*c + (24*(-I*a^2*b - a*b^2)*
(d*sqrt(x) + c)^2 + 10*(-I*a^2*b - a*b^2)*c^2 + 15*(-I*a*b^2 - b^3 + 2*(I*a^2*b + a*b^2)*c)*(d*sqrt(x) + c) +
10*(I*a*b^2 + b^3)*c)*cos(2*d*sqrt(x) + 2*c) + (24*(a^2*b - I*a*b^2)*(d*sqrt(x) + c)^2 + 10*(a^2*b - I*a*b^2)*
c^2 + 15*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*sqrt(x) + c) - 10*(a*b^2 - I*b^3)*c)*sin(2*d*sqrt(x) + 2*c
))*polylog(4, (I*a + b)*e^(2*I*d*sqrt(x) + 2*I*c)/(-I*a + b)) + 30*(32*(a^2*b + I*a*b^2)*(d*sqrt(x) + c)^3 - 1
0*(a^2*b + I*a*b^2)*c^3 + 30*(a*b^2 + I*b^3 - 2*(a^2*b + I*a*b^2)*c)*(d*sqrt(x) + c)^2 + 15*(a*b^2 + I*b^3)*c^
2 + 40*((a^2*b + I*a*b^2)*c^2 - (a*b^2 + I*b^3)*c)*(d*sqrt(x) + c) + (32*(a^2*b - I*a*b^2)*(d*sqrt(x) + c)^3 -
10*(a^2*b - I*a*b^2)*c^3 + 30*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*sqrt(x) + c)^2 + 15*(a*b^2 - I*b^3)*
c^2 + 40*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) - (32*(-I*a^2*b -
a*b^2)*(d*sqrt(x) + c)^3 + 10*(I*a^2*b + a*b^2)*c^3 + 30*(-I*a*b^2 - b^3 + 2*(I*a^2*b + a*b^2)*c)*(d*sqrt(x)
+ c)^2 + 15*(-I*a*b^2 - b^3)*c^2 + 40*((-I*a^2*b - a*b^2)*c^2 + (I*a*b^2 + b^3)*c)*(d*sqrt(x) + c))*sin(2*d*sq
rt(x) + 2*c))*polylog(3, (I*a + b)*e^(2*I*d*sqrt(x) + 2*I*c)/(-I*a + b)) - 5*((-I*a^3 - 3*a^2*b + 3*I*a*b^2 +
b^3)*(d*sqrt(x) + c)^6 - 6*(2*a*b^2 - 2*I*b^3 - (I*a^3 + 3*a^2*b - 3*I*a*b^2 - b^3)*c)*(d*sqrt(x) + c)^5 - 60*
(a*b^2 - I*b^3)*(d*sqrt(x) + c)*c^4 + 15*((-I*a^3 - 3*a^2*b + 3*I*a*b^2 + b^3)*c^2 + 4*(a*b^2 - I*b^3)*c)*(d*s
qrt(x) + c)^4 + 20*((I*a^3 + 3*a^2*b - 3*I*a*b^2 - b^3)*c^3 - 6*(a*b^2 - I*b^3)*c^2)*(d*sqrt(x) + c)^3 + 15*((
-I*a^3 - 3*a^2*b + 3*I*a*b^2 + b^3)*c^4 + 8*(a*b^2 - I*b^3)*c^3)*(d*sqrt(x) + c)^2)*sin(2*d*sqrt(x) + 2*c))/(a
^5 + I*a^4*b + 2*a^3*b^2 + 2*I*a^2*b^3 + a*b^4 + I*b^5 + (a^5 - I*a^4*b + 2*a^3*b^2 - 2*I*a^2*b^3 + a*b^4 - I*
b^5)*cos(2*d*sqrt(x) + 2*c) - (-I*a^5 - a^4*b - 2*I*a^3*b^2 - 2*a^2*b^3 - I*a*b^4 - b^5)*sin(2*d*sqrt(x) + 2*c
)))/d^6

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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