### 3.1048 $$\int \coth ^3(x) \text{csch}(x) \sqrt{1+\text{csch}(x)} \, dx$$

Optimal. Leaf size=37 $-\frac{2}{7} (\text{csch}(x)+1)^{7/2}+\frac{4}{5} (\text{csch}(x)+1)^{5/2}-\frac{4}{3} (\text{csch}(x)+1)^{3/2}$

[Out]

(-4*(1 + Csch[x])^(3/2))/3 + (4*(1 + Csch[x])^(5/2))/5 - (2*(1 + Csch[x])^(7/2))/7

________________________________________________________________________________________

Rubi [A]  time = 0.105672, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.267, Rules used = {4372, 1570, 1469, 697} $-\frac{2}{7} (\text{csch}(x)+1)^{7/2}+\frac{4}{5} (\text{csch}(x)+1)^{5/2}-\frac{4}{3} (\text{csch}(x)+1)^{3/2}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[x]^3*Csch[x]*Sqrt[1 + Csch[x]],x]

[Out]

(-4*(1 + Csch[x])^(3/2))/3 + (4*(1 + Csch[x])^(5/2))/5 - (2*(1 + Csch[x])^(7/2))/7

Rule 4372

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist
[1/(b*c*d^(n - 1)), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2)/x^n, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*
(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && N
onsumQ[u] && (EqQ[F, Cot] || EqQ[F, cot])

Rule 1570

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Int[x^(m - 2*n
*p)*(d + e*x^n)^q*(c + a*x^(2*n))^p, x] /; FreeQ[{a, c, d, e, m, n, q}, x] && EqQ[mn2, -2*n] && IntegerQ[p]

Rule 1469

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Sim
plify[m - n + 1], 0]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \coth ^3(x) \text{csch}(x) \sqrt{1+\text{csch}(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{1}{x}} \left (1+x^2\right )}{x^4} \, dx,x,\sinh (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{\left (1+\frac{1}{x^2}\right ) \sqrt{1+\frac{1}{x}}}{x^2} \, dx,x,\sinh (x)\right )\\ &=-\operatorname{Subst}\left (\int \sqrt{1+x} \left (1+x^2\right ) \, dx,x,\text{csch}(x)\right )\\ &=-\operatorname{Subst}\left (\int \left (2 \sqrt{1+x}-2 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,\text{csch}(x)\right )\\ &=-\frac{4}{3} (1+\text{csch}(x))^{3/2}+\frac{4}{5} (1+\text{csch}(x))^{5/2}-\frac{2}{7} (1+\text{csch}(x))^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0585229, size = 34, normalized size = 0.92 $-\frac{1}{210} \text{csch}^3(x) \sqrt{\text{csch}(x)+1} (-117 \sinh (x)+43 \sinh (3 x)+62 \cosh (2 x)-2)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Coth[x]^3*Csch[x]*Sqrt[1 + Csch[x]],x]

[Out]

-(Csch[x]^3*Sqrt[1 + Csch[x]]*(-2 + 62*Cosh[2*x] - 117*Sinh[x] + 43*Sinh[3*x]))/210

________________________________________________________________________________________

Maple [F]  time = 0.113, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (x\right ) \right ) ^{3}{\rm csch} \left (x\right )\sqrt{1+{\rm csch} \left (x\right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3*csch(x)*(1+csch(x))^(1/2),x)

[Out]

int(coth(x)^3*csch(x)*(1+csch(x))^(1/2),x)

________________________________________________________________________________________

Maxima [B]  time = 1.44684, size = 525, normalized size = 14.19 \begin{align*} \frac{124 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-x\right )}}{105 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac{78 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-2 \, x\right )}}{35 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac{8 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-3 \, x\right )}}{105 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{78 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-4 \, x\right )}}{35 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{124 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-5 \, x\right )}}{105 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac{86 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-6 \, x\right )}}{105 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{86 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1}}{105 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3*csch(x)*(1+csch(x))^(1/2),x, algorithm="maxima")

[Out]

124/105*sqrt(-2*e^(-x) + e^(-2*x) - 1)*e^(-x)/(sqrt(e^(-x) + 1)*sqrt(e^(-x) - 1)*(3*e^(-2*x) - 3*e^(-4*x) + e^
(-6*x) - 1)) - 78/35*sqrt(-2*e^(-x) + e^(-2*x) - 1)*e^(-2*x)/(sqrt(e^(-x) + 1)*sqrt(e^(-x) - 1)*(3*e^(-2*x) -
3*e^(-4*x) + e^(-6*x) - 1)) - 8/105*sqrt(-2*e^(-x) + e^(-2*x) - 1)*e^(-3*x)/(sqrt(e^(-x) + 1)*sqrt(e^(-x) - 1)
*(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1)) + 78/35*sqrt(-2*e^(-x) + e^(-2*x) - 1)*e^(-4*x)/(sqrt(e^(-x) + 1)*s
qrt(e^(-x) - 1)*(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1)) + 124/105*sqrt(-2*e^(-x) + e^(-2*x) - 1)*e^(-5*x)/(s
qrt(e^(-x) + 1)*sqrt(e^(-x) - 1)*(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1)) - 86/105*sqrt(-2*e^(-x) + e^(-2*x)
- 1)*e^(-6*x)/(sqrt(e^(-x) + 1)*sqrt(e^(-x) - 1)*(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1)) + 86/105*sqrt(-2*e^
(-x) + e^(-2*x) - 1)/(sqrt(e^(-x) + 1)*sqrt(e^(-x) - 1)*(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1))

________________________________________________________________________________________

Fricas [B]  time = 2.09575, size = 953, normalized size = 25.76 \begin{align*} -\frac{2 \,{\left (43 \, \cosh \left (x\right )^{6} + 2 \,{\left (129 \, \cosh \left (x\right ) + 31\right )} \sinh \left (x\right )^{5} + 43 \, \sinh \left (x\right )^{6} + 62 \, \cosh \left (x\right )^{5} +{\left (645 \, \cosh \left (x\right )^{2} + 310 \, \cosh \left (x\right ) - 117\right )} \sinh \left (x\right )^{4} - 117 \, \cosh \left (x\right )^{4} + 4 \,{\left (215 \, \cosh \left (x\right )^{3} + 155 \, \cosh \left (x\right )^{2} - 117 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} - 4 \, \cosh \left (x\right )^{3} +{\left (645 \, \cosh \left (x\right )^{4} + 620 \, \cosh \left (x\right )^{3} - 702 \, \cosh \left (x\right )^{2} - 12 \, \cosh \left (x\right ) + 117\right )} \sinh \left (x\right )^{2} + 117 \, \cosh \left (x\right )^{2} + 2 \,{\left (129 \, \cosh \left (x\right )^{5} + 155 \, \cosh \left (x\right )^{4} - 234 \, \cosh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{2} + 117 \, \cosh \left (x\right ) + 31\right )} \sinh \left (x\right ) + 62 \, \cosh \left (x\right ) - 43\right )} \sqrt{\frac{\sinh \left (x\right ) + 1}{\sinh \left (x\right )}}}{105 \,{\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \,{\left (5 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{4} + 4 \,{\left (5 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \cosh \left (x\right )^{4} - 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \,{\left (\cosh \left (x\right )^{5} - 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3*csch(x)*(1+csch(x))^(1/2),x, algorithm="fricas")

[Out]

-2/105*(43*cosh(x)^6 + 2*(129*cosh(x) + 31)*sinh(x)^5 + 43*sinh(x)^6 + 62*cosh(x)^5 + (645*cosh(x)^2 + 310*cos
h(x) - 117)*sinh(x)^4 - 117*cosh(x)^4 + 4*(215*cosh(x)^3 + 155*cosh(x)^2 - 117*cosh(x) - 1)*sinh(x)^3 - 4*cosh
(x)^3 + (645*cosh(x)^4 + 620*cosh(x)^3 - 702*cosh(x)^2 - 12*cosh(x) + 117)*sinh(x)^2 + 117*cosh(x)^2 + 2*(129*
cosh(x)^5 + 155*cosh(x)^4 - 234*cosh(x)^3 - 6*cosh(x)^2 + 117*cosh(x) + 31)*sinh(x) + 62*cosh(x) - 43)*sqrt((s
inh(x) + 1)/sinh(x))/(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 - 1)*sinh(x)^4 - 3*cosh(x)^
4 + 4*(5*cosh(x)^3 - 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 - 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cos
h(x)^5 - 2*cosh(x)^3 + cosh(x))*sinh(x) - 1)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**3*csch(x)*(1+csch(x))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\operatorname{csch}\left (x\right ) + 1} \coth \left (x\right )^{3} \operatorname{csch}\left (x\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3*csch(x)*(1+csch(x))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(csch(x) + 1)*coth(x)^3*csch(x), x)