\(\int x^2 (a+b \text {csch}(c+d \sqrt {x}))^2 \, dx\) [37]

Optimal result

Integrand size = 20, antiderivative size = 441 \[ \int x^2 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {2 b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 a b x^{5/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{5/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 a b x^2 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 a b x^2 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 b^2 x^{3/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {30 b^2 x \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {240 a b x \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {240 a b x \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {30 b^2 \sqrt {x} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {15 b^2 \operatorname {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {480 a b \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {480 a b \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6} \]

[Out]

-2*b^2*x^(5/2)/d+1/3*a^2*x^3-8*a*b*x^(5/2)*arctanh(exp(c+d*x^(1/2)))/d-2*b^2*x^(5/2)*coth(c+d*x^(1/2))/d+10*b^
2*x^2*ln(1-exp(2*c+2*d*x^(1/2)))/d^2-20*a*b*x^2*polylog(2,-exp(c+d*x^(1/2)))/d^2+20*a*b*x^2*polylog(2,exp(c+d*
x^(1/2)))/d^2+20*b^2*x^(3/2)*polylog(2,exp(2*c+2*d*x^(1/2)))/d^3+80*a*b*x^(3/2)*polylog(3,-exp(c+d*x^(1/2)))/d
^3-80*a*b*x^(3/2)*polylog(3,exp(c+d*x^(1/2)))/d^3-30*b^2*x*polylog(3,exp(2*c+2*d*x^(1/2)))/d^4-240*a*b*x*polyl
og(4,-exp(c+d*x^(1/2)))/d^4+240*a*b*x*polylog(4,exp(c+d*x^(1/2)))/d^4-15*b^2*polylog(5,exp(2*c+2*d*x^(1/2)))/d
^6-480*a*b*polylog(6,-exp(c+d*x^(1/2)))/d^6+480*a*b*polylog(6,exp(c+d*x^(1/2)))/d^6+30*b^2*polylog(4,exp(2*c+2
*d*x^(1/2)))*x^(1/2)/d^5+480*a*b*polylog(5,-exp(c+d*x^(1/2)))*x^(1/2)/d^5-480*a*b*polylog(5,exp(c+d*x^(1/2)))*
x^(1/2)/d^5

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5545, 4275, 4267, 2611, 6744, 2320, 6724, 4269, 3797, 2221} \[ \int x^2 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^3}{3}-\frac {8 a b x^{5/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {480 a b \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {480 a b \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6}+\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {240 a b x \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {240 a b x \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {20 a b x^2 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 a b x^2 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}-\frac {15 b^2 \operatorname {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {30 b^2 \sqrt {x} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {30 b^2 x \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {20 b^2 x^{3/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 b^2 x^{5/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {2 b^2 x^{5/2}}{d} \]

[In]

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

[Out]

(-2*b^2*x^(5/2))/d + (a^2*x^3)/3 - (8*a*b*x^(5/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (2*b^2*x^(5/2)*Coth[c + d*Sq
rt[x]])/d + (10*b^2*x^2*Log[1 - E^(2*(c + d*Sqrt[x]))])/d^2 - (20*a*b*x^2*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2
+ (20*a*b*x^2*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (20*b^2*x^(3/2)*PolyLog[2, E^(2*(c + d*Sqrt[x]))])/d^3 + (8
0*a*b*x^(3/2)*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (80*a*b*x^(3/2)*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (30*b
^2*x*PolyLog[3, E^(2*(c + d*Sqrt[x]))])/d^4 - (240*a*b*x*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (240*a*b*x*Poly
Log[4, E^(c + d*Sqrt[x])])/d^4 + (30*b^2*Sqrt[x]*PolyLog[4, E^(2*(c + d*Sqrt[x]))])/d^5 + (480*a*b*Sqrt[x]*Pol
yLog[5, -E^(c + d*Sqrt[x])])/d^5 - (480*a*b*Sqrt[x]*PolyLog[5, E^(c + d*Sqrt[x])])/d^5 - (15*b^2*PolyLog[5, E^
(2*(c + d*Sqrt[x]))])/d^6 - (480*a*b*PolyLog[6, -E^(c + d*Sqrt[x])])/d^6 + (480*a*b*PolyLog[6, E^(c + d*Sqrt[x
])])/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 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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4275

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

Rule 5545

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Csch[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*} \text {integral}& = 2 \text {Subst}\left (\int x^5 (a+b \text {csch}(c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^5+2 a b x^5 \text {csch}(c+d x)+b^2 x^5 \text {csch}^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^3}{3}+(4 a b) \text {Subst}\left (\int x^5 \text {csch}(c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^5 \text {csch}^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^3}{3}-\frac {8 a b x^{5/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{5/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {(20 a b) \text {Subst}\left (\int x^4 \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(20 a b) \text {Subst}\left (\int x^4 \log \left (1+e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (10 b^2\right ) \text {Subst}\left (\int x^4 \coth (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 a b x^{5/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{5/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {20 a b x^2 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 a b x^2 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(80 a b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(80 a b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (20 b^2\right ) \text {Subst}\left (\int \frac {e^{2 (c+d x)} x^4}{1-e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 a b x^{5/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{5/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 a b x^2 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 a b x^2 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(240 a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {(240 a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (40 b^2\right ) \text {Subst}\left (\int x^3 \log \left (1-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = -\frac {2 b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 a b x^{5/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{5/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 a b x^2 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 a b x^2 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 b^2 x^{3/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {240 a b x \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {240 a b x \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {(480 a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (4,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(480 a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (4,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {\left (60 b^2\right ) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = -\frac {2 b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 a b x^{5/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{5/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 a b x^2 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 a b x^2 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 b^2 x^{3/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {30 b^2 x \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {240 a b x \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {240 a b x \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {(480 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (5,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^5}+\frac {(480 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (5,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^5}+\frac {\left (60 b^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4} \\ & = -\frac {2 b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 a b x^{5/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{5/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 a b x^2 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 a b x^2 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 b^2 x^{3/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {30 b^2 x \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {240 a b x \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {240 a b x \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {30 b^2 \sqrt {x} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {(480 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(5,-x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^6}+\frac {(480 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(5,x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^6}-\frac {\left (30 b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (4,e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5} \\ & = -\frac {2 b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 a b x^{5/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{5/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 a b x^2 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 a b x^2 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 b^2 x^{3/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {30 b^2 x \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {240 a b x \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {240 a b x \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {30 b^2 \sqrt {x} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {480 a b \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {480 a b \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,x)}{x} \, dx,x,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6} \\ & = -\frac {2 b^2 x^{5/2}}{d}+\frac {a^2 x^3}{3}-\frac {8 a b x^{5/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{5/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {20 a b x^2 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 a b x^2 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {20 b^2 x^{3/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {80 a b x^{3/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {30 b^2 x \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {240 a b x \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {240 a b x \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {30 b^2 \sqrt {x} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {480 a b \sqrt {x} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {15 b^2 \operatorname {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {480 a b \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {480 a b \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1017\) vs. \(2(441)=882\).

Time = 7.15 (sec) , antiderivative size = 1017, normalized size of antiderivative = 2.31 \[ \int x^2 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \sinh ^2\left (c+d \sqrt {x}\right )}{3 \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )^2}-\frac {2 b \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \left (2 b d^5 x^{5/2}-5 b d^4 \left (-1+e^{2 c}\right ) x^2 \log \left (1-e^{-c-d \sqrt {x}}\right )-2 a d^5 \left (-1+e^{2 c}\right ) x^{5/2} \log \left (1-e^{-c-d \sqrt {x}}\right )-5 b d^4 \left (-1+e^{2 c}\right ) x^2 \log \left (1+e^{-c-d \sqrt {x}}\right )+2 a d^5 \left (-1+e^{2 c}\right ) x^{5/2} \log \left (1+e^{-c-d \sqrt {x}}\right )+20 b d^3 \left (-1+e^{2 c}\right ) x^{3/2} \operatorname {PolyLog}\left (2,-e^{-c-d \sqrt {x}}\right )-10 a d^4 \left (-1+e^{2 c}\right ) x^2 \operatorname {PolyLog}\left (2,-e^{-c-d \sqrt {x}}\right )+20 b d^3 \left (-1+e^{2 c}\right ) x^{3/2} \operatorname {PolyLog}\left (2,e^{-c-d \sqrt {x}}\right )+10 a d^4 \left (-1+e^{2 c}\right ) x^2 \operatorname {PolyLog}\left (2,e^{-c-d \sqrt {x}}\right )+60 b d^2 \left (-1+e^{2 c}\right ) x \operatorname {PolyLog}\left (3,-e^{-c-d \sqrt {x}}\right )-40 a d^3 \left (-1+e^{2 c}\right ) x^{3/2} \operatorname {PolyLog}\left (3,-e^{-c-d \sqrt {x}}\right )+60 b d^2 \left (-1+e^{2 c}\right ) x \operatorname {PolyLog}\left (3,e^{-c-d \sqrt {x}}\right )+40 a d^3 \left (-1+e^{2 c}\right ) x^{3/2} \operatorname {PolyLog}\left (3,e^{-c-d \sqrt {x}}\right )+120 b d \left (-1+e^{2 c}\right ) \sqrt {x} \operatorname {PolyLog}\left (4,-e^{-c-d \sqrt {x}}\right )-120 a d^2 \left (-1+e^{2 c}\right ) x \operatorname {PolyLog}\left (4,-e^{-c-d \sqrt {x}}\right )+120 b d \left (-1+e^{2 c}\right ) \sqrt {x} \operatorname {PolyLog}\left (4,e^{-c-d \sqrt {x}}\right )+120 a d^2 \left (-1+e^{2 c}\right ) x \operatorname {PolyLog}\left (4,e^{-c-d \sqrt {x}}\right )+120 b \left (-1+e^{2 c}\right ) \operatorname {PolyLog}\left (5,-e^{-c-d \sqrt {x}}\right )-240 a d \left (-1+e^{2 c}\right ) \sqrt {x} \operatorname {PolyLog}\left (5,-e^{-c-d \sqrt {x}}\right )+120 b \left (-1+e^{2 c}\right ) \operatorname {PolyLog}\left (5,e^{-c-d \sqrt {x}}\right )+240 a d \left (-1+e^{2 c}\right ) \sqrt {x} \operatorname {PolyLog}\left (5,e^{-c-d \sqrt {x}}\right )-240 a \left (-1+e^{2 c}\right ) \operatorname {PolyLog}\left (6,-e^{-c-d \sqrt {x}}\right )+240 a \left (-1+e^{2 c}\right ) \operatorname {PolyLog}\left (6,e^{-c-d \sqrt {x}}\right )\right ) \sinh ^2\left (c+d \sqrt {x}\right )}{d^6 \left (-1+e^{2 c}\right ) \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )^2}+\frac {b^2 x^{5/2} \text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right ) \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \sinh ^2\left (c+d \sqrt {x}\right ) \sinh \left (\frac {d \sqrt {x}}{2}\right )}{d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )^2}-\frac {b^2 x^{5/2} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right ) \sinh ^2\left (c+d \sqrt {x}\right ) \sinh \left (\frac {d \sqrt {x}}{2}\right )}{d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )^2} \]

[In]

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

[Out]

(a^2*x^3*(a + b*Csch[c + d*Sqrt[x]])^2*Sinh[c + d*Sqrt[x]]^2)/(3*(b + a*Sinh[c + d*Sqrt[x]])^2) - (2*b*(a + b*
Csch[c + d*Sqrt[x]])^2*(2*b*d^5*x^(5/2) - 5*b*d^4*(-1 + E^(2*c))*x^2*Log[1 - E^(-c - d*Sqrt[x])] - 2*a*d^5*(-1
+ E^(2*c))*x^(5/2)*Log[1 - E^(-c - d*Sqrt[x])] - 5*b*d^4*(-1 + E^(2*c))*x^2*Log[1 + E^(-c - d*Sqrt[x])] + 2*a
*d^5*(-1 + E^(2*c))*x^(5/2)*Log[1 + E^(-c - d*Sqrt[x])] + 20*b*d^3*(-1 + E^(2*c))*x^(3/2)*PolyLog[2, -E^(-c -
d*Sqrt[x])] - 10*a*d^4*(-1 + E^(2*c))*x^2*PolyLog[2, -E^(-c - d*Sqrt[x])] + 20*b*d^3*(-1 + E^(2*c))*x^(3/2)*Po
lyLog[2, E^(-c - d*Sqrt[x])] + 10*a*d^4*(-1 + E^(2*c))*x^2*PolyLog[2, E^(-c - d*Sqrt[x])] + 60*b*d^2*(-1 + E^(
2*c))*x*PolyLog[3, -E^(-c - d*Sqrt[x])] - 40*a*d^3*(-1 + E^(2*c))*x^(3/2)*PolyLog[3, -E^(-c - d*Sqrt[x])] + 60
*b*d^2*(-1 + E^(2*c))*x*PolyLog[3, E^(-c - d*Sqrt[x])] + 40*a*d^3*(-1 + E^(2*c))*x^(3/2)*PolyLog[3, E^(-c - d*
Sqrt[x])] + 120*b*d*(-1 + E^(2*c))*Sqrt[x]*PolyLog[4, -E^(-c - d*Sqrt[x])] - 120*a*d^2*(-1 + E^(2*c))*x*PolyLo
g[4, -E^(-c - d*Sqrt[x])] + 120*b*d*(-1 + E^(2*c))*Sqrt[x]*PolyLog[4, E^(-c - d*Sqrt[x])] + 120*a*d^2*(-1 + E^
(2*c))*x*PolyLog[4, E^(-c - d*Sqrt[x])] + 120*b*(-1 + E^(2*c))*PolyLog[5, -E^(-c - d*Sqrt[x])] - 240*a*d*(-1 +
E^(2*c))*Sqrt[x]*PolyLog[5, -E^(-c - d*Sqrt[x])] + 120*b*(-1 + E^(2*c))*PolyLog[5, E^(-c - d*Sqrt[x])] + 240*
a*d*(-1 + E^(2*c))*Sqrt[x]*PolyLog[5, E^(-c - d*Sqrt[x])] - 240*a*(-1 + E^(2*c))*PolyLog[6, -E^(-c - d*Sqrt[x]
)] + 240*a*(-1 + E^(2*c))*PolyLog[6, E^(-c - d*Sqrt[x])])*Sinh[c + d*Sqrt[x]]^2)/(d^6*(-1 + E^(2*c))*(b + a*Si
nh[c + d*Sqrt[x]])^2) + (b^2*x^(5/2)*Csch[c/2]*Csch[c/2 + (d*Sqrt[x])/2]*(a + b*Csch[c + d*Sqrt[x]])^2*Sinh[c
+ d*Sqrt[x]]^2*Sinh[(d*Sqrt[x])/2])/(d*(b + a*Sinh[c + d*Sqrt[x]])^2) - (b^2*x^(5/2)*(a + b*Csch[c + d*Sqrt[x]
])^2*Sech[c/2]*Sech[c/2 + (d*Sqrt[x])/2]*Sinh[c + d*Sqrt[x]]^2*Sinh[(d*Sqrt[x])/2])/(d*(b + a*Sinh[c + d*Sqrt[
x]])^2)

Maple [F]

\[\int x^{2} \left (a +b \,\operatorname {csch}\left (c +d \sqrt {x}\right )\right )^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.12 \[ \int x^2 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {1}{3} \, a^{2} x^{3} - \frac {4 \, b^{2} x^{\frac {5}{2}}}{d e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} - d} - \frac {4 \, {\left (d^{5} x^{\frac {5}{2}} \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 5 \, d^{4} x^{2} {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 20 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 60 \, d^{2} x {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )}) - 120 \, d \sqrt {x} {\rm Li}_{5}(-e^{\left (d \sqrt {x} + c\right )}) + 120 \, {\rm Li}_{6}(-e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{6}} + \frac {4 \, {\left (d^{5} x^{\frac {5}{2}} \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 5 \, d^{4} x^{2} {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 20 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 60 \, d^{2} x {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )}) - 120 \, d \sqrt {x} {\rm Li}_{5}(e^{\left (d \sqrt {x} + c\right )}) + 120 \, {\rm Li}_{6}(e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{6}} + \frac {10 \, {\left (d^{4} x^{2} \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 4 \, d^{3} x^{\frac {3}{2}} {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 12 \, d^{2} x {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 24 \, d \sqrt {x} {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )}) - 24 \, {\rm Li}_{5}(-e^{\left (d \sqrt {x} + c\right )})\right )} b^{2}}{d^{6}} + \frac {10 \, {\left (d^{4} x^{2} \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 4 \, d^{3} x^{\frac {3}{2}} {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 12 \, d^{2} x {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 24 \, d \sqrt {x} {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )}) - 24 \, {\rm Li}_{5}(e^{\left (d \sqrt {x} + c\right )})\right )} b^{2}}{d^{6}} - \frac {2 \, {\left (a b d^{6} x^{3} + 3 \, b^{2} d^{5} x^{\frac {5}{2}}\right )}}{3 \, d^{6}} + \frac {2 \, {\left (a b d^{6} x^{3} - 3 \, b^{2} d^{5} x^{\frac {5}{2}}\right )}}{3 \, d^{6}} \]

[In]

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

[Out]

1/3*a^2*x^3 - 4*b^2*x^(5/2)/(d*e^(2*d*sqrt(x) + 2*c) - d) - 4*(d^5*x^(5/2)*log(e^(d*sqrt(x) + c) + 1) + 5*d^4*
x^2*dilog(-e^(d*sqrt(x) + c)) - 20*d^3*x^(3/2)*polylog(3, -e^(d*sqrt(x) + c)) + 60*d^2*x*polylog(4, -e^(d*sqrt
(x) + c)) - 120*d*sqrt(x)*polylog(5, -e^(d*sqrt(x) + c)) + 120*polylog(6, -e^(d*sqrt(x) + c)))*a*b/d^6 + 4*(d^
5*x^(5/2)*log(-e^(d*sqrt(x) + c) + 1) + 5*d^4*x^2*dilog(e^(d*sqrt(x) + c)) - 20*d^3*x^(3/2)*polylog(3, e^(d*sq
rt(x) + c)) + 60*d^2*x*polylog(4, e^(d*sqrt(x) + c)) - 120*d*sqrt(x)*polylog(5, e^(d*sqrt(x) + c)) + 120*polyl
og(6, e^(d*sqrt(x) + c)))*a*b/d^6 + 10*(d^4*x^2*log(e^(d*sqrt(x) + c) + 1) + 4*d^3*x^(3/2)*dilog(-e^(d*sqrt(x)
+ c)) - 12*d^2*x*polylog(3, -e^(d*sqrt(x) + c)) + 24*d*sqrt(x)*polylog(4, -e^(d*sqrt(x) + c)) - 24*polylog(5,
-e^(d*sqrt(x) + c)))*b^2/d^6 + 10*(d^4*x^2*log(-e^(d*sqrt(x) + c) + 1) + 4*d^3*x^(3/2)*dilog(e^(d*sqrt(x) + c
)) - 12*d^2*x*polylog(3, e^(d*sqrt(x) + c)) + 24*d*sqrt(x)*polylog(4, e^(d*sqrt(x) + c)) - 24*polylog(5, e^(d*
sqrt(x) + c)))*b^2/d^6 - 2/3*(a*b*d^6*x^3 + 3*b^2*d^5*x^(5/2))/d^6 + 2/3*(a*b*d^6*x^3 - 3*b^2*d^5*x^(5/2))/d^6

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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