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]

-1/5*coth(x)^5+2/7*coth(x)^7-1/9*coth(x)^9

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Rubi [A]  time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.03, 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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fricas [B]  time = 0.38, size = 430, normalized size = 17.20 \[ -\frac {16 \, {\left (211 \, \cosh \relax (x)^{6} + 1254 \, \cosh \relax (x) \sinh \relax (x)^{5} + 211 \, \sinh \relax (x)^{6} + 3 \, {\left (1055 \, \cosh \relax (x)^{2} + 102\right )} \sinh \relax (x)^{4} + 306 \, \cosh \relax (x)^{4} + 4 \, {\left (1045 \, \cosh \relax (x)^{3} + 324 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (1055 \, \cosh \relax (x)^{4} + 612 \, \cosh \relax (x)^{2} + 159\right )} \sinh \relax (x)^{2} + 477 \, \cosh \relax (x)^{2} + 6 \, {\left (209 \, \cosh \relax (x)^{5} + 216 \, \cosh \relax (x)^{3} + 135 \, \cosh \relax (x)\right )} \sinh \relax (x) + 126\right )}}{315 \, {\left (\cosh \relax (x)^{12} + 12 \, \cosh \relax (x) \sinh \relax (x)^{11} + \sinh \relax (x)^{12} + 3 \, {\left (22 \, \cosh \relax (x)^{2} - 3\right )} \sinh \relax (x)^{10} - 9 \, \cosh \relax (x)^{10} + 10 \, {\left (22 \, \cosh \relax (x)^{3} - 9 \, \cosh \relax (x)\right )} \sinh \relax (x)^{9} + 9 \, {\left (55 \, \cosh \relax (x)^{4} - 45 \, \cosh \relax (x)^{2} + 4\right )} \sinh \relax (x)^{8} + 36 \, \cosh \relax (x)^{8} + 72 \, {\left (11 \, \cosh \relax (x)^{5} - 15 \, \cosh \relax (x)^{3} + 4 \, \cosh \relax (x)\right )} \sinh \relax (x)^{7} + {\left (924 \, \cosh \relax (x)^{6} - 1890 \, \cosh \relax (x)^{4} + 1008 \, \cosh \relax (x)^{2} - 85\right )} \sinh \relax (x)^{6} - 85 \, \cosh \relax (x)^{6} + 6 \, {\left (132 \, \cosh \relax (x)^{7} - 378 \, \cosh \relax (x)^{5} + 336 \, \cosh \relax (x)^{3} - 83 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 15 \, {\left (33 \, \cosh \relax (x)^{8} - 126 \, \cosh \relax (x)^{6} + 168 \, \cosh \relax (x)^{4} - 85 \, \cosh \relax (x)^{2} + 9\right )} \sinh \relax (x)^{4} + 135 \, \cosh \relax (x)^{4} + 4 \, {\left (55 \, \cosh \relax (x)^{9} - 270 \, \cosh \relax (x)^{7} + 504 \, \cosh \relax (x)^{5} - 415 \, \cosh \relax (x)^{3} + 117 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (22 \, \cosh \relax (x)^{10} - 135 \, \cosh \relax (x)^{8} + 336 \, \cosh \relax (x)^{6} - 425 \, \cosh \relax (x)^{4} + 270 \, \cosh \relax (x)^{2} - 54\right )} \sinh \relax (x)^{2} - 162 \, \cosh \relax (x)^{2} + 6 \, {\left (2 \, \cosh \relax (x)^{11} - 15 \, \cosh \relax (x)^{9} + 48 \, \cosh \relax (x)^{7} - 83 \, \cosh \relax (x)^{5} + 78 \, \cosh \relax (x)^{3} - 30 \, \cosh \relax (x)\right )} \sinh \relax (x) + 84\right )}} \]

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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giac [B]  time = 0.12, size = 48, normalized size = 1.92 \[ -\frac {16 \, {\left (210 \, e^{\left (12 \, x\right )} + 315 \, e^{\left (10 \, x\right )} + 441 \, e^{\left (8 \, x\right )} + 126 \, e^{\left (6 \, x\right )} + 36 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} + 1\right )}}{315 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{9}} \]

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

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maple [B]  time = 0.34, size = 50, normalized size = 2.00 \[ -\frac {\cosh ^{3}\relax (x )}{6 \sinh \relax (x )^{9}}+\frac {\cosh \relax (x )}{16 \sinh \relax (x )^{9}}+\frac {\left (-\frac {128}{315}-\frac {\mathrm {csch}\relax (x )^{8}}{9}+\frac {8 \mathrm {csch}\relax (x )^{6}}{63}-\frac {16 \mathrm {csch}\relax (x )^{4}}{105}+\frac {64 \mathrm {csch}\relax (x )^{2}}{315}\right ) \coth \relax (x )}{16} \]

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 = 0.31, size = 431, normalized size = 17.24 \[ -\frac {16 \, e^{\left (-2 \, x\right )}}{35 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac {64 \, e^{\left (-4 \, x\right )}}{35 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac {32 \, e^{\left (-6 \, x\right )}}{5 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac {112 \, e^{\left (-8 \, x\right )}}{5 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac {16 \, e^{\left (-10 \, x\right )}}{9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1} + \frac {32 \, e^{\left (-12 \, x\right )}}{3 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac {16}{315 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} \]

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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mupad [B]  time = 1.39, size = 413, normalized size = 16.52 \[ -\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{9}+\frac {16\,{\mathrm {e}}^{4\,x}}{3}+\frac {32\,{\mathrm {e}}^{6\,x}}{3}+\frac {80\,{\mathrm {e}}^{8\,x}}{9}+\frac {8\,{\mathrm {e}}^{10\,x}}{3}}{28\,{\mathrm {e}}^{4\,x}-8\,{\mathrm {e}}^{2\,x}-56\,{\mathrm {e}}^{6\,x}+70\,{\mathrm {e}}^{8\,x}-56\,{\mathrm {e}}^{10\,x}+28\,{\mathrm {e}}^{12\,x}-8\,{\mathrm {e}}^{14\,x}+{\mathrm {e}}^{16\,x}+1}-\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{21}+\frac {16}{63}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {8}{63\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {\frac {64\,{\mathrm {e}}^{2\,x}}{63}+\frac {16\,{\mathrm {e}}^{4\,x}}{21}+\frac {32}{105}}{5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1}-\frac {\frac {32\,{\mathrm {e}}^{2\,x}}{21}+\frac {160\,{\mathrm {e}}^{4\,x}}{63}+\frac {80\,{\mathrm {e}}^{6\,x}}{63}+\frac {16}{63}}{15\,{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}-20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}-6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1}-\frac {\frac {32\,{\mathrm {e}}^{4\,x}}{9}+\frac {128\,{\mathrm {e}}^{6\,x}}{9}+\frac {64\,{\mathrm {e}}^{8\,x}}{3}+\frac {128\,{\mathrm {e}}^{10\,x}}{9}+\frac {32\,{\mathrm {e}}^{12\,x}}{9}}{9\,{\mathrm {e}}^{2\,x}-36\,{\mathrm {e}}^{4\,x}+84\,{\mathrm {e}}^{6\,x}-126\,{\mathrm {e}}^{8\,x}+126\,{\mathrm {e}}^{10\,x}-84\,{\mathrm {e}}^{12\,x}+36\,{\mathrm {e}}^{14\,x}-9\,{\mathrm {e}}^{16\,x}+{\mathrm {e}}^{18\,x}-1}-\frac {\frac {32\,{\mathrm {e}}^{2\,x}}{21}+\frac {32\,{\mathrm {e}}^{4\,x}}{7}+\frac {320\,{\mathrm {e}}^{6\,x}}{63}+\frac {40\,{\mathrm {e}}^{8\,x}}{21}+\frac {8}{63}}{7\,{\mathrm {e}}^{2\,x}-21\,{\mathrm {e}}^{4\,x}+35\,{\mathrm {e}}^{6\,x}-35\,{\mathrm {e}}^{8\,x}+21\,{\mathrm {e}}^{10\,x}-7\,{\mathrm {e}}^{12\,x}+{\mathrm {e}}^{14\,x}-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^4/sinh(x)^6,x)

[Out]

- ((8*exp(2*x))/9 + (16*exp(4*x))/3 + (32*exp(6*x))/3 + (80*exp(8*x))/9 + (8*exp(10*x))/3)/(28*exp(4*x) - 8*ex
p(2*x) - 56*exp(6*x) + 70*exp(8*x) - 56*exp(10*x) + 28*exp(12*x) - 8*exp(14*x) + exp(16*x) + 1) - ((8*exp(2*x)
)/21 + 16/63)/(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1) - 8/(63*(3*exp(2*x) - 3*exp(4*x) + exp(6*x
) - 1)) - ((64*exp(2*x))/63 + (16*exp(4*x))/21 + 32/105)/(5*exp(2*x) - 10*exp(4*x) + 10*exp(6*x) - 5*exp(8*x)
+ exp(10*x) - 1) - ((32*exp(2*x))/21 + (160*exp(4*x))/63 + (80*exp(6*x))/63 + 16/63)/(15*exp(4*x) - 6*exp(2*x)
- 20*exp(6*x) + 15*exp(8*x) - 6*exp(10*x) + exp(12*x) + 1) - ((32*exp(4*x))/9 + (128*exp(6*x))/9 + (64*exp(8*
x))/3 + (128*exp(10*x))/9 + (32*exp(12*x))/9)/(9*exp(2*x) - 36*exp(4*x) + 84*exp(6*x) - 126*exp(8*x) + 126*exp
(10*x) - 84*exp(12*x) + 36*exp(14*x) - 9*exp(16*x) + exp(18*x) - 1) - ((32*exp(2*x))/21 + (32*exp(4*x))/7 + (3
20*exp(6*x))/63 + (40*exp(8*x))/21 + 8/63)/(7*exp(2*x) - 21*exp(4*x) + 35*exp(6*x) - 35*exp(8*x) + 21*exp(10*x
) - 7*exp(12*x) + exp(14*x) - 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{4}{\relax (x )} \operatorname {csch}^{6}{\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(x)**4*csch(x)**6, x)

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