3.52 \(\int \sqrt{x} (a+b \text{csch}(c+d \sqrt{x})) \, dx\)

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

[Out]

(2*a*x^(3/2))/3 - (4*b*x*ArcTanh[E^(c + d*Sqrt[x])])/d - (4*b*Sqrt[x]*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (4
*b*Sqrt[x]*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (4*b*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (4*b*PolyLog[3, E^(
c + d*Sqrt[x])])/d^3

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Rubi [A]  time = 0.117621, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {14, 5437, 4182, 2531, 2282, 6589} \[ -\frac{4 b \sqrt{x} \text{PolyLog}\left (2,-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{4 b \sqrt{x} \text{PolyLog}\left (2,e^{c+d \sqrt{x}}\right )}{d^2}+\frac{4 b \text{PolyLog}\left (3,-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{4 b \text{PolyLog}\left (3,e^{c+d \sqrt{x}}\right )}{d^3}+\frac{2}{3} a x^{3/2}-\frac{4 b x \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*x^(3/2))/3 - (4*b*x*ArcTanh[E^(c + d*Sqrt[x])])/d - (4*b*Sqrt[x]*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (4
*b*Sqrt[x]*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (4*b*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (4*b*PolyLog[3, E^(
c + d*Sqrt[x])])/d^3

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 5437

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 4182

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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

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

Mathematica [A]  time = 9.3006, size = 142, normalized size = 1.18 \[ \frac{2 \left (-6 b d \sqrt{x} \text{PolyLog}\left (2,-e^{c+d \sqrt{x}}\right )+6 b d \sqrt{x} \text{PolyLog}\left (2,e^{c+d \sqrt{x}}\right )+6 b \text{PolyLog}\left (3,-e^{c+d \sqrt{x}}\right )-6 b \text{PolyLog}\left (3,e^{c+d \sqrt{x}}\right )+a d^3 x^{3/2}+3 b d^2 x \log \left (1-e^{c+d \sqrt{x}}\right )-3 b d^2 x \log \left (e^{c+d \sqrt{x}}+1\right )\right )}{3 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*d^3*x^(3/2) + 3*b*d^2*x*Log[1 - E^(c + d*Sqrt[x])] - 3*b*d^2*x*Log[1 + E^(c + d*Sqrt[x])] - 6*b*d*Sqrt[x
]*PolyLog[2, -E^(c + d*Sqrt[x])] + 6*b*d*Sqrt[x]*PolyLog[2, E^(c + d*Sqrt[x])] + 6*b*PolyLog[3, -E^(c + d*Sqrt
[x])] - 6*b*PolyLog[3, E^(c + d*Sqrt[x])]))/(3*d^3)

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm csch} \left (c+d\sqrt{x}\right ) \right ) \sqrt{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.08514, size = 174, normalized size = 1.45 \begin{align*} \frac{2}{3} \, a x^{\frac{3}{2}} - \frac{2 \,{\left (\log \left (e^{\left (d \sqrt{x} + c\right )} + 1\right ) \log \left (e^{\left (d \sqrt{x}\right )}\right )^{2} + 2 \,{\rm Li}_2\left (-e^{\left (d \sqrt{x} + c\right )}\right ) \log \left (e^{\left (d \sqrt{x}\right )}\right ) - 2 \,{\rm Li}_{3}(-e^{\left (d \sqrt{x} + c\right )})\right )} b}{d^{3}} + \frac{2 \,{\left (\log \left (-e^{\left (d \sqrt{x} + c\right )} + 1\right ) \log \left (e^{\left (d \sqrt{x}\right )}\right )^{2} + 2 \,{\rm Li}_2\left (e^{\left (d \sqrt{x} + c\right )}\right ) \log \left (e^{\left (d \sqrt{x}\right )}\right ) - 2 \,{\rm Li}_{3}(e^{\left (d \sqrt{x} + c\right )})\right )} b}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*a*x^(3/2) - 2*(log(e^(d*sqrt(x) + c) + 1)*log(e^(d*sqrt(x)))^2 + 2*dilog(-e^(d*sqrt(x) + c))*log(e^(d*sqrt
(x))) - 2*polylog(3, -e^(d*sqrt(x) + c)))*b/d^3 + 2*(log(-e^(d*sqrt(x) + c) + 1)*log(e^(d*sqrt(x)))^2 + 2*dilo
g(e^(d*sqrt(x) + c))*log(e^(d*sqrt(x))) - 2*polylog(3, e^(d*sqrt(x) + c)))*b/d^3

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b \sqrt{x} \operatorname{csch}\left (d \sqrt{x} + c\right ) + a \sqrt{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{csch}\left (d \sqrt{x} + c\right ) + a\right )} \sqrt{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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