### 3.128 $$\int \coth ^4(x) \text{csch}^6(x) \, dx$$

Optimal. Leaf size=25 $-\frac{1}{9} \coth ^9(x)+\frac{2 \coth ^7(x)}{7}-\frac{\coth ^5(x)}{5}$

[Out]

-Coth[x]^5/5 + (2*Coth[x]^7)/7 - Coth[x]^9/9

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Rubi [A]  time = 0.0293946, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {2607, 270} $-\frac{1}{9} \coth ^9(x)+\frac{2 \coth ^7(x)}{7}-\frac{\coth ^5(x)}{5}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[x]^4*Csch[x]^6,x]

[Out]

-Coth[x]^5/5 + (2*Coth[x]^7)/7 - Coth[x]^9/9

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \coth ^4(x) \text{csch}^6(x) \, dx &=i \operatorname{Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,i \coth (x)\right )\\ &=i \operatorname{Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,i \coth (x)\right )\\ &=-\frac{1}{5} \coth ^5(x)+\frac{2 \coth ^7(x)}{7}-\frac{\coth ^9(x)}{9}\\ \end{align*}

Mathematica [A]  time = 0.027158, size = 47, normalized size = 1.88 $-\frac{8 \coth (x)}{315}-\frac{1}{9} \coth (x) \text{csch}^8(x)-\frac{10}{63} \coth (x) \text{csch}^6(x)-\frac{1}{105} \coth (x) \text{csch}^4(x)+\frac{4}{315} \coth (x) \text{csch}^2(x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Coth[x]^4*Csch[x]^6,x]

[Out]

(-8*Coth[x])/315 + (4*Coth[x]*Csch[x]^2)/315 - (Coth[x]*Csch[x]^4)/105 - (10*Coth[x]*Csch[x]^6)/63 - (Coth[x]*
Csch[x]^8)/9

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Maple [B]  time = 0.014, size = 50, normalized size = 2. \begin{align*} -{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{3}}{6\, \left ( \sinh \left ( x \right ) \right ) ^{9}}}+{\frac{\cosh \left ( x \right ) }{16\, \left ( \sinh \left ( x \right ) \right ) ^{9}}}+{\frac{{\rm coth} \left (x\right )}{16} \left ( -{\frac{128}{315}}-{\frac{ \left ({\rm csch} \left (x\right ) \right ) ^{8}}{9}}+{\frac{8\, \left ({\rm csch} \left (x\right ) \right ) ^{6}}{63}}-{\frac{16\, \left ({\rm csch} \left (x\right ) \right ) ^{4}}{105}}+{\frac{64\, \left ({\rm csch} \left (x\right ) \right ) ^{2}}{315}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^4*csch(x)^6,x)

[Out]

-1/6/sinh(x)^9*cosh(x)^3+1/16/sinh(x)^9*cosh(x)+1/16*(-128/315-1/9*csch(x)^8+8/63*csch(x)^6-16/105*csch(x)^4+6
4/315*csch(x)^2)*coth(x)

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Maxima [B]  time = 1.25276, size = 582, normalized size = 23.28 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^4*csch(x)^6,x, algorithm="maxima")

[Out]

-16/35*e^(-2*x)/(9*e^(-2*x) - 36*e^(-4*x) + 84*e^(-6*x) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*e^(
-14*x) - 9*e^(-16*x) + e^(-18*x) - 1) + 64/35*e^(-4*x)/(9*e^(-2*x) - 36*e^(-4*x) + 84*e^(-6*x) - 126*e^(-8*x)
+ 126*e^(-10*x) - 84*e^(-12*x) + 36*e^(-14*x) - 9*e^(-16*x) + e^(-18*x) - 1) + 32/5*e^(-6*x)/(9*e^(-2*x) - 36*
e^(-4*x) + 84*e^(-6*x) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*e^(-14*x) - 9*e^(-16*x) + e^(-18*x)
- 1) + 112/5*e^(-8*x)/(9*e^(-2*x) - 36*e^(-4*x) + 84*e^(-6*x) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x) +
36*e^(-14*x) - 9*e^(-16*x) + e^(-18*x) - 1) + 16*e^(-10*x)/(9*e^(-2*x) - 36*e^(-4*x) + 84*e^(-6*x) - 126*e^(-8
*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*e^(-14*x) - 9*e^(-16*x) + e^(-18*x) - 1) + 32/3*e^(-12*x)/(9*e^(-2*x)
- 36*e^(-4*x) + 84*e^(-6*x) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*e^(-14*x) - 9*e^(-16*x) + e^(-1
8*x) - 1) + 16/315/(9*e^(-2*x) - 36*e^(-4*x) + 84*e^(-6*x) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*
e^(-14*x) - 9*e^(-16*x) + e^(-18*x) - 1)

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Fricas [B]  time = 1.90838, size = 1516, normalized size = 60.64 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^4*csch(x)^6,x, algorithm="fricas")

[Out]

-16/315*(211*cosh(x)^6 + 1254*cosh(x)*sinh(x)^5 + 211*sinh(x)^6 + 3*(1055*cosh(x)^2 + 102)*sinh(x)^4 + 306*cos
h(x)^4 + 4*(1045*cosh(x)^3 + 324*cosh(x))*sinh(x)^3 + 3*(1055*cosh(x)^4 + 612*cosh(x)^2 + 159)*sinh(x)^2 + 477
*cosh(x)^2 + 6*(209*cosh(x)^5 + 216*cosh(x)^3 + 135*cosh(x))*sinh(x) + 126)/(cosh(x)^12 + 12*cosh(x)*sinh(x)^1
1 + sinh(x)^12 + 3*(22*cosh(x)^2 - 3)*sinh(x)^10 - 9*cosh(x)^10 + 10*(22*cosh(x)^3 - 9*cosh(x))*sinh(x)^9 + 9*
(55*cosh(x)^4 - 45*cosh(x)^2 + 4)*sinh(x)^8 + 36*cosh(x)^8 + 72*(11*cosh(x)^5 - 15*cosh(x)^3 + 4*cosh(x))*sinh
(x)^7 + (924*cosh(x)^6 - 1890*cosh(x)^4 + 1008*cosh(x)^2 - 85)*sinh(x)^6 - 85*cosh(x)^6 + 6*(132*cosh(x)^7 - 3
78*cosh(x)^5 + 336*cosh(x)^3 - 83*cosh(x))*sinh(x)^5 + 15*(33*cosh(x)^8 - 126*cosh(x)^6 + 168*cosh(x)^4 - 85*c
osh(x)^2 + 9)*sinh(x)^4 + 135*cosh(x)^4 + 4*(55*cosh(x)^9 - 270*cosh(x)^7 + 504*cosh(x)^5 - 415*cosh(x)^3 + 11
7*cosh(x))*sinh(x)^3 + 3*(22*cosh(x)^10 - 135*cosh(x)^8 + 336*cosh(x)^6 - 425*cosh(x)^4 + 270*cosh(x)^2 - 54)*
sinh(x)^2 - 162*cosh(x)^2 + 6*(2*cosh(x)^11 - 15*cosh(x)^9 + 48*cosh(x)^7 - 83*cosh(x)^5 + 78*cosh(x)^3 - 30*c
osh(x))*sinh(x) + 84)

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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)**4*csch(x)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.17559, size = 65, normalized size = 2.6 \begin{align*} -\frac{16 \,{\left (210 \, e^{\left (12 \, x\right )} + 315 \, e^{\left (10 \, x\right )} + 441 \, e^{\left (8 \, x\right )} + 126 \, e^{\left (6 \, x\right )} + 36 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} + 1\right )}}{315 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^4*csch(x)^6,x, algorithm="giac")

[Out]

-16/315*(210*e^(12*x) + 315*e^(10*x) + 441*e^(8*x) + 126*e^(6*x) + 36*e^(4*x) - 9*e^(2*x) + 1)/(e^(2*x) - 1)^9