3.130 \(\int \coth ^7(x) \text {csch}^3(x) \, dx\)

Optimal. Leaf size=33 \[ -\frac {1}{9} \text {csch}^9(x)-\frac {3 \text {csch}^7(x)}{7}-\frac {3 \text {csch}^5(x)}{5}-\frac {\text {csch}^3(x)}{3} \]

[Out]

-1/3*csch(x)^3-3/5*csch(x)^5-3/7*csch(x)^7-1/9*csch(x)^9

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Rubi [A]  time = 0.03, antiderivative size = 33, 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 = {2606, 270} \[ -\frac {1}{9} \text {csch}^9(x)-\frac {3 \text {csch}^7(x)}{7}-\frac {3 \text {csch}^5(x)}{5}-\frac {\text {csch}^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^7*Csch[x]^3,x]

[Out]

-Csch[x]^3/3 - (3*Csch[x]^5)/5 - (3*Csch[x]^7)/7 - Csch[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 2606

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

Rubi steps

\begin {align*} \int \coth ^7(x) \text {csch}^3(x) \, dx &=-\left (i \operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,-i \text {csch}(x)\right )\right )\\ &=-\left (i \operatorname {Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,-i \text {csch}(x)\right )\right )\\ &=-\frac {1}{3} \text {csch}^3(x)-\frac {3 \text {csch}^5(x)}{5}-\frac {3 \text {csch}^7(x)}{7}-\frac {\text {csch}^9(x)}{9}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.00 \[ -\frac {1}{9} \text {csch}^9(x)-\frac {3 \text {csch}^7(x)}{7}-\frac {3 \text {csch}^5(x)}{5}-\frac {\text {csch}^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^7*Csch[x]^3,x]

[Out]

-1/3*Csch[x]^3 - (3*Csch[x]^5)/5 - (3*Csch[x]^7)/7 - Csch[x]^9/9

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fricas [B]  time = 0.68, size = 442, normalized size = 13.39 \[ -\frac {8 \, {\left (105 \, \cosh \relax (x)^{8} + 840 \, \cosh \relax (x) \sinh \relax (x)^{7} + 105 \, \sinh \relax (x)^{8} + 42 \, {\left (70 \, \cosh \relax (x)^{2} + 3\right )} \sinh \relax (x)^{6} + 126 \, \cosh \relax (x)^{6} + 84 \, {\left (70 \, \cosh \relax (x)^{3} + 9 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 6 \, {\left (1225 \, \cosh \relax (x)^{4} + 315 \, \cosh \relax (x)^{2} + 136\right )} \sinh \relax (x)^{4} + 816 \, \cosh \relax (x)^{4} + 24 \, {\left (245 \, \cosh \relax (x)^{5} + 105 \, \cosh \relax (x)^{3} + 101 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 2 \, {\left (1470 \, \cosh \relax (x)^{6} + 945 \, \cosh \relax (x)^{4} + 2448 \, \cosh \relax (x)^{2} + 241\right )} \sinh \relax (x)^{2} + 482 \, \cosh \relax (x)^{2} + 4 \, {\left (210 \, \cosh \relax (x)^{7} + 189 \, \cosh \relax (x)^{5} + 606 \, \cosh \relax (x)^{3} + 115 \, \cosh \relax (x)\right )} \sinh \relax (x) + 711\right )}}{315 \, {\left (\cosh \relax (x)^{11} + 11 \, \cosh \relax (x) \sinh \relax (x)^{10} + \sinh \relax (x)^{11} + {\left (55 \, \cosh \relax (x)^{2} - 9\right )} \sinh \relax (x)^{9} - 9 \, \cosh \relax (x)^{9} + 3 \, {\left (55 \, \cosh \relax (x)^{3} - 27 \, \cosh \relax (x)\right )} \sinh \relax (x)^{8} + {\left (330 \, \cosh \relax (x)^{4} - 324 \, \cosh \relax (x)^{2} + 37\right )} \sinh \relax (x)^{7} + 35 \, \cosh \relax (x)^{7} + 7 \, {\left (66 \, \cosh \relax (x)^{5} - 108 \, \cosh \relax (x)^{3} + 35 \, \cosh \relax (x)\right )} \sinh \relax (x)^{6} + 3 \, {\left (154 \, \cosh \relax (x)^{6} - 378 \, \cosh \relax (x)^{4} + 259 \, \cosh \relax (x)^{2} - 31\right )} \sinh \relax (x)^{5} - 75 \, \cosh \relax (x)^{5} + {\left (330 \, \cosh \relax (x)^{7} - 1134 \, \cosh \relax (x)^{5} + 1225 \, \cosh \relax (x)^{3} - 375 \, \cosh \relax (x)\right )} \sinh \relax (x)^{4} + {\left (165 \, \cosh \relax (x)^{8} - 756 \, \cosh \relax (x)^{6} + 1295 \, \cosh \relax (x)^{4} - 930 \, \cosh \relax (x)^{2} + 162\right )} \sinh \relax (x)^{3} + 90 \, \cosh \relax (x)^{3} + {\left (55 \, \cosh \relax (x)^{9} - 324 \, \cosh \relax (x)^{7} + 735 \, \cosh \relax (x)^{5} - 750 \, \cosh \relax (x)^{3} + 270 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + {\left (11 \, \cosh \relax (x)^{10} - 81 \, \cosh \relax (x)^{8} + 259 \, \cosh \relax (x)^{6} - 465 \, \cosh \relax (x)^{4} + 486 \, \cosh \relax (x)^{2} - 210\right )} \sinh \relax (x) - 42 \, \cosh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^7*csch(x)^3,x, algorithm="fricas")

[Out]

-8/315*(105*cosh(x)^8 + 840*cosh(x)*sinh(x)^7 + 105*sinh(x)^8 + 42*(70*cosh(x)^2 + 3)*sinh(x)^6 + 126*cosh(x)^
6 + 84*(70*cosh(x)^3 + 9*cosh(x))*sinh(x)^5 + 6*(1225*cosh(x)^4 + 315*cosh(x)^2 + 136)*sinh(x)^4 + 816*cosh(x)
^4 + 24*(245*cosh(x)^5 + 105*cosh(x)^3 + 101*cosh(x))*sinh(x)^3 + 2*(1470*cosh(x)^6 + 945*cosh(x)^4 + 2448*cos
h(x)^2 + 241)*sinh(x)^2 + 482*cosh(x)^2 + 4*(210*cosh(x)^7 + 189*cosh(x)^5 + 606*cosh(x)^3 + 115*cosh(x))*sinh
(x) + 711)/(cosh(x)^11 + 11*cosh(x)*sinh(x)^10 + sinh(x)^11 + (55*cosh(x)^2 - 9)*sinh(x)^9 - 9*cosh(x)^9 + 3*(
55*cosh(x)^3 - 27*cosh(x))*sinh(x)^8 + (330*cosh(x)^4 - 324*cosh(x)^2 + 37)*sinh(x)^7 + 35*cosh(x)^7 + 7*(66*c
osh(x)^5 - 108*cosh(x)^3 + 35*cosh(x))*sinh(x)^6 + 3*(154*cosh(x)^6 - 378*cosh(x)^4 + 259*cosh(x)^2 - 31)*sinh
(x)^5 - 75*cosh(x)^5 + (330*cosh(x)^7 - 1134*cosh(x)^5 + 1225*cosh(x)^3 - 375*cosh(x))*sinh(x)^4 + (165*cosh(x
)^8 - 756*cosh(x)^6 + 1295*cosh(x)^4 - 930*cosh(x)^2 + 162)*sinh(x)^3 + 90*cosh(x)^3 + (55*cosh(x)^9 - 324*cos
h(x)^7 + 735*cosh(x)^5 - 750*cosh(x)^3 + 270*cosh(x))*sinh(x)^2 + (11*cosh(x)^10 - 81*cosh(x)^8 + 259*cosh(x)^
6 - 465*cosh(x)^4 + 486*cosh(x)^2 - 210)*sinh(x) - 42*cosh(x))

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giac [B]  time = 0.13, size = 54, normalized size = 1.64 \[ \frac {8 \, {\left (105 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{6} + 756 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 2160 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 2240\right )}}{315 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^7*csch(x)^3,x, algorithm="giac")

[Out]

8/315*(105*(e^(-x) - e^x)^6 + 756*(e^(-x) - e^x)^4 + 2160*(e^(-x) - e^x)^2 + 2240)/(e^(-x) - e^x)^9

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maple [A]  time = 0.11, size = 38, normalized size = 1.15 \[ -\frac {\cosh ^{6}\relax (x )}{3 \sinh \relax (x )^{9}}+\frac {2 \left (\cosh ^{4}\relax (x )\right )}{5 \sinh \relax (x )^{9}}-\frac {8 \left (\cosh ^{2}\relax (x )\right )}{35 \sinh \relax (x )^{9}}+\frac {16}{315 \sinh \relax (x )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^7*csch(x)^3,x)

[Out]

-1/3/sinh(x)^9*cosh(x)^6+2/5/sinh(x)^9*cosh(x)^4-8/35/sinh(x)^9*cosh(x)^2+16/315/sinh(x)^9

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maxima [B]  time = 0.32, size = 435, normalized size = 13.18 \[ \frac {8 \, e^{\left (-3 \, 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 \, e^{\left (-5 \, 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 {632 \, e^{\left (-7 \, 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 {2848 \, e^{\left (-9 \, x\right )}}{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 )}} + \frac {632 \, e^{\left (-11 \, 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 {16 \, e^{\left (-13 \, 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 {8 \, e^{\left (-15 \, 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 )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^7*csch(x)^3,x, algorithm="maxima")

[Out]

8/3*e^(-3*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) + 16/5*e^(-5*x)/(9*e^(-2*x) - 36*e^(-4*x) + 84*e^(-6*x) - 126*e^(-8*x) + 12
6*e^(-10*x) - 84*e^(-12*x) + 36*e^(-14*x) - 9*e^(-16*x) + e^(-18*x) - 1) + 632/35*e^(-7*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) + 2848/315*e^(-9*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) + 632/35*e^(-11*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) + 16/5*e^(-13*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) + 8/3*e^(-15*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)

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mupad [B]  time = 1.41, size = 372, normalized size = 11.27 \[ -\frac {5872\,{\mathrm {e}}^x}{105\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {\frac {28\,{\mathrm {e}}^{3\,x}}{9}+\frac {28\,{\mathrm {e}}^{5\,x}}{3}+\frac {140\,{\mathrm {e}}^{7\,x}}{9}+\frac {140\,{\mathrm {e}}^{9\,x}}{9}+\frac {28\,{\mathrm {e}}^{11\,x}}{3}+\frac {28\,{\mathrm {e}}^{13\,x}}{9}+\frac {4\,{\mathrm {e}}^{15\,x}}{9}+\frac {4\,{\mathrm {e}}^x}{9}}{9\,{\mathrm {e}}^{2\,x}-36\,{\mathrm {e}}^{4\,x}+84\,{\mathrm {e}}^{6\,x}-126\,{\mathrm {e}}^{8\,x}+126\,{\mathrm {e}}^{10\,x}-84\,{\mathrm {e}}^{12\,x}+36\,{\mathrm {e}}^{14\,x}-9\,{\mathrm {e}}^{16\,x}+{\mathrm {e}}^{18\,x}-1}-\frac {3008\,{\mathrm {e}}^x}{21\,\left (15\,{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}-20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}-6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1\right )}-\frac {704\,{\mathrm {e}}^x}{45\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {256\,{\mathrm {e}}^x}{9\,\left (28\,{\mathrm {e}}^{4\,x}-8\,{\mathrm {e}}^{2\,x}-56\,{\mathrm {e}}^{6\,x}+70\,{\mathrm {e}}^{8\,x}-56\,{\mathrm {e}}^{10\,x}+28\,{\mathrm {e}}^{12\,x}-8\,{\mathrm {e}}^{14\,x}+{\mathrm {e}}^{16\,x}+1\right )}-\frac {36608\,{\mathrm {e}}^x}{315\,\left (5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1\right )}-\frac {20\,{\mathrm {e}}^x}{9\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {2048\,{\mathrm {e}}^x}{21\,\left (7\,{\mathrm {e}}^{2\,x}-21\,{\mathrm {e}}^{4\,x}+35\,{\mathrm {e}}^{6\,x}-35\,{\mathrm {e}}^{8\,x}+21\,{\mathrm {e}}^{10\,x}-7\,{\mathrm {e}}^{12\,x}+{\mathrm {e}}^{14\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^7/sinh(x)^3,x)

[Out]

- (5872*exp(x))/(105*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1)) - ((28*exp(3*x))/9 + (28*exp(5*x))
/3 + (140*exp(7*x))/9 + (140*exp(9*x))/9 + (28*exp(11*x))/3 + (28*exp(13*x))/9 + (4*exp(15*x))/9 + (4*exp(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*ex
p(16*x) + exp(18*x) - 1) - (3008*exp(x))/(21*(15*exp(4*x) - 6*exp(2*x) - 20*exp(6*x) + 15*exp(8*x) - 6*exp(10*
x) + exp(12*x) + 1)) - (704*exp(x))/(45*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (256*exp(x))/(9*(28*exp(4*
x) - 8*exp(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)) - (3
6608*exp(x))/(315*(5*exp(2*x) - 10*exp(4*x) + 10*exp(6*x) - 5*exp(8*x) + exp(10*x) - 1)) - (20*exp(x))/(9*(exp
(4*x) - 2*exp(2*x) + 1)) - (2048*exp(x))/(21*(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 \[ \int \coth ^{7}{\relax (x )} \operatorname {csch}^{3}{\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**7*csch(x)**3,x)

[Out]

Integral(coth(x)**7*csch(x)**3, x)

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