3.1.43 \(\int \frac {x^2}{a+b \text {sech}(c+d \sqrt {x})} \, dx\) [43]

Optimal. Leaf size=721 \[ \frac {x^3}{3 a}-\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {10 b x^2 \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {10 b x^2 \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {40 b x^{3/2} \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {40 b x^{3/2} \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {120 b x \text {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {120 b x \text {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {240 b \sqrt {x} \text {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}-\frac {240 b \sqrt {x} \text {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}-\frac {240 b \text {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {240 b \text {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6} \]

[Out]

1/3*x^3/a-2*b*x^(5/2)*ln(1+a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a/d/(-a^2+b^2)^(1/2)+2*b*x^(5/2)*ln(1+a*ex
p(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a/d/(-a^2+b^2)^(1/2)-10*b*x^2*polylog(2,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)
^(1/2)))/a/d^2/(-a^2+b^2)^(1/2)+10*b*x^2*polylog(2,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a/d^2/(-a^2+b^2)^
(1/2)+40*b*x^(3/2)*polylog(3,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a/d^3/(-a^2+b^2)^(1/2)-40*b*x^(3/2)*pol
ylog(3,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a/d^3/(-a^2+b^2)^(1/2)-120*b*x*polylog(4,-a*exp(c+d*x^(1/2))/
(b-(-a^2+b^2)^(1/2)))/a/d^4/(-a^2+b^2)^(1/2)+120*b*x*polylog(4,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a/d^4
/(-a^2+b^2)^(1/2)-240*b*polylog(6,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a/d^6/(-a^2+b^2)^(1/2)+240*b*polyl
og(6,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a/d^6/(-a^2+b^2)^(1/2)+240*b*polylog(5,-a*exp(c+d*x^(1/2))/(b-(
-a^2+b^2)^(1/2)))*x^(1/2)/a/d^5/(-a^2+b^2)^(1/2)-240*b*polylog(5,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))*x^(
1/2)/a/d^5/(-a^2+b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.77, antiderivative size = 721, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5544, 4276, 3401, 2296, 2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {240 b \text {Li}_6\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^6 \sqrt {b^2-a^2}}+\frac {240 b \text {Li}_6\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^6 \sqrt {b^2-a^2}}+\frac {240 b \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^5 \sqrt {b^2-a^2}}-\frac {240 b \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^5 \sqrt {b^2-a^2}}-\frac {120 b x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^4 \sqrt {b^2-a^2}}+\frac {120 b x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^4 \sqrt {b^2-a^2}}+\frac {40 b x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {40 b x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}+\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}-\frac {2 b x^{5/2} \log \left (\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}+1\right )}{a d \sqrt {b^2-a^2}}+\frac {2 b x^{5/2} \log \left (\frac {a e^{c+d \sqrt {x}}}{\sqrt {b^2-a^2}+b}+1\right )}{a d \sqrt {b^2-a^2}}+\frac {x^3}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^3/(3*a) - (2*b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (2*b*
x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - (10*b*x^2*PolyLog[2, -
((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (10*b*x^2*PolyLog[2, -((a*E^(c + d
*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (40*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]
))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (40*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b +
Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (120*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b
^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (120*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sq
rt[-a^2 + b^2]*d^4) + (240*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2
+ b^2]*d^5) - (240*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]
*d^5) - (240*b*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6) + (240*b*
PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6)

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 2296

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 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 3401

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)/(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))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4276

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 5544

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 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*} \int \frac {x^2}{a+b \text {sech}\left (c+d \sqrt {x}\right )} \, dx &=2 \text {Subst}\left (\int \frac {x^5}{a+b \text {sech}(c+d x)} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {x^5}{a}-\frac {b x^5}{a (b+a \cosh (c+d x))}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {x^3}{3 a}-\frac {(2 b) \text {Subst}\left (\int \frac {x^5}{b+a \cosh (c+d x)} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {x^3}{3 a}-\frac {(4 b) \text {Subst}\left (\int \frac {e^{c+d x} x^5}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {x^3}{3 a}-\frac {(4 b) \text {Subst}\left (\int \frac {e^{c+d x} x^5}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^2+b^2}}+\frac {(4 b) \text {Subst}\left (\int \frac {e^{c+d x} x^5}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^2+b^2}}\\ &=\frac {x^3}{3 a}-\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {(10 b) \text {Subst}\left (\int x^4 \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 \sqrt {-a^2+b^2} d}-\frac {(10 b) \text {Subst}\left (\int x^4 \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 \sqrt {-a^2+b^2} d}\\ &=\frac {x^3}{3 a}-\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {(40 b) \text {Subst}\left (\int x^3 \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 \sqrt {-a^2+b^2} d^2}-\frac {(40 b) \text {Subst}\left (\int x^3 \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 \sqrt {-a^2+b^2} d^2}\\ &=\frac {x^3}{3 a}-\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {40 b x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {40 b x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {(120 b) \text {Subst}\left (\int x^2 \text {Li}_3\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^3}+\frac {(120 b) \text {Subst}\left (\int x^2 \text {Li}_3\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^3}\\ &=\frac {x^3}{3 a}-\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {40 b x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {40 b x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {120 b x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {120 b x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {(240 b) \text {Subst}\left (\int x \text {Li}_4\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {(240 b) \text {Subst}\left (\int x \text {Li}_4\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^4}\\ &=\frac {x^3}{3 a}-\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {40 b x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {40 b x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {120 b x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {120 b x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {240 b \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}-\frac {240 b \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}-\frac {(240 b) \text {Subst}\left (\int \text {Li}_5\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {(240 b) \text {Subst}\left (\int \text {Li}_5\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^5}\\ &=\frac {x^3}{3 a}-\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {40 b x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {40 b x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {120 b x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {120 b x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {240 b \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}-\frac {240 b \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}-\frac {(240 b) \text {Subst}\left (\int \frac {\text {Li}_5\left (\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {(240 b) \text {Subst}\left (\int \frac {\text {Li}_5\left (-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a \sqrt {-a^2+b^2} d^6}\\ &=\frac {x^3}{3 a}-\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {10 b x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {40 b x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {40 b x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {120 b x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {120 b x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {240 b \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}-\frac {240 b \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}-\frac {240 b \text {Li}_6\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {240 b \text {Li}_6\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}\\ \end {align*}

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Mathematica [A]
time = 1.42, size = 744, normalized size = 1.03 \begin {gather*} \frac {d^6 \sqrt {\left (-a^2+b^2\right ) e^{2 c}} x^3-6 b d^5 e^c x^{5/2} \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+6 b d^5 e^c x^{5/2} \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-30 b d^4 e^c x^2 \text {PolyLog}\left (2,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+30 b d^4 e^c x^2 \text {PolyLog}\left (2,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+120 b d^3 e^c x^{3/2} \text {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-120 b d^3 e^c x^{3/2} \text {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-360 b d^2 e^c x \text {PolyLog}\left (4,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+360 b d^2 e^c x \text {PolyLog}\left (4,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+720 b d e^c \sqrt {x} \text {PolyLog}\left (5,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-720 b d e^c \sqrt {x} \text {PolyLog}\left (5,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-720 b e^c \text {PolyLog}\left (6,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+720 b e^c \text {PolyLog}\left (6,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )}{3 a d^6 \sqrt {\left (-a^2+b^2\right ) e^{2 c}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^6*Sqrt[(-a^2 + b^2)*E^(2*c)]*x^3 - 6*b*d^5*E^c*x^(5/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2
+ b^2)*E^(2*c)])] + 6*b*d^5*E^c*x^(5/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])]
- 30*b*d^4*E^c*x^2*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 30*b*d^4*E^c*
x^2*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 120*b*d^3*E^c*x^(3/2)*PolyLo
g[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 120*b*d^3*E^c*x^(3/2)*PolyLog[3, -((a*
E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 360*b*d^2*E^c*x*PolyLog[4, -((a*E^(2*c + d*Sqrt[
x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 360*b*d^2*E^c*x*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sq
rt[(-a^2 + b^2)*E^(2*c)]))] + 720*b*d*E^c*Sqrt[x]*PolyLog[5, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b
^2)*E^(2*c)]))] - 720*b*d*E^c*Sqrt[x]*PolyLog[5, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]
))] - 720*b*E^c*PolyLog[6, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 720*b*E^c*PolyLo
g[6, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))])/(3*a*d^6*Sqrt[(-a^2 + b^2)*E^(2*c)])

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Maple [F]
time = 3.06, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{a +b \,\mathrm {sech}\left (c +d \sqrt {x}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a-b>0)', see `assume?` for mor
e details)Is

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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