\(\int x^3 (a+b \text {csch}(c+d \sqrt {x}))^2 \, dx\) [36]

Optimal result

Integrand size = 20, antiderivative size = 597 \[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 a b x \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6}+\frac {315 b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{c+d \sqrt {x}}\right )}{d^7}-\frac {315 b^2 \operatorname {PolyLog}\left (7,e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {20160 a b \operatorname {PolyLog}\left (8,-e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 a b \operatorname {PolyLog}\left (8,e^{c+d \sqrt {x}}\right )}{d^8} \]

[Out]

14*b^2*x^3*ln(1-exp(2*c+2*d*x^(1/2)))/d^2+42*b^2*x^(5/2)*polylog(2,exp(2*c+2*d*x^(1/2)))/d^3-105*b^2*x^2*polyl
og(3,exp(2*c+2*d*x^(1/2)))/d^4-315/2*b^2*polylog(7,exp(2*c+2*d*x^(1/2)))/d^8-2*b^2*x^(7/2)/d+1/4*a^2*x^4-20160
*a*b*polylog(7,exp(c+d*x^(1/2)))*x^(1/2)/d^7-10080*a*b*x*polylog(6,-exp(c+d*x^(1/2)))/d^6+10080*a*b*x*polylog(
6,exp(c+d*x^(1/2)))/d^6+20160*a*b*polylog(7,-exp(c+d*x^(1/2)))*x^(1/2)/d^7+168*a*b*x^(5/2)*polylog(3,-exp(c+d*
x^(1/2)))/d^3-168*a*b*x^(5/2)*polylog(3,exp(c+d*x^(1/2)))/d^3-840*a*b*x^2*polylog(4,-exp(c+d*x^(1/2)))/d^4+840
*a*b*x^2*polylog(4,exp(c+d*x^(1/2)))/d^4+3360*a*b*x^(3/2)*polylog(5,-exp(c+d*x^(1/2)))/d^5-3360*a*b*x^(3/2)*po
lylog(5,exp(c+d*x^(1/2)))/d^5-8*a*b*x^(7/2)*arctanh(exp(c+d*x^(1/2)))/d-28*a*b*x^3*polylog(2,-exp(c+d*x^(1/2))
)/d^2+28*a*b*x^3*polylog(2,exp(c+d*x^(1/2)))/d^2+210*b^2*x^(3/2)*polylog(4,exp(2*c+2*d*x^(1/2)))/d^5-315*b^2*x
*polylog(5,exp(2*c+2*d*x^(1/2)))/d^6-20160*a*b*polylog(8,-exp(c+d*x^(1/2)))/d^8+20160*a*b*polylog(8,exp(c+d*x^
(1/2)))/d^8+315*b^2*polylog(6,exp(2*c+2*d*x^(1/2)))*x^(1/2)/d^7-2*b^2*x^(7/2)*coth(c+d*x^(1/2))/d

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.00, number of steps used = 30, 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^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {20160 a b \operatorname {PolyLog}\left (8,-e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 a b \operatorname {PolyLog}\left (8,e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{c+d \sqrt {x}}\right )}{d^7}-\frac {10080 a b x \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {840 a b x^2 \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {28 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}-\frac {315 b^2 \operatorname {PolyLog}\left (7,e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8}+\frac {315 b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {2 b^2 x^{7/2}}{d} \]

[In]

Int[x^3*(a + b*Csch[c + d*Sqrt[x]])^2,x]

[Out]

(-2*b^2*x^(7/2))/d + (a^2*x^4)/4 - (8*a*b*x^(7/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (2*b^2*x^(7/2)*Coth[c + d*Sq
rt[x]])/d + (14*b^2*x^3*Log[1 - E^(2*(c + d*Sqrt[x]))])/d^2 - (28*a*b*x^3*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2
+ (28*a*b*x^3*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (42*b^2*x^(5/2)*PolyLog[2, E^(2*(c + d*Sqrt[x]))])/d^3 + (1
68*a*b*x^(5/2)*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (168*a*b*x^(5/2)*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (10
5*b^2*x^2*PolyLog[3, E^(2*(c + d*Sqrt[x]))])/d^4 - (840*a*b*x^2*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (840*a*b
*x^2*PolyLog[4, E^(c + d*Sqrt[x])])/d^4 + (210*b^2*x^(3/2)*PolyLog[4, E^(2*(c + d*Sqrt[x]))])/d^5 + (3360*a*b*
x^(3/2)*PolyLog[5, -E^(c + d*Sqrt[x])])/d^5 - (3360*a*b*x^(3/2)*PolyLog[5, E^(c + d*Sqrt[x])])/d^5 - (315*b^2*
x*PolyLog[5, E^(2*(c + d*Sqrt[x]))])/d^6 - (10080*a*b*x*PolyLog[6, -E^(c + d*Sqrt[x])])/d^6 + (10080*a*b*x*Pol
yLog[6, E^(c + d*Sqrt[x])])/d^6 + (315*b^2*Sqrt[x]*PolyLog[6, E^(2*(c + d*Sqrt[x]))])/d^7 + (20160*a*b*Sqrt[x]
*PolyLog[7, -E^(c + d*Sqrt[x])])/d^7 - (20160*a*b*Sqrt[x]*PolyLog[7, E^(c + d*Sqrt[x])])/d^7 - (315*b^2*PolyLo
g[7, E^(2*(c + d*Sqrt[x]))])/(2*d^8) - (20160*a*b*PolyLog[8, -E^(c + d*Sqrt[x])])/d^8 + (20160*a*b*PolyLog[8,
E^(c + d*Sqrt[x])])/d^8

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^7 (a+b \text {csch}(c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^7+2 a b x^7 \text {csch}(c+d x)+b^2 x^7 \text {csch}^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^4}{4}+(4 a b) \text {Subst}\left (\int x^7 \text {csch}(c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^7 \text {csch}^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {(28 a b) \text {Subst}\left (\int x^6 \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(28 a b) \text {Subst}\left (\int x^6 \log \left (1+e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (14 b^2\right ) \text {Subst}\left (\int x^6 \coth (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {28 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(168 a b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(168 a b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (28 b^2\right ) \text {Subst}\left (\int \frac {e^{2 (c+d x)} x^6}{1-e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(840 a b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {(840 a b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (84 b^2\right ) \text {Subst}\left (\int x^5 \log \left (1-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = -\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {840 a b x^2 \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {(3360 a b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(3360 a b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {\left (210 b^2\right ) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = -\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {(10080 a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^5}+\frac {(10080 a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^5}+\frac {\left (420 b^2\right ) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4} \\ & = -\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {10080 a b x \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6}+\frac {(20160 a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^6}-\frac {(20160 a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^6}-\frac {\left (630 b^2\right ) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (4,e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5} \\ & = -\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 a b x \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{c+d \sqrt {x}}\right )}{d^7}-\frac {(20160 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^7}+\frac {(20160 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^7}+\frac {\left (630 b^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (5,e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6} \\ & = -\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 a b x \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6}+\frac {315 b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{c+d \sqrt {x}}\right )}{d^7}-\frac {(20160 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(7,-x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^8}+\frac {(20160 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(7,x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^8}-\frac {\left (315 b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (6,e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7} \\ & = -\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 a b x \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6}+\frac {315 b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \operatorname {PolyLog}\left (8,-e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 a b \operatorname {PolyLog}\left (8,e^{c+d \sqrt {x}}\right )}{d^8}-\frac {\left (315 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(6,x)}{x} \, dx,x,e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8} \\ & = -\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 a b x \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6}+\frac {315 b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{c+d \sqrt {x}}\right )}{d^7}-\frac {315 b^2 \operatorname {PolyLog}\left (7,e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {20160 a b \operatorname {PolyLog}\left (8,-e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 a b \operatorname {PolyLog}\left (8,e^{c+d \sqrt {x}}\right )}{d^8} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1289\) vs. \(2(597)=1194\).

Time = 7.40 (sec) , antiderivative size = 1289, normalized size of antiderivative = 2.16 \[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^4 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \sinh ^2\left (c+d \sqrt {x}\right )}{4 \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^7 x^{7/2}-7 b d^6 \left (-1+e^{2 c}\right ) x^3 \log \left (1-e^{-c-d \sqrt {x}}\right )-2 a d^7 \left (-1+e^{2 c}\right ) x^{7/2} \log \left (1-e^{-c-d \sqrt {x}}\right )-7 b d^6 \left (-1+e^{2 c}\right ) x^3 \log \left (1+e^{-c-d \sqrt {x}}\right )+2 a d^7 \left (-1+e^{2 c}\right ) x^{7/2} \log \left (1+e^{-c-d \sqrt {x}}\right )+42 b d^5 \left (-1+e^{2 c}\right ) x^{5/2} \operatorname {PolyLog}\left (2,-e^{-c-d \sqrt {x}}\right )-14 a d^6 \left (-1+e^{2 c}\right ) x^3 \operatorname {PolyLog}\left (2,-e^{-c-d \sqrt {x}}\right )+42 b d^5 \left (-1+e^{2 c}\right ) x^{5/2} \operatorname {PolyLog}\left (2,e^{-c-d \sqrt {x}}\right )+14 a d^6 \left (-1+e^{2 c}\right ) x^3 \operatorname {PolyLog}\left (2,e^{-c-d \sqrt {x}}\right )+210 b d^4 \left (-1+e^{2 c}\right ) x^2 \operatorname {PolyLog}\left (3,-e^{-c-d \sqrt {x}}\right )-84 a d^5 \left (-1+e^{2 c}\right ) x^{5/2} \operatorname {PolyLog}\left (3,-e^{-c-d \sqrt {x}}\right )+210 b d^4 \left (-1+e^{2 c}\right ) x^2 \operatorname {PolyLog}\left (3,e^{-c-d \sqrt {x}}\right )+84 a d^5 \left (-1+e^{2 c}\right ) x^{5/2} \operatorname {PolyLog}\left (3,e^{-c-d \sqrt {x}}\right )+840 b d^3 \left (-1+e^{2 c}\right ) x^{3/2} \operatorname {PolyLog}\left (4,-e^{-c-d \sqrt {x}}\right )-420 a d^4 \left (-1+e^{2 c}\right ) x^2 \operatorname {PolyLog}\left (4,-e^{-c-d \sqrt {x}}\right )+840 b d^3 \left (-1+e^{2 c}\right ) x^{3/2} \operatorname {PolyLog}\left (4,e^{-c-d \sqrt {x}}\right )+420 a d^4 \left (-1+e^{2 c}\right ) x^2 \operatorname {PolyLog}\left (4,e^{-c-d \sqrt {x}}\right )+2520 b d^2 \left (-1+e^{2 c}\right ) x \operatorname {PolyLog}\left (5,-e^{-c-d \sqrt {x}}\right )-1680 a d^3 \left (-1+e^{2 c}\right ) x^{3/2} \operatorname {PolyLog}\left (5,-e^{-c-d \sqrt {x}}\right )+2520 b d^2 \left (-1+e^{2 c}\right ) x \operatorname {PolyLog}\left (5,e^{-c-d \sqrt {x}}\right )+1680 a d^3 \left (-1+e^{2 c}\right ) x^{3/2} \operatorname {PolyLog}\left (5,e^{-c-d \sqrt {x}}\right )+5040 b d \left (-1+e^{2 c}\right ) \sqrt {x} \operatorname {PolyLog}\left (6,-e^{-c-d \sqrt {x}}\right )-5040 a d^2 \left (-1+e^{2 c}\right ) x \operatorname {PolyLog}\left (6,-e^{-c-d \sqrt {x}}\right )+5040 b d \left (-1+e^{2 c}\right ) \sqrt {x} \operatorname {PolyLog}\left (6,e^{-c-d \sqrt {x}}\right )+5040 a d^2 \left (-1+e^{2 c}\right ) x \operatorname {PolyLog}\left (6,e^{-c-d \sqrt {x}}\right )+5040 b \left (-1+e^{2 c}\right ) \operatorname {PolyLog}\left (7,-e^{-c-d \sqrt {x}}\right )-10080 a d \left (-1+e^{2 c}\right ) \sqrt {x} \operatorname {PolyLog}\left (7,-e^{-c-d \sqrt {x}}\right )+5040 b \left (-1+e^{2 c}\right ) \operatorname {PolyLog}\left (7,e^{-c-d \sqrt {x}}\right )+10080 a d \left (-1+e^{2 c}\right ) \sqrt {x} \operatorname {PolyLog}\left (7,e^{-c-d \sqrt {x}}\right )-10080 a \left (-1+e^{2 c}\right ) \operatorname {PolyLog}\left (8,-e^{-c-d \sqrt {x}}\right )+10080 a \left (-1+e^{2 c}\right ) \operatorname {PolyLog}\left (8,e^{-c-d \sqrt {x}}\right )\right ) \sinh ^2\left (c+d \sqrt {x}\right )}{d^8 \left (-1+e^{2 c}\right ) \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )^2}+\frac {b^2 x^{7/2} \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^{7/2} \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^3*(a + b*Csch[c + d*Sqrt[x]])^2,x]

[Out]

(a^2*x^4*(a + b*Csch[c + d*Sqrt[x]])^2*Sinh[c + d*Sqrt[x]]^2)/(4*(b + a*Sinh[c + d*Sqrt[x]])^2) - (2*b*(a + b*
Csch[c + d*Sqrt[x]])^2*(2*b*d^7*x^(7/2) - 7*b*d^6*(-1 + E^(2*c))*x^3*Log[1 - E^(-c - d*Sqrt[x])] - 2*a*d^7*(-1
+ E^(2*c))*x^(7/2)*Log[1 - E^(-c - d*Sqrt[x])] - 7*b*d^6*(-1 + E^(2*c))*x^3*Log[1 + E^(-c - d*Sqrt[x])] + 2*a
*d^7*(-1 + E^(2*c))*x^(7/2)*Log[1 + E^(-c - d*Sqrt[x])] + 42*b*d^5*(-1 + E^(2*c))*x^(5/2)*PolyLog[2, -E^(-c -
d*Sqrt[x])] - 14*a*d^6*(-1 + E^(2*c))*x^3*PolyLog[2, -E^(-c - d*Sqrt[x])] + 42*b*d^5*(-1 + E^(2*c))*x^(5/2)*Po
lyLog[2, E^(-c - d*Sqrt[x])] + 14*a*d^6*(-1 + E^(2*c))*x^3*PolyLog[2, E^(-c - d*Sqrt[x])] + 210*b*d^4*(-1 + E^
(2*c))*x^2*PolyLog[3, -E^(-c - d*Sqrt[x])] - 84*a*d^5*(-1 + E^(2*c))*x^(5/2)*PolyLog[3, -E^(-c - d*Sqrt[x])] +
210*b*d^4*(-1 + E^(2*c))*x^2*PolyLog[3, E^(-c - d*Sqrt[x])] + 84*a*d^5*(-1 + E^(2*c))*x^(5/2)*PolyLog[3, E^(-
c - d*Sqrt[x])] + 840*b*d^3*(-1 + E^(2*c))*x^(3/2)*PolyLog[4, -E^(-c - d*Sqrt[x])] - 420*a*d^4*(-1 + E^(2*c))*
x^2*PolyLog[4, -E^(-c - d*Sqrt[x])] + 840*b*d^3*(-1 + E^(2*c))*x^(3/2)*PolyLog[4, E^(-c - d*Sqrt[x])] + 420*a*
d^4*(-1 + E^(2*c))*x^2*PolyLog[4, E^(-c - d*Sqrt[x])] + 2520*b*d^2*(-1 + E^(2*c))*x*PolyLog[5, -E^(-c - d*Sqrt
[x])] - 1680*a*d^3*(-1 + E^(2*c))*x^(3/2)*PolyLog[5, -E^(-c - d*Sqrt[x])] + 2520*b*d^2*(-1 + E^(2*c))*x*PolyLo
g[5, E^(-c - d*Sqrt[x])] + 1680*a*d^3*(-1 + E^(2*c))*x^(3/2)*PolyLog[5, E^(-c - d*Sqrt[x])] + 5040*b*d*(-1 + E
^(2*c))*Sqrt[x]*PolyLog[6, -E^(-c - d*Sqrt[x])] - 5040*a*d^2*(-1 + E^(2*c))*x*PolyLog[6, -E^(-c - d*Sqrt[x])]
+ 5040*b*d*(-1 + E^(2*c))*Sqrt[x]*PolyLog[6, E^(-c - d*Sqrt[x])] + 5040*a*d^2*(-1 + E^(2*c))*x*PolyLog[6, E^(-
c - d*Sqrt[x])] + 5040*b*(-1 + E^(2*c))*PolyLog[7, -E^(-c - d*Sqrt[x])] - 10080*a*d*(-1 + E^(2*c))*Sqrt[x]*Pol
yLog[7, -E^(-c - d*Sqrt[x])] + 5040*b*(-1 + E^(2*c))*PolyLog[7, E^(-c - d*Sqrt[x])] + 10080*a*d*(-1 + E^(2*c))
*Sqrt[x]*PolyLog[7, E^(-c - d*Sqrt[x])] - 10080*a*(-1 + E^(2*c))*PolyLog[8, -E^(-c - d*Sqrt[x])] + 10080*a*(-1
+ E^(2*c))*PolyLog[8, E^(-c - d*Sqrt[x])])*Sinh[c + d*Sqrt[x]]^2)/(d^8*(-1 + E^(2*c))*(b + a*Sinh[c + d*Sqrt[
x]])^2) + (b^2*x^(7/2)*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^(7/2)*(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]

\[\int x^{3} \left (a +b \,\operatorname {csch}\left (c +d \sqrt {x}\right )\right )^{2}d x\]

[In]

int(x^3*(a+b*csch(c+d*x^(1/2)))^2,x)

[Out]

int(x^3*(a+b*csch(c+d*x^(1/2)))^2,x)

Fricas [F]

\[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{3} \,d x } \]

[In]

integrate(x^3*(a+b*csch(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^3*csch(d*sqrt(x) + c)^2 + 2*a*b*x^3*csch(d*sqrt(x) + c) + a^2*x^3, x)

Sympy [F]

\[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^{3} \left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]

[In]

integrate(x**3*(a+b*csch(c+d*x**(1/2)))**2,x)

[Out]

Integral(x**3*(a + b*csch(c + d*sqrt(x)))**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 648, normalized size of antiderivative = 1.09 \[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {1}{4} \, a^{2} x^{4} - \frac {4 \, b^{2} x^{\frac {7}{2}}}{d e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} - d} - \frac {4 \, {\left (d^{7} x^{\frac {7}{2}} \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 7 \, d^{6} x^{3} {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 42 \, d^{5} x^{\frac {5}{2}} {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 210 \, d^{4} x^{2} {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )}) - 840 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{5}(-e^{\left (d \sqrt {x} + c\right )}) + 2520 \, d^{2} x {\rm Li}_{6}(-e^{\left (d \sqrt {x} + c\right )}) - 5040 \, d \sqrt {x} {\rm Li}_{7}(-e^{\left (d \sqrt {x} + c\right )}) + 5040 \, {\rm Li}_{8}(-e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{8}} + \frac {4 \, {\left (d^{7} x^{\frac {7}{2}} \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 7 \, d^{6} x^{3} {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 42 \, d^{5} x^{\frac {5}{2}} {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 210 \, d^{4} x^{2} {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )}) - 840 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{5}(e^{\left (d \sqrt {x} + c\right )}) + 2520 \, d^{2} x {\rm Li}_{6}(e^{\left (d \sqrt {x} + c\right )}) - 5040 \, d \sqrt {x} {\rm Li}_{7}(e^{\left (d \sqrt {x} + c\right )}) + 5040 \, {\rm Li}_{8}(e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{8}} + \frac {14 \, {\left (d^{6} x^{3} \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 6 \, d^{5} x^{\frac {5}{2}} {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 30 \, d^{4} x^{2} {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 120 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )}) - 360 \, d^{2} x {\rm Li}_{5}(-e^{\left (d \sqrt {x} + c\right )}) + 720 \, d \sqrt {x} {\rm Li}_{6}(-e^{\left (d \sqrt {x} + c\right )}) - 720 \, {\rm Li}_{7}(-e^{\left (d \sqrt {x} + c\right )})\right )} b^{2}}{d^{8}} + \frac {14 \, {\left (d^{6} x^{3} \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 6 \, d^{5} x^{\frac {5}{2}} {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 30 \, d^{4} x^{2} {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 120 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )}) - 360 \, d^{2} x {\rm Li}_{5}(e^{\left (d \sqrt {x} + c\right )}) + 720 \, d \sqrt {x} {\rm Li}_{6}(e^{\left (d \sqrt {x} + c\right )}) - 720 \, {\rm Li}_{7}(e^{\left (d \sqrt {x} + c\right )})\right )} b^{2}}{d^{8}} - \frac {a b d^{8} x^{4} + 4 \, b^{2} d^{7} x^{\frac {7}{2}}}{2 \, d^{8}} + \frac {a b d^{8} x^{4} - 4 \, b^{2} d^{7} x^{\frac {7}{2}}}{2 \, d^{8}} \]

[In]

integrate(x^3*(a+b*csch(c+d*x^(1/2)))^2,x, algorithm="maxima")

[Out]

1/4*a^2*x^4 - 4*b^2*x^(7/2)/(d*e^(2*d*sqrt(x) + 2*c) - d) - 4*(d^7*x^(7/2)*log(e^(d*sqrt(x) + c) + 1) + 7*d^6*
x^3*dilog(-e^(d*sqrt(x) + c)) - 42*d^5*x^(5/2)*polylog(3, -e^(d*sqrt(x) + c)) + 210*d^4*x^2*polylog(4, -e^(d*s
qrt(x) + c)) - 840*d^3*x^(3/2)*polylog(5, -e^(d*sqrt(x) + c)) + 2520*d^2*x*polylog(6, -e^(d*sqrt(x) + c)) - 50
40*d*sqrt(x)*polylog(7, -e^(d*sqrt(x) + c)) + 5040*polylog(8, -e^(d*sqrt(x) + c)))*a*b/d^8 + 4*(d^7*x^(7/2)*lo
g(-e^(d*sqrt(x) + c) + 1) + 7*d^6*x^3*dilog(e^(d*sqrt(x) + c)) - 42*d^5*x^(5/2)*polylog(3, e^(d*sqrt(x) + c))
+ 210*d^4*x^2*polylog(4, e^(d*sqrt(x) + c)) - 840*d^3*x^(3/2)*polylog(5, e^(d*sqrt(x) + c)) + 2520*d^2*x*polyl
og(6, e^(d*sqrt(x) + c)) - 5040*d*sqrt(x)*polylog(7, e^(d*sqrt(x) + c)) + 5040*polylog(8, e^(d*sqrt(x) + c)))*
a*b/d^8 + 14*(d^6*x^3*log(e^(d*sqrt(x) + c) + 1) + 6*d^5*x^(5/2)*dilog(-e^(d*sqrt(x) + c)) - 30*d^4*x^2*polylo
g(3, -e^(d*sqrt(x) + c)) + 120*d^3*x^(3/2)*polylog(4, -e^(d*sqrt(x) + c)) - 360*d^2*x*polylog(5, -e^(d*sqrt(x)
+ c)) + 720*d*sqrt(x)*polylog(6, -e^(d*sqrt(x) + c)) - 720*polylog(7, -e^(d*sqrt(x) + c)))*b^2/d^8 + 14*(d^6*
x^3*log(-e^(d*sqrt(x) + c) + 1) + 6*d^5*x^(5/2)*dilog(e^(d*sqrt(x) + c)) - 30*d^4*x^2*polylog(3, e^(d*sqrt(x)
+ c)) + 120*d^3*x^(3/2)*polylog(4, e^(d*sqrt(x) + c)) - 360*d^2*x*polylog(5, e^(d*sqrt(x) + c)) + 720*d*sqrt(x
)*polylog(6, e^(d*sqrt(x) + c)) - 720*polylog(7, e^(d*sqrt(x) + c)))*b^2/d^8 - 1/2*(a*b*d^8*x^4 + 4*b^2*d^7*x^
(7/2))/d^8 + 1/2*(a*b*d^8*x^4 - 4*b^2*d^7*x^(7/2))/d^8

Giac [F]

\[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{3} \,d x } \]

[In]

integrate(x^3*(a+b*csch(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

integrate((b*csch(d*sqrt(x) + c) + a)^2*x^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^3\,{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]

[In]

int(x^3*(a + b/sinh(c + d*x^(1/2)))^2,x)

[Out]

int(x^3*(a + b/sinh(c + d*x^(1/2)))^2, x)