3.68 \(\int \frac{\sqrt{x}}{(a+b \text{sech}(c+d \sqrt{x}))^2} \, dx\)

Optimal. Leaf size=1027 \[ \text{result too large to display} \]

[Out]

(2*b^2*x)/(a^2*(a^2 - b^2)*d) + (2*x^(3/2))/(3*a^2) - (4*b^2*Sqrt[x]*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-
a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) + (2*b^3*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a
^2 + b^2)^(3/2)*d) - (4*b*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) -
(4*b^2*Sqrt[x]*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (2*b^3*x*Log[1 +
(a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (4*b*x*Log[1 + (a*E^(c + d*Sqrt[x
]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (4*b^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a
^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (4*b^3*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])
)])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (8*b*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(
a^2*Sqrt[-a^2 + b^2]*d^2) - (4*b^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^
2)*d^3) - (4*b^3*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*
d^2) + (8*b*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) -
(4*b^3*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + (8*b*PolyLo
g[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (4*b^3*PolyLog[3, -((a*E^(
c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (8*b*PolyLog[3, -((a*E^(c + d*Sqrt[x]
))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (2*b^2*x*Sinh[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b +
a*Cosh[c + d*Sqrt[x]]))

________________________________________________________________________________________

Rubi [A]  time = 1.89769, antiderivative size = 1027, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {5436, 4191, 3324, 3320, 2264, 2190, 2531, 2282, 6589, 5562, 2279, 2391} \[ \frac{2 x \log \left (\frac{e^{c+d \sqrt{x}} a}{b-\sqrt{b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac{2 x \log \left (\frac{e^{c+d \sqrt{x}} a}{b+\sqrt{b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac{4 \sqrt{x} \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac{4 \sqrt{x} \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac{4 \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac{4 \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac{2 x b^2}{a^2 \left (a^2-b^2\right ) d}-\frac{4 \sqrt{x} \log \left (\frac{e^{c+d \sqrt{x}} a}{b-\sqrt{b^2-a^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac{4 \sqrt{x} \log \left (\frac{e^{c+d \sqrt{x}} a}{b+\sqrt{b^2-a^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac{4 \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}-\frac{4 \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}+\frac{2 x \sinh \left (c+d \sqrt{x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{4 x \log \left (\frac{e^{c+d \sqrt{x}} a}{b-\sqrt{b^2-a^2}}+1\right ) b}{a^2 \sqrt{b^2-a^2} d}+\frac{4 x \log \left (\frac{e^{c+d \sqrt{x}} a}{b+\sqrt{b^2-a^2}}+1\right ) b}{a^2 \sqrt{b^2-a^2} d}-\frac{8 \sqrt{x} \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{8 \sqrt{x} \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{8 \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{8 \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{b^2-a^2}}\right ) b}{a^2 \sqrt{b^2-a^2} d^3}+\frac{2 x^{3/2}}{3 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^2*x)/(a^2*(a^2 - b^2)*d) + (2*x^(3/2))/(3*a^2) - (4*b^2*Sqrt[x]*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-
a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) + (2*b^3*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a
^2 + b^2)^(3/2)*d) - (4*b*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) -
(4*b^2*Sqrt[x]*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (2*b^3*x*Log[1 +
(a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (4*b*x*Log[1 + (a*E^(c + d*Sqrt[x
]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (4*b^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a
^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (4*b^3*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])
)])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (8*b*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(
a^2*Sqrt[-a^2 + b^2]*d^2) - (4*b^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^
2)*d^3) - (4*b^3*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*
d^2) + (8*b*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) -
(4*b^3*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + (8*b*PolyLo
g[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (4*b^3*PolyLog[3, -((a*E^(
c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (8*b*PolyLog[3, -((a*E^(c + d*Sqrt[x]
))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (2*b^2*x*Sinh[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b +
a*Cosh[c + d*Sqrt[x]]))

Rule 5436

Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rule 4191

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

Rule 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3320

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(E^(I*Pi*(k - 1/2))*(b + (2*a*E^(-(I*e) + f*fz*x))/E^(I*Pi*(k
- 1/2)) - (b*E^(2*(-(I*e) + f*fz*x)))/E^(2*I*k*Pi))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 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 5562

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

Rule 2279

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{\left (a+b \text{sech}\left (c+d \sqrt{x}\right )\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^2}{(a+b \text{sech}(c+d x))^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^2}{a^2}+\frac{b^2 x^2}{a^2 (b+a \cosh (c+d x))^2}-\frac{2 b x^2}{a^2 (b+a \cosh (c+d x))}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 x^{3/2}}{3 a^2}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{x^2}{b+a \cosh (c+d x)} \, dx,x,\sqrt{x}\right )}{a^2}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{(b+a \cosh (c+d x))^2} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=\frac{2 x^{3/2}}{3 a^2}+\frac{2 b^2 x \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt{x}\right )}{a^2}-\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{b+a \cosh (c+d x)} \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right )}-\frac{\left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{x \sinh (c+d x)}{b+a \cosh (c+d x)} \, dx,x,\sqrt{x}\right )}{a \left (a^2-b^2\right ) d}\\ &=\frac{2 b^2 x}{a^2 \left (a^2-b^2\right ) d}+\frac{2 x^{3/2}}{3 a^2}+\frac{2 b^2 x \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right )}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2}}+\frac{(8 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2}}-\frac{\left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{b-\sqrt{-a^2+b^2}+a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \left (a^2-b^2\right ) d}-\frac{\left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{b+\sqrt{-a^2+b^2}+a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \left (a^2-b^2\right ) d}\\ &=\frac{2 b^2 x}{a^2 \left (a^2-b^2\right ) d}+\frac{2 x^{3/2}}{3 a^2}-\frac{4 b^2 \sqrt{x} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{4 b x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{4 b^2 \sqrt{x} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{4 b x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{2 b^2 x \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \left (-a^2+b^2\right )^{3/2}}+\frac{\left (4 b^2\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{\left (4 b^2\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{(8 b) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{(8 b) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d}\\ &=\frac{2 b^2 x}{a^2 \left (a^2-b^2\right ) d}+\frac{2 x^{3/2}}{3 a^2}-\frac{4 b^2 \sqrt{x} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{4 b^2 \sqrt{x} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{8 b \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{8 b \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{2 b^2 x \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{b-\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{\left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{(8 b) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{(8 b) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}\\ &=\frac{2 b^2 x}{a^2 \left (a^2-b^2\right ) d}+\frac{2 x^{3/2}}{3 a^2}-\frac{4 b^2 \sqrt{x} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{4 b^2 \sqrt{x} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{4 b^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{4 b^3 \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{8 b \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{4 b^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{4 b^3 \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{8 b \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{2 b^2 x \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{(8 b) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}\\ &=\frac{2 b^2 x}{a^2 \left (a^2-b^2\right ) d}+\frac{2 x^{3/2}}{3 a^2}-\frac{4 b^2 \sqrt{x} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{4 b^2 \sqrt{x} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{4 b^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{4 b^3 \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{8 b \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{4 b^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{4 b^3 \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{8 b \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{8 b \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{8 b \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{2 b^2 x \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}\\ &=\frac{2 b^2 x}{a^2 \left (a^2-b^2\right ) d}+\frac{2 x^{3/2}}{3 a^2}-\frac{4 b^2 \sqrt{x} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{4 b^2 \sqrt{x} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{4 b^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{4 b^3 \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{8 b \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{4 b^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{4 b^3 \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{8 b \sqrt{x} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{4 b^3 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{8 b \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{4 b^3 \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{8 b \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{2 b^2 x \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}\\ \end{align*}

Mathematica [A]  time = 13.0337, size = 986, normalized size = 0.96 \[ \frac{2 \left (b+a \cosh \left (c+d \sqrt{x}\right )\right ) \text{sech}^2\left (c+d \sqrt{x}\right ) \left (\frac{3 x \text{sech}(c) \left (b \sinh (c)-a \sinh \left (d \sqrt{x}\right )\right ) b^2}{\left (b^2-a^2\right ) d}+\frac{3 e^c \left (b+a \cosh \left (c+d \sqrt{x}\right )\right ) \left (2 b e^c x-\frac{e^{-c} \left (1+e^{2 c}\right ) \left (2 d^2 e^c x \log \left (\frac{e^{2 c+d \sqrt{x}} a}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}+1\right ) a^2-2 d^2 e^c x \log \left (\frac{e^{2 c+d \sqrt{x}} a}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}+1\right ) a^2-4 e^c \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) a^2+4 e^c \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) a^2-b^2 d^2 e^c x \log \left (\frac{e^{2 c+d \sqrt{x}} a}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}+1\right )+2 b d \sqrt{\left (b^2-a^2\right ) e^{2 c}} \sqrt{x} \log \left (\frac{e^{2 c+d \sqrt{x}} a}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}+1\right )+b^2 d^2 e^c x \log \left (\frac{e^{2 c+d \sqrt{x}} a}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}+1\right )+2 b d \sqrt{\left (b^2-a^2\right ) e^{2 c}} \sqrt{x} \log \left (\frac{e^{2 c+d \sqrt{x}} a}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}+1\right )+2 \left (2 d e^c \sqrt{x} a^2+b \sqrt{\left (b^2-a^2\right ) e^{2 c}}-b^2 d e^c \sqrt{x}\right ) \text{PolyLog}\left (2,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right )+2 \left (-2 d e^c \sqrt{x} a^2+b \sqrt{\left (b^2-a^2\right ) e^{2 c}}+b^2 d e^c \sqrt{x}\right ) \text{PolyLog}\left (2,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right )+2 b^2 e^c \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right )-2 b^2 e^c \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{e^c b+\sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right )\right )}{d^2 \sqrt{\left (b^2-a^2\right ) e^{2 c}}}\right ) b}{\left (a^2-b^2\right ) d \left (1+e^{2 c}\right )}+x^{3/2} \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )\right )}{3 a^2 \left (a+b \text{sech}\left (c+d \sqrt{x}\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + a*Cosh[c + d*Sqrt[x]])*Sech[c + d*Sqrt[x]]^2*(x^(3/2)*(b + a*Cosh[c + d*Sqrt[x]]) + (3*b*E^c*(b + a*Co
sh[c + d*Sqrt[x]])*(2*b*E^c*x - ((1 + E^(2*c))*(2*b*d*Sqrt[(-a^2 + b^2)*E^(2*c)]*Sqrt[x]*Log[1 + (a*E^(2*c + d
*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])] + 2*a^2*d^2*E^c*x*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sq
rt[(-a^2 + b^2)*E^(2*c)])] - b^2*d^2*E^c*x*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]
)] + 2*b*d*Sqrt[(-a^2 + b^2)*E^(2*c)]*Sqrt[x]*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*
c)])] - 2*a^2*d^2*E^c*x*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] + b^2*d^2*E^c*x*
Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] + 2*(b*Sqrt[(-a^2 + b^2)*E^(2*c)] + 2*a^
2*d*E^c*Sqrt[x] - b^2*d*E^c*Sqrt[x])*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])
)] + 2*(b*Sqrt[(-a^2 + b^2)*E^(2*c)] - 2*a^2*d*E^c*Sqrt[x] + b^2*d*E^c*Sqrt[x])*PolyLog[2, -((a*E^(2*c + d*Sqr
t[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 4*a^2*E^c*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(
-a^2 + b^2)*E^(2*c)]))] + 2*b^2*E^c*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))
] + 4*a^2*E^c*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 2*b^2*E^c*PolyLog[
3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))]))/(d^2*E^c*Sqrt[(-a^2 + b^2)*E^(2*c)])))/(
(a^2 - b^2)*d*(1 + E^(2*c))) + (3*b^2*x*Sech[c]*(b*Sinh[c] - a*Sinh[d*Sqrt[x]]))/((-a^2 + b^2)*d)))/(3*a^2*(a
+ b*Sech[c + d*Sqrt[x]])^2)

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Maple [F]  time = 0.099, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{x} \left ( a+b{\rm sech} \left (c+d\sqrt{x}\right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x}}{b^{2} \operatorname{sech}\left (d \sqrt{x} + c\right )^{2} + 2 \, a b \operatorname{sech}\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^(1/2)/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x}}{{\left (b \operatorname{sech}\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^(1/2)/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

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