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]

-4/3*(1+csch(x))^(3/2)+4/5*(1+csch(x))^(5/2)-2/7*(1+csch(x))^(7/2)

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Rubi [A]  time = 0.11, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 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]

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 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 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])

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*}

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Mathematica [A]  time = 0.05, 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]

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

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fricas [B]  time = 0.41, size = 271, normalized size = 7.32 \[ -\frac {2 \, {\left (43 \, \cosh \relax (x)^{6} + 2 \, {\left (129 \, \cosh \relax (x) + 31\right )} \sinh \relax (x)^{5} + 43 \, \sinh \relax (x)^{6} + 62 \, \cosh \relax (x)^{5} + {\left (645 \, \cosh \relax (x)^{2} + 310 \, \cosh \relax (x) - 117\right )} \sinh \relax (x)^{4} - 117 \, \cosh \relax (x)^{4} + 4 \, {\left (215 \, \cosh \relax (x)^{3} + 155 \, \cosh \relax (x)^{2} - 117 \, \cosh \relax (x) - 1\right )} \sinh \relax (x)^{3} - 4 \, \cosh \relax (x)^{3} + {\left (645 \, \cosh \relax (x)^{4} + 620 \, \cosh \relax (x)^{3} - 702 \, \cosh \relax (x)^{2} - 12 \, \cosh \relax (x) + 117\right )} \sinh \relax (x)^{2} + 117 \, \cosh \relax (x)^{2} + 2 \, {\left (129 \, \cosh \relax (x)^{5} + 155 \, \cosh \relax (x)^{4} - 234 \, \cosh \relax (x)^{3} - 6 \, \cosh \relax (x)^{2} + 117 \, \cosh \relax (x) + 31\right )} \sinh \relax (x) + 62 \, \cosh \relax (x) - 43\right )} \sqrt {\frac {\sinh \relax (x) + 1}{\sinh \relax (x)}}}{105 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{4} - 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} - 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)^{2} + 6 \, {\left (\cosh \relax (x)^{5} - 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) - 1\right )}} \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {csch}\relax (x) + 1} \coth \relax (x)^{3} \operatorname {csch}\relax (x)\,{d x} \]

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)

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maple [F]  time = 0.45, size = 0, normalized size = 0.00 \[ \int \left (\coth ^{3}\relax (x )\right ) \mathrm {csch}\relax (x ) \sqrt {1+\mathrm {csch}\relax (x )}\, dx \]

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)

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maxima [B]  time = 0.42, size = 389, normalized size = 10.51 \[ \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 )}} \]

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))

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mupad [B]  time = 2.41, size = 207, normalized size = 5.59 \[ -\frac {8\,\sqrt {1-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}}{35\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {8\,\sqrt {1-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}}{35\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {86\,\sqrt {1-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}}{105}-\frac {16\,{\mathrm {e}}^x\,\sqrt {1-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}}{7\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {16\,{\mathrm {e}}^x\,\sqrt {1-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}}{7\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {124\,{\mathrm {e}}^x\,\sqrt {1-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}}{105\,\left ({\mathrm {e}}^{2\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (8*(1 - 1/(exp(-x)/2 - exp(x)/2))^(1/2))/(35*(exp(2*x) - 1)) - (8*(1 - 1/(exp(-x)/2 - exp(x)/2))^(1/2))/(35*
(exp(4*x) - 2*exp(2*x) + 1)) - (86*(1 - 1/(exp(-x)/2 - exp(x)/2))^(1/2))/105 - (16*exp(x)*(1 - 1/(exp(-x)/2 -
exp(x)/2))^(1/2))/(7*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (16*exp(x)*(1 - 1/(exp(-x)/2 - exp(x)/2))^(1/
2))/(7*(exp(4*x) - 2*exp(2*x) + 1)) - (124*exp(x)*(1 - 1/(exp(-x)/2 - exp(x)/2))^(1/2))/(105*(exp(2*x) - 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {csch}{\relax (x )} + 1} \coth ^{3}{\relax (x )} \operatorname {csch}{\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(csch(x) + 1)*coth(x)**3*csch(x), x)

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