\(\int x (a+b \text {csch}(c+d \sqrt {x}))^2 \, dx\) [38]

Optimal result

Integrand size = 18, antiderivative size = 287 \[ \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {2 b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{3/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {6 b^2 x \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 a b x \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 a b x \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 b^2 \sqrt {x} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {3 b^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {24 a b \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {24 a b \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4} \]

[Out]

-2*b^2*x^(3/2)/d+1/2*a^2*x^2-8*a*b*x^(3/2)*arctanh(exp(c+d*x^(1/2)))/d-2*b^2*x^(3/2)*coth(c+d*x^(1/2))/d+6*b^2
*x*ln(1-exp(2*c+2*d*x^(1/2)))/d^2-12*a*b*x*polylog(2,-exp(c+d*x^(1/2)))/d^2+12*a*b*x*polylog(2,exp(c+d*x^(1/2)
))/d^2-3*b^2*polylog(3,exp(2*c+2*d*x^(1/2)))/d^4-24*a*b*polylog(4,-exp(c+d*x^(1/2)))/d^4+24*a*b*polylog(4,exp(
c+d*x^(1/2)))/d^4+6*b^2*polylog(2,exp(2*c+2*d*x^(1/2)))*x^(1/2)/d^3+24*a*b*polylog(3,-exp(c+d*x^(1/2)))*x^(1/2
)/d^3-24*a*b*polylog(3,exp(c+d*x^(1/2)))*x^(1/2)/d^3

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5545, 4275, 4267, 2611, 6744, 2320, 6724, 4269, 3797, 2221} \[ \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {24 a b \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {24 a b \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 a b x \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 a b x \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}-\frac {3 b^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {6 b^2 \sqrt {x} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {6 b^2 x \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 b^2 x^{3/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {2 b^2 x^{3/2}}{d} \]

[In]

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

[Out]

(-2*b^2*x^(3/2))/d + (a^2*x^2)/2 - (8*a*b*x^(3/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (2*b^2*x^(3/2)*Coth[c + d*Sq
rt[x]])/d + (6*b^2*x*Log[1 - E^(2*(c + d*Sqrt[x]))])/d^2 - (12*a*b*x*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (12
*a*b*x*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (6*b^2*Sqrt[x]*PolyLog[2, E^(2*(c + d*Sqrt[x]))])/d^3 + (24*a*b*Sq
rt[x]*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (24*a*b*Sqrt[x]*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (3*b^2*PolyLo
g[3, E^(2*(c + d*Sqrt[x]))])/d^4 - (24*a*b*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (24*a*b*PolyLog[4, E^(c + d*S
qrt[x])])/d^4

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^3 (a+b \text {csch}(c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^3+2 a b x^3 \text {csch}(c+d x)+b^2 x^3 \text {csch}^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^2}{2}+(4 a b) \text {Subst}\left (\int x^3 \text {csch}(c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^3 \text {csch}^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{3/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {(12 a b) \text {Subst}\left (\int x^2 \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(12 a b) \text {Subst}\left (\int x^2 \log \left (1+e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int x^2 \coth (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{3/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {12 a b x \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 a b x \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(24 a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(24 a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {e^{2 (c+d x)} x^2}{1-e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{3/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {6 b^2 x \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 a b x \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 a b x \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(24 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {(24 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int x \log \left (1-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = -\frac {2 b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{3/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {6 b^2 x \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 a b x \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 a b x \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 b^2 \sqrt {x} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(24 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {(24 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^4}-\frac {\left (6 b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = -\frac {2 b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{3/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {6 b^2 x \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 a b x \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 a b x \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 b^2 \sqrt {x} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {24 a b \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {24 a b \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4} \\ & = -\frac {2 b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{3/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {6 b^2 x \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 a b x \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 a b x \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 b^2 \sqrt {x} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {3 b^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {24 a b \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {24 a b \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.98 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.33 \[ \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {4 b^2 d^3 x^{3/2}+a^2 d^4 x^2-4 b^2 d^3 x^{3/2} \coth \left (c+d \sqrt {x}\right )+12 b^2 d^2 x \log \left (1-e^{-c-d \sqrt {x}}\right )+8 a b d^3 x^{3/2} \log \left (1-e^{-c-d \sqrt {x}}\right )+12 b^2 d^2 x \log \left (1+e^{-c-d \sqrt {x}}\right )-8 a b d^3 x^{3/2} \log \left (1+e^{-c-d \sqrt {x}}\right )+24 \left (-b^2 d \sqrt {x}+a b d^2 x\right ) \operatorname {PolyLog}\left (2,-e^{-c-d \sqrt {x}}\right )-24 b d \left (b+a d \sqrt {x}\right ) \sqrt {x} \operatorname {PolyLog}\left (2,e^{-c-d \sqrt {x}}\right )-24 b^2 \operatorname {PolyLog}\left (3,-e^{-c-d \sqrt {x}}\right )+48 a b d \sqrt {x} \operatorname {PolyLog}\left (3,-e^{-c-d \sqrt {x}}\right )-24 b^2 \operatorname {PolyLog}\left (3,e^{-c-d \sqrt {x}}\right )-48 a b d \sqrt {x} \operatorname {PolyLog}\left (3,e^{-c-d \sqrt {x}}\right )+48 a b \operatorname {PolyLog}\left (4,-e^{-c-d \sqrt {x}}\right )-48 a b \operatorname {PolyLog}\left (4,e^{-c-d \sqrt {x}}\right )}{2 d^4} \]

[In]

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

[Out]

(4*b^2*d^3*x^(3/2) + a^2*d^4*x^2 - 4*b^2*d^3*x^(3/2)*Coth[c + d*Sqrt[x]] + 12*b^2*d^2*x*Log[1 - E^(-c - d*Sqrt
[x])] + 8*a*b*d^3*x^(3/2)*Log[1 - E^(-c - d*Sqrt[x])] + 12*b^2*d^2*x*Log[1 + E^(-c - d*Sqrt[x])] - 8*a*b*d^3*x
^(3/2)*Log[1 + E^(-c - d*Sqrt[x])] + 24*(-(b^2*d*Sqrt[x]) + a*b*d^2*x)*PolyLog[2, -E^(-c - d*Sqrt[x])] - 24*b*
d*(b + a*d*Sqrt[x])*Sqrt[x]*PolyLog[2, E^(-c - d*Sqrt[x])] - 24*b^2*PolyLog[3, -E^(-c - d*Sqrt[x])] + 48*a*b*d
*Sqrt[x]*PolyLog[3, -E^(-c - d*Sqrt[x])] - 24*b^2*PolyLog[3, E^(-c - d*Sqrt[x])] - 48*a*b*d*Sqrt[x]*PolyLog[3,
E^(-c - d*Sqrt[x])] + 48*a*b*PolyLog[4, -E^(-c - d*Sqrt[x])] - 48*a*b*PolyLog[4, E^(-c - d*Sqrt[x])])/(2*d^4)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.20 \[ \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {1}{2} \, a^{2} x^{2} - \frac {4 \, b^{2} x^{\frac {3}{2}}}{d e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} - d} - \frac {4 \, {\left (d^{3} x^{\frac {3}{2}} \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 3 \, d^{2} x {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 6 \, d \sqrt {x} {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{4}} + \frac {4 \, {\left (d^{3} x^{\frac {3}{2}} \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 3 \, d^{2} x {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 6 \, d \sqrt {x} {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 6 \, {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{4}} + \frac {6 \, {\left (d^{2} x \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 2 \, d \sqrt {x} {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )})\right )} b^{2}}{d^{4}} + \frac {6 \, {\left (d^{2} x \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 2 \, d \sqrt {x} {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )})\right )} b^{2}}{d^{4}} - \frac {a b d^{4} x^{2} + 2 \, b^{2} d^{3} x^{\frac {3}{2}}}{d^{4}} + \frac {a b d^{4} x^{2} - 2 \, b^{2} d^{3} x^{\frac {3}{2}}}{d^{4}} \]

[In]

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

[Out]

1/2*a^2*x^2 - 4*b^2*x^(3/2)/(d*e^(2*d*sqrt(x) + 2*c) - d) - 4*(d^3*x^(3/2)*log(e^(d*sqrt(x) + c) + 1) + 3*d^2*
x*dilog(-e^(d*sqrt(x) + c)) - 6*d*sqrt(x)*polylog(3, -e^(d*sqrt(x) + c)) + 6*polylog(4, -e^(d*sqrt(x) + c)))*a
*b/d^4 + 4*(d^3*x^(3/2)*log(-e^(d*sqrt(x) + c) + 1) + 3*d^2*x*dilog(e^(d*sqrt(x) + c)) - 6*d*sqrt(x)*polylog(3
, e^(d*sqrt(x) + c)) + 6*polylog(4, e^(d*sqrt(x) + c)))*a*b/d^4 + 6*(d^2*x*log(e^(d*sqrt(x) + c) + 1) + 2*d*sq
rt(x)*dilog(-e^(d*sqrt(x) + c)) - 2*polylog(3, -e^(d*sqrt(x) + c)))*b^2/d^4 + 6*(d^2*x*log(-e^(d*sqrt(x) + c)
+ 1) + 2*d*sqrt(x)*dilog(e^(d*sqrt(x) + c)) - 2*polylog(3, e^(d*sqrt(x) + c)))*b^2/d^4 - (a*b*d^4*x^2 + 2*b^2*
d^3*x^(3/2))/d^4 + (a*b*d^4*x^2 - 2*b^2*d^3*x^(3/2))/d^4

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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