### 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]

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

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Rubi [A]  time = 0.0336737, antiderivative size = 33, 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 = {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 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])

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 ^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*}

Mathematica [A]  time = 0.0117182, size = 33, normalized size = 1. $-\frac{1}{9} \text{csch}^9(x)-\frac{3 \text{csch}^7(x)}{7}-\frac{3 \text{csch}^5(x)}{5}-\frac{\text{csch}^3(x)}{3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.015, size = 76, normalized size = 2.3 \begin{align*} -{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{6}}{3\, \left ( \sinh \left ( x \right ) \right ) ^{9}}}+{\frac{2\, \left ( \cosh \left ( x \right ) \right ) ^{4}}{5\, \left ( \sinh \left ( x \right ) \right ) ^{9}}}-{\frac{8\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{45\, \left ( \sinh \left ( x \right ) \right ) ^{9}}}-{\frac{16\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{315\, \left ( \sinh \left ( x \right ) \right ) ^{7}}}+{\frac{16\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{315\, \left ( \sinh \left ( x \right ) \right ) ^{5}}}-{\frac{16\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{315\, \left ( \sinh \left ( x \right ) \right ) ^{3}}}+{\frac{16\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{315\,\sinh \left ( x \right ) }}-{\frac{16\,\sinh \left ( x \right ) }{315}} \end{align*}

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)^4-8/45/sinh(x)^9*cosh(x)^2-16/315/sinh(x)^7*cosh(x)^2+16/315/si
nh(x)^5*cosh(x)^2-16/315/sinh(x)^3*cosh(x)^2+16/315/sinh(x)*cosh(x)^2-16/315*sinh(x)

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Maxima [B]  time = 1.06554, size = 587, normalized size = 17.79 \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [B]  time = 1.77903, size = 1571, normalized size = 47.61 \begin{align*} \text{result too large to display} \end{align*}

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

[Out]

Timed out

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Giac [B]  time = 1.17492, size = 73, normalized size = 2.21 \begin{align*} \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}} \end{align*}

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