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

Optimal. Leaf size=787 \[ -\frac{6 b^2 x \text{PolyLog}\left (2,-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac{6 i b^2 \sqrt{x} \text{PolyLog}\left (2,-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}-\frac{6 i b^2 \sqrt{x} \text{PolyLog}\left (3,-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}+\frac{3 b^2 \text{PolyLog}\left (3,-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^4 \left (a^2+b^2\right )^2}+\frac{3 b^2 \text{PolyLog}\left (4,-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^4 \left (a^2+b^2\right )^2}+\frac{6 b x \text{PolyLog}\left (2,-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^2 (-b+i a) (a-i b)^2}+\frac{6 b \sqrt{x} \text{PolyLog}\left (3,-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^3 (a-i b)^2 (a+i b)}-\frac{3 b \text{PolyLog}\left (4,-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^4 (-b+i a) (a-i b)^2}+\frac{6 b^2 x \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac{4 i b^2 x^{3/2} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d \left (a^2+b^2\right )^2}-\frac{4 i b^2 x^{3/2}}{d \left (a^2+b^2\right )^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{4 b^2 x^{3/2}}{d (a+i b) (b+i a)^2 \left ((b+i a) e^{2 i \left (c+d \sqrt{x}\right )}+i a-b\right )}+\frac{4 b x^{3/2} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d (a-i b)^2 (a+i b)}+\frac{2 b x^2}{(-b+i a) (a-i b)^2}+\frac{x^2}{2 (a-i b)^2} \]

[Out]

((-4*I)*b^2*x^(3/2))/((a^2 + b^2)^2*d) + (4*b^2*x^(3/2))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^((2*I
)*(c + d*Sqrt[x])))) + x^2/(2*(a - I*b)^2) + (2*b*x^2)/((I*a - b)*(a - I*b)^2) - (2*b^2*x^2)/(a^2 + b^2)^2 + (
6*b^2*x*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^(3/2)*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^(3/2)*Log[1 + ((a -
I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b)])/((a^2 + b^2)^2*d) - ((6*I)*b^2*Sqrt[x]*PolyLog[2, -(((a - I*b)*E^(
(2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (6*b*x*PolyLog[2, -(((a - I*b)*E^((2*I)*(c + d*Sqrt
[x])))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^2) - (6*b^2*x*PolyLog[2, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/
(a + I*b))])/((a^2 + b^2)^2*d^2) + (3*b^2*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^
2 + b^2)^2*d^4) + (6*b*Sqrt[x]*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a - I*b)^2*(a
+ I*b)*d^3) - ((6*I)*b^2*Sqrt[x]*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)
^2*d^3) - (3*b*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^4) + (
3*b^2*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^4)

________________________________________________________________________________________

Rubi [A]  time = 1.71086, antiderivative size = 787, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {3747, 3734, 2185, 2184, 2190, 2531, 6609, 2282, 6589, 2191} \[ -\frac{6 b^2 x \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac{6 i b^2 \sqrt{x} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}-\frac{6 i b^2 \sqrt{x} \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}+\frac{3 b^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^4 \left (a^2+b^2\right )^2}+\frac{3 b^2 \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^4 \left (a^2+b^2\right )^2}+\frac{6 b^2 x \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac{4 i b^2 x^{3/2} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d \left (a^2+b^2\right )^2}-\frac{4 i b^2 x^{3/2}}{d \left (a^2+b^2\right )^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{4 b^2 x^{3/2}}{d (a+i b) (b+i a)^2 \left ((b+i a) e^{2 i \left (c+d \sqrt{x}\right )}+i a-b\right )}+\frac{6 b x \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^2 (-b+i a) (a-i b)^2}+\frac{6 b \sqrt{x} \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^3 (a-i b)^2 (a+i b)}-\frac{3 b \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d^4 (-b+i a) (a-i b)^2}+\frac{4 b x^{3/2} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt{x}\right )}}{a+i b}\right )}{d (a-i b)^2 (a+i b)}+\frac{2 b x^2}{(-b+i a) (a-i b)^2}+\frac{x^2}{2 (a-i b)^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Tan[c + d*Sqrt[x]])^2,x]

[Out]

((-4*I)*b^2*x^(3/2))/((a^2 + b^2)^2*d) + (4*b^2*x^(3/2))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^((2*I
)*(c + d*Sqrt[x])))) + x^2/(2*(a - I*b)^2) + (2*b*x^2)/((I*a - b)*(a - I*b)^2) - (2*b^2*x^2)/(a^2 + b^2)^2 + (
6*b^2*x*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^(3/2)*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^(3/2)*Log[1 + ((a -
I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b)])/((a^2 + b^2)^2*d) - ((6*I)*b^2*Sqrt[x]*PolyLog[2, -(((a - I*b)*E^(
(2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (6*b*x*PolyLog[2, -(((a - I*b)*E^((2*I)*(c + d*Sqrt
[x])))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^2) - (6*b^2*x*PolyLog[2, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/
(a + I*b))])/((a^2 + b^2)^2*d^2) + (3*b^2*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^
2 + b^2)^2*d^4) + (6*b*Sqrt[x]*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a - I*b)^2*(a
+ I*b)*d^3) - ((6*I)*b^2*Sqrt[x]*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)
^2*d^3) - (3*b*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^4) + (
3*b^2*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^4)

Rule 3747

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 3734

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 2185

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 2184

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 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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6609

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

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 2191

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]

Rubi steps

\begin{align*} \int \frac{x}{\left (a+b \tan \left (c+d \sqrt{x}\right )\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{(a+b \tan (c+d x))^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^3}{(a-i b)^2}-\frac{4 b^2 x^3}{(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^3}{(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^2}{2 (a-i b)^2}+\frac{(8 b) \operatorname{Subst}\left (\int \frac{x^3}{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 ) \operatorname{Subst}\left (\int \frac{x^3}{\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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}+\frac{\left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{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) \operatorname{Subst}\left (\int \frac{e^{2 i c+2 i d x} x^3}{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 ) \operatorname{Subst}\left (\int \frac{e^{2 i c+2 i d x} x^3}{\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^{3/2}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{4 b x^{3/2} \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 ) \operatorname{Subst}\left (\int \frac{e^{2 i c+2 i d x} x^3}{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{(12 b) \operatorname{Subst}\left (\int x^2 \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 (12 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{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^{3/2}}{\left (a^2+b^2\right )^2 d}-\frac{4 b^2 x^{3/2}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{4 b x^{3/2} \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^{3/2} \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{6 b x \text{Li}_2\left (-\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{(12 b) \operatorname{Subst}\left (\int x \text{Li}_2\left (-\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 (12 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 i c+2 i d x} x^2}{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 (12 i b^2\right ) \operatorname{Subst}\left (\int x^2 \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^{3/2}}{\left (a^2+b^2\right )^2 d}-\frac{4 b^2 x^{3/2}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{6 b^2 x \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^{3/2} \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^{3/2} \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{6 b x \text{Li}_2\left (-\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{6 b^2 x \text{Li}_2\left (-\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{6 b \sqrt{x} \text{Li}_3\left (-\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{(6 b) \operatorname{Subst}\left (\int \text{Li}_3\left (-\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 (12 b^2\right ) \operatorname{Subst}\left (\int x \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 (12 b^2\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-\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^{3/2}}{\left (a^2+b^2\right )^2 d}-\frac{4 b^2 x^{3/2}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{6 b^2 x \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^{3/2} \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^{3/2} \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{6 i b^2 \sqrt{x} \text{Li}_2\left (-\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{6 b x \text{Li}_2\left (-\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{6 b^2 x \text{Li}_2\left (-\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{6 b \sqrt{x} \text{Li}_3\left (-\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{6 i b^2 \sqrt{x} \text{Li}_3\left (-\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{(3 b) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\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^4}+\frac{\left (6 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\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 (6 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-\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^{3/2}}{\left (a^2+b^2\right )^2 d}-\frac{4 b^2 x^{3/2}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{6 b^2 x \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^{3/2} \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^{3/2} \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{6 i b^2 \sqrt{x} \text{Li}_2\left (-\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{6 b x \text{Li}_2\left (-\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{6 b^2 x \text{Li}_2\left (-\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{6 b \sqrt{x} \text{Li}_3\left (-\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{6 i b^2 \sqrt{x} \text{Li}_3\left (-\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{3 b \text{Li}_4\left (-\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{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\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^4}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\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^4}\\ &=-\frac{4 i b^2 x^{3/2}}{\left (a^2+b^2\right )^2 d}-\frac{4 b^2 x^{3/2}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{6 b^2 x \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^{3/2} \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^{3/2} \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{6 i b^2 \sqrt{x} \text{Li}_2\left (-\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{6 b x \text{Li}_2\left (-\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{6 b^2 x \text{Li}_2\left (-\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{3 b^2 \text{Li}_3\left (-\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{6 b \sqrt{x} \text{Li}_3\left (-\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{6 i b^2 \sqrt{x} \text{Li}_3\left (-\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{3 b \text{Li}_4\left (-\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{3 b^2 \text{Li}_4\left (-\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}\\ \end{align*}

Mathematica [A]  time = 3.72311, size = 633, normalized size = 0.8 \[ \frac{\frac{2 b \left (\frac{3 b \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) \left (2 d \sqrt{x} \text{PolyLog}\left (2,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt{x}\right )}}{a-i b}\right )-i \text{PolyLog}\left (3,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt{x}\right )}}{a-i b}\right )\right )}{d^3 \left (a^2+b^2\right )}+\frac{3 a \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) \left (2 d^2 x \text{PolyLog}\left (2,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt{x}\right )}}{a-i b}\right )-2 i d \sqrt{x} \text{PolyLog}\left (3,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt{x}\right )}}{a-i b}\right )-\text{PolyLog}\left (4,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt{x}\right )}}{a-i b}\right )\right )}{d^3 \left (a^2+b^2\right )}+\frac{4 a x^{3/2} \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) \log \left (1+\frac{(a+i b) e^{-2 i \left (c+d \sqrt{x}\right )}}{a-i b}\right )}{(a+i b) (b+i a)}+\frac{6 b x \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) \log \left (1+\frac{(a+i b) e^{-2 i \left (c+d \sqrt{x}\right )}}{a-i b}\right )}{d (a+i b) (b+i a)}+\frac{2 a d x^2}{a-i b}+\frac{4 b x^{3/2}}{a-i b}\right )}{d \left (-i a \left (1+e^{2 i c}\right )+b \left (-e^{2 i c}\right )+b\right )}+\frac{4 b^2 x^{3/2} \sin \left (d \sqrt{x}\right )}{d (a \cos (c)+b \sin (c)) \left (a \cos \left (c+d \sqrt{x}\right )+b \sin \left (c+d \sqrt{x}\right )\right )}+\frac{x^2 (a \cos (c)-b \sin (c))}{a \cos (c)+b \sin (c)}}{2 \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.28, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b\tan \left ( c+d\sqrt{x} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 4.80826, size = 3357, normalized size = 4.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((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)*tan(d*sq
rt(x) + c)))*c^3 - ((3*a^3 - 3*I*a^2*b + 3*a*b^2 - 3*I*b^3)*(d*sqrt(x) + c)^4 - (12*a^3 - 12*I*a^2*b + 12*a*b^
2 - 12*I*b^3)*(d*sqrt(x) + c)^3*c + (18*a^3 - 18*I*a^2*b + 18*a*b^2 - 18*I*b^3)*(d*sqrt(x) + c)^2*c^2 - (36*(-
I*a*b^2 - b^3)*c^2*cos(2*d*sqrt(x) + 2*c) + (36*a*b^2 - 36*I*b^3)*c^2*sin(2*d*sqrt(x) + 2*c) + 36*(-I*a*b^2 +
b^3)*c^2)*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) + ((-32*I*a^2*b + 32*a*b^2)*(d*sqrt(x) + c)^3 + (-36*I*a*b^2 + 36*b^3 + (72*I*a^2*b - 7
2*a*b^2)*c)*(d*sqrt(x) + c)^2 + ((-72*I*a^2*b + 72*a*b^2)*c^2 - 72*(-I*a*b^2 + b^3)*c)*(d*sqrt(x) + c) + ((-32
*I*a^2*b - 32*a*b^2)*(d*sqrt(x) + c)^3 + (-36*I*a*b^2 - 36*b^3 + (72*I*a^2*b + 72*a*b^2)*c)*(d*sqrt(x) + c)^2
+ ((-72*I*a^2*b - 72*a*b^2)*c^2 - 72*(-I*a*b^2 - b^3)*c)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + (32*(a^2*b
- I*a*b^2)*(d*sqrt(x) + c)^3 + (36*a*b^2 - 36*I*b^3 - 72*(a^2*b - I*a*b^2)*c)*(d*sqrt(x) + c)^2 + (72*(a^2*b -
I*a*b^2)*c^2 - (72*a*b^2 - 72*I*b^3)*c)*(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)) + ((3*a^3 - 9*I*a^2*b - 9*a*b^2 + 3*I*b^3)*(d*sqrt(x) + c)^4 + (-2
4*I*a*b^2 - 24*b^3 - (12*a^3 - 36*I*a^2*b - 36*a*b^2 + 12*I*b^3)*c)*(d*sqrt(x) + c)^3 - 72*(I*a*b^2 + b^3)*(d*
sqrt(x) + c)*c^2 + ((18*a^3 - 54*I*a^2*b - 54*a*b^2 + 18*I*b^3)*c^2 - 72*(-I*a*b^2 - b^3)*c)*(d*sqrt(x) + c)^2
)*cos(2*d*sqrt(x) + 2*c) + ((-48*I*a^2*b + 48*a*b^2)*(d*sqrt(x) + c)^2 + (-36*I*a^2*b + 36*a*b^2)*c^2 + (-36*I
*a*b^2 + 36*b^3 + (72*I*a^2*b - 72*a*b^2)*c)*(d*sqrt(x) + c) - 36*(-I*a*b^2 + b^3)*c + ((-48*I*a^2*b - 48*a*b^
2)*(d*sqrt(x) + c)^2 + (-36*I*a^2*b - 36*a*b^2)*c^2 + (-36*I*a*b^2 - 36*b^3 + (72*I*a^2*b + 72*a*b^2)*c)*(d*sq
rt(x) + c) - 36*(-I*a*b^2 - b^3)*c)*cos(2*d*sqrt(x) + 2*c) + (48*(a^2*b - I*a*b^2)*(d*sqrt(x) + c)^2 + 36*(a^2
*b - I*a*b^2)*c^2 + (36*a*b^2 - 36*I*b^3 - 72*(a^2*b - I*a*b^2)*c)*(d*sqrt(x) + c) - (36*a*b^2 - 36*I*b^3)*c)*
sin(2*d*sqrt(x) + 2*c))*dilog((I*a + b)*e^(2*I*d*sqrt(x) + 2*I*c)/(-I*a + b)) + ((18*a*b^2 - 18*I*b^3)*c^2*cos
(2*d*sqrt(x) + 2*c) - 18*(-I*a*b^2 - b^3)*c^2*sin(2*d*sqrt(x) + 2*c) + (18*a*b^2 + 18*I*b^3)*c^2)*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)) + (16*(a^2*b + I*a*b^2)*(d*sqrt(x) + c)^3 + (18*a*b^2 + 18*I*b^3 - 36
*(a^2*b + I*a*b^2)*c)*(d*sqrt(x) + c)^2 + (36*(a^2*b + I*a*b^2)*c^2 - (36*a*b^2 + 36*I*b^3)*c)*(d*sqrt(x) + c)
+ (16*(a^2*b - I*a*b^2)*(d*sqrt(x) + c)^3 + (18*a*b^2 - 18*I*b^3 - 36*(a^2*b - I*a*b^2)*c)*(d*sqrt(x) + c)^2
+ (36*(a^2*b - I*a*b^2)*c^2 - (36*a*b^2 - 36*I*b^3)*c)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + ((16*I*a^2*b
+ 16*a*b^2)*(d*sqrt(x) + c)^3 + (18*I*a*b^2 + 18*b^3 + (-36*I*a^2*b - 36*a*b^2)*c)*(d*sqrt(x) + c)^2 + ((36*I*
a^2*b + 36*a*b^2)*c^2 - 36*(I*a*b^2 + b^3)*c)*(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)) + (24*I*a^2*b - 24*a*b^2 + (24*I*a^2*b + 24*a*b^2)*cos(2*d*sqrt(x)
+ 2*c) - 24*(a^2*b - I*a*b^2)*sin(2*d*sqrt(x) + 2*c))*polylog(4, (I*a + b)*e^(2*I*d*sqrt(x) + 2*I*c)/(-I*a +
b)) + (18*a*b^2 + 18*I*b^3 + 48*(a^2*b + I*a*b^2)*(d*sqrt(x) + c) - 36*(a^2*b + I*a*b^2)*c + (18*a*b^2 - 18*I*
b^3 + 48*(a^2*b - I*a*b^2)*(d*sqrt(x) + c) - 36*(a^2*b - I*a*b^2)*c)*cos(2*d*sqrt(x) + 2*c) + (18*I*a*b^2 + 18
*b^3 + (48*I*a^2*b + 48*a*b^2)*(d*sqrt(x) + c) + (-36*I*a^2*b - 36*a*b^2)*c)*sin(2*d*sqrt(x) + 2*c))*polylog(3
, (I*a + b)*e^(2*I*d*sqrt(x) + 2*I*c)/(-I*a + b)) + ((3*I*a^3 + 9*a^2*b - 9*I*a*b^2 - 3*b^3)*(d*sqrt(x) + c)^4
+ (24*a*b^2 - 24*I*b^3 + (-12*I*a^3 - 36*a^2*b + 36*I*a*b^2 + 12*b^3)*c)*(d*sqrt(x) + c)^3 + (72*a*b^2 - 72*I
*b^3)*(d*sqrt(x) + c)*c^2 + ((18*I*a^3 + 54*a^2*b - 54*I*a*b^2 - 18*b^3)*c^2 - (72*a*b^2 - 72*I*b^3)*c)*(d*sqr
t(x) + c)^2)*sin(2*d*sqrt(x) + 2*c))/(12*a^5 + 12*I*a^4*b + 24*a^3*b^2 + 24*I*a^2*b^3 + 12*a*b^4 + 12*I*b^5 +
(12*a^5 - 12*I*a^4*b + 24*a^3*b^2 - 24*I*a^2*b^3 + 12*a*b^4 - 12*I*b^5)*cos(2*d*sqrt(x) + 2*c) + (12*I*a^5 + 1
2*a^4*b + 24*I*a^3*b^2 + 24*a^2*b^3 + 12*I*a*b^4 + 12*b^5)*sin(2*d*sqrt(x) + 2*c)))/d^4

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{b^{2} \tan \left (d \sqrt{x} + c\right )^{2} + 2 \, a b \tan \left (d \sqrt{x} + c\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (a + b \tan{\left (c + d \sqrt{x} \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (b \tan \left (d \sqrt{x} + c\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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