### 3.653 $$\int \frac{1}{(a \coth (x)+b \text{csch}(x))^5} \, dx$$

Optimal. Leaf size=98 $-\frac{\left (a^2-b^2\right )^2}{4 a^5 (a \cosh (x)+b)^4}-\frac{4 b \left (a^2-b^2\right )}{3 a^5 (a \cosh (x)+b)^3}+\frac{a^2-3 b^2}{a^5 (a \cosh (x)+b)^2}+\frac{4 b}{a^5 (a \cosh (x)+b)}+\frac{\log (a \cosh (x)+b)}{a^5}$

[Out]

-(a^2 - b^2)^2/(4*a^5*(b + a*Cosh[x])^4) - (4*b*(a^2 - b^2))/(3*a^5*(b + a*Cosh[x])^3) + (a^2 - 3*b^2)/(a^5*(b
+ a*Cosh[x])^2) + (4*b)/(a^5*(b + a*Cosh[x])) + Log[b + a*Cosh[x]]/a^5

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Rubi [A]  time = 0.15647, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.273, Rules used = {4392, 2668, 697} $-\frac{\left (a^2-b^2\right )^2}{4 a^5 (a \cosh (x)+b)^4}-\frac{4 b \left (a^2-b^2\right )}{3 a^5 (a \cosh (x)+b)^3}+\frac{a^2-3 b^2}{a^5 (a \cosh (x)+b)^2}+\frac{4 b}{a^5 (a \cosh (x)+b)}+\frac{\log (a \cosh (x)+b)}{a^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*Coth[x] + b*Csch[x])^(-5),x]

[Out]

-(a^2 - b^2)^2/(4*a^5*(b + a*Cosh[x])^4) - (4*b*(a^2 - b^2))/(3*a^5*(b + a*Cosh[x])^3) + (a^2 - 3*b^2)/(a^5*(b
+ a*Cosh[x])^2) + (4*b)/(a^5*(b + a*Cosh[x])) + Log[b + a*Cosh[x]]/a^5

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

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]

Rubi steps

\begin{align*} \int \frac{1}{(a \coth (x)+b \text{csch}(x))^5} \, dx &=i \int \frac{\sinh ^5(x)}{(i b+i a \cosh (x))^5} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (-a^2-x^2\right )^2}{(i b+x)^5} \, dx,x,i a \cosh (x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\left (a^2-b^2\right )^2}{(i b+x)^5}+\frac{4 i b \left (-a^2+b^2\right )}{(i b+x)^4}+\frac{2 \left (a^2-3 b^2\right )}{(i b+x)^3}-\frac{4 i b}{(i b+x)^2}+\frac{1}{i b+x}\right ) \, dx,x,i a \cosh (x)\right )}{a^5}\\ &=-\frac{\left (a^2-b^2\right )^2}{4 a^5 (b+a \cosh (x))^4}-\frac{4 b \left (a^2-b^2\right )}{3 a^5 (b+a \cosh (x))^3}+\frac{a^2-3 b^2}{a^5 (b+a \cosh (x))^2}+\frac{4 b}{a^5 (b+a \cosh (x))}+\frac{\log (b+a \cosh (x))}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.305308, size = 138, normalized size = 1.41 $\frac{12 a^2 \cosh ^2(x) \left (a^2+6 b^2 \log (a \cosh (x)+b)+9 b^2\right )+8 a b \cosh (x) \left (a^2+6 b^2 \log (a \cosh (x)+b)+11 b^2\right )+2 a^2 b^2+12 a^4 \cosh ^4(x) \log (a \cosh (x)+b)+48 a^3 b \cosh ^3(x) (\log (a \cosh (x)+b)+1)-3 a^4+12 b^4 \log (a \cosh (x)+b)+25 b^4}{12 a^5 (a \cosh (x)+b)^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*Coth[x] + b*Csch[x])^(-5),x]

[Out]

(-3*a^4 + 2*a^2*b^2 + 25*b^4 + 12*b^4*Log[b + a*Cosh[x]] + 12*a^4*Cosh[x]^4*Log[b + a*Cosh[x]] + 48*a^3*b*Cosh
[x]^3*(1 + Log[b + a*Cosh[x]]) + 12*a^2*Cosh[x]^2*(a^2 + 9*b^2 + 6*b^2*Log[b + a*Cosh[x]]) + 8*a*b*Cosh[x]*(a^
2 + 11*b^2 + 6*b^2*Log[b + a*Cosh[x]]))/(12*a^5*(b + a*Cosh[x])^4)

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Maple [B]  time = 0.07, size = 309, normalized size = 3.2 \begin{align*} -{\frac{1}{{a}^{5}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-2\,{\frac{1}{{a}^{4} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+a+b \right ) }}-4\,{\frac{a}{ \left ( a-b \right ) ^{2} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+a+b \right ) ^{4}}}-8\,{\frac{b}{ \left ( a-b \right ) ^{2} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+a+b \right ) ^{4}}}-4\,{\frac{{b}^{2}}{a \left ( a-b \right ) ^{2} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+a+b \right ) ^{4}}}-2\,{\frac{1}{{a}^{3} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+a+b \right ) ^{2}}}+{\frac{1}{{a}^{5}}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+a+b \right ) }+8\,{\frac{1}{ \left ( a-b \right ) ^{2} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+a+b \right ) ^{3}}}+{\frac{16\,b}{3\,a \left ( a-b \right ) ^{2}} \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+a+b \right ) ^{-3}}-{\frac{8\,{b}^{2}}{3\,{a}^{2} \left ( a-b \right ) ^{2}} \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+a+b \right ) ^{-3}}-{\frac{1}{{a}^{5}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*coth(x)+b*csch(x))^5,x)

[Out]

-1/a^5*ln(tanh(1/2*x)+1)-2/a^4/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)-4*a/(a-b)^2/(a*tanh(1/2*x)^2-tanh(1/2*x)^
2*b+a+b)^4-8/(a-b)^2/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)^4*b-4/a/(a-b)^2/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+
b)^4*b^2-2/a^3/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)^2+1/a^5*ln(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)+8/(a-b)^2
/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)^3+16/3/a/(a-b)^2/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)^3*b-8/3/a^2/(a-b
)^2/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)^3*b^2-1/a^5*ln(tanh(1/2*x)-1)

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Maxima [B]  time = 1.24709, size = 385, normalized size = 3.93 \begin{align*} \frac{4 \,{\left (6 \, a^{3} b e^{\left (-x\right )} + 6 \, a^{3} b e^{\left (-7 \, x\right )} + 3 \,{\left (a^{4} + 9 \, a^{2} b^{2}\right )} e^{\left (-2 \, x\right )} + 22 \,{\left (a^{3} b + 2 \, a b^{3}\right )} e^{\left (-3 \, x\right )} +{\left (3 \, a^{4} + 56 \, a^{2} b^{2} + 25 \, b^{4}\right )} e^{\left (-4 \, x\right )} + 22 \,{\left (a^{3} b + 2 \, a b^{3}\right )} e^{\left (-5 \, x\right )} + 3 \,{\left (a^{4} + 9 \, a^{2} b^{2}\right )} e^{\left (-6 \, x\right )}\right )}}{3 \,{\left (8 \, a^{8} b e^{\left (-x\right )} + 8 \, a^{8} b e^{\left (-7 \, x\right )} + a^{9} e^{\left (-8 \, x\right )} + a^{9} + 4 \,{\left (a^{9} + 6 \, a^{7} b^{2}\right )} e^{\left (-2 \, x\right )} + 8 \,{\left (3 \, a^{8} b + 4 \, a^{6} b^{3}\right )} e^{\left (-3 \, x\right )} + 2 \,{\left (3 \, a^{9} + 24 \, a^{7} b^{2} + 8 \, a^{5} b^{4}\right )} e^{\left (-4 \, x\right )} + 8 \,{\left (3 \, a^{8} b + 4 \, a^{6} b^{3}\right )} e^{\left (-5 \, x\right )} + 4 \,{\left (a^{9} + 6 \, a^{7} b^{2}\right )} e^{\left (-6 \, x\right )}\right )}} + \frac{x}{a^{5}} + \frac{\log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*coth(x)+b*csch(x))^5,x, algorithm="maxima")

[Out]

4/3*(6*a^3*b*e^(-x) + 6*a^3*b*e^(-7*x) + 3*(a^4 + 9*a^2*b^2)*e^(-2*x) + 22*(a^3*b + 2*a*b^3)*e^(-3*x) + (3*a^4
+ 56*a^2*b^2 + 25*b^4)*e^(-4*x) + 22*(a^3*b + 2*a*b^3)*e^(-5*x) + 3*(a^4 + 9*a^2*b^2)*e^(-6*x))/(8*a^8*b*e^(-
x) + 8*a^8*b*e^(-7*x) + a^9*e^(-8*x) + a^9 + 4*(a^9 + 6*a^7*b^2)*e^(-2*x) + 8*(3*a^8*b + 4*a^6*b^3)*e^(-3*x) +
2*(3*a^9 + 24*a^7*b^2 + 8*a^5*b^4)*e^(-4*x) + 8*(3*a^8*b + 4*a^6*b^3)*e^(-5*x) + 4*(a^9 + 6*a^7*b^2)*e^(-6*x)
) + x/a^5 + log(2*b*e^(-x) + a*e^(-2*x) + a)/a^5

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Fricas [B]  time = 2.57025, size = 6251, normalized size = 63.79 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*coth(x)+b*csch(x))^5,x, algorithm="fricas")

[Out]

-1/3*(3*a^4*x*cosh(x)^8 + 3*a^4*x*sinh(x)^8 + 24*(a^3*b*x - a^3*b)*cosh(x)^7 + 24*(a^4*x*cosh(x) + a^3*b*x - a
^3*b)*sinh(x)^7 - 12*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^2*b^2)*x)*cosh(x)^6 + 12*(7*a^4*x*cosh(x)^2 - a^4 - 9*a^2*b
^2 + (a^4 + 6*a^2*b^2)*x + 14*(a^3*b*x - a^3*b)*cosh(x))*sinh(x)^6 - 8*(11*a^3*b + 22*a*b^3 - 3*(3*a^3*b + 4*a
*b^3)*x)*cosh(x)^5 + 8*(21*a^4*x*cosh(x)^3 - 11*a^3*b - 22*a*b^3 + 63*(a^3*b*x - a^3*b)*cosh(x)^2 + 3*(3*a^3*b
+ 4*a*b^3)*x - 9*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^2*b^2)*x)*cosh(x))*sinh(x)^5 + 3*a^4*x - 2*(6*a^4 + 112*a^2*b^
2 + 50*b^4 - 3*(3*a^4 + 24*a^2*b^2 + 8*b^4)*x)*cosh(x)^4 + 2*(105*a^4*x*cosh(x)^4 - 6*a^4 - 112*a^2*b^2 - 50*b
^4 + 420*(a^3*b*x - a^3*b)*cosh(x)^3 - 90*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^2*b^2)*x)*cosh(x)^2 + 3*(3*a^4 + 24*a^
2*b^2 + 8*b^4)*x - 20*(11*a^3*b + 22*a*b^3 - 3*(3*a^3*b + 4*a*b^3)*x)*cosh(x))*sinh(x)^4 - 8*(11*a^3*b + 22*a*
b^3 - 3*(3*a^3*b + 4*a*b^3)*x)*cosh(x)^3 + 8*(21*a^4*x*cosh(x)^5 + 105*(a^3*b*x - a^3*b)*cosh(x)^4 - 11*a^3*b
- 22*a*b^3 - 30*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^2*b^2)*x)*cosh(x)^3 - 10*(11*a^3*b + 22*a*b^3 - 3*(3*a^3*b + 4*a
*b^3)*x)*cosh(x)^2 + 3*(3*a^3*b + 4*a*b^3)*x - (6*a^4 + 112*a^2*b^2 + 50*b^4 - 3*(3*a^4 + 24*a^2*b^2 + 8*b^4)*
x)*cosh(x))*sinh(x)^3 - 12*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^2*b^2)*x)*cosh(x)^2 + 4*(21*a^4*x*cosh(x)^6 + 126*(a^
3*b*x - a^3*b)*cosh(x)^5 - 45*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^2*b^2)*x)*cosh(x)^4 - 3*a^4 - 27*a^2*b^2 - 20*(11*
a^3*b + 22*a*b^3 - 3*(3*a^3*b + 4*a*b^3)*x)*cosh(x)^3 - 3*(6*a^4 + 112*a^2*b^2 + 50*b^4 - 3*(3*a^4 + 24*a^2*b^
2 + 8*b^4)*x)*cosh(x)^2 + 3*(a^4 + 6*a^2*b^2)*x - 6*(11*a^3*b + 22*a*b^3 - 3*(3*a^3*b + 4*a*b^3)*x)*cosh(x))*s
inh(x)^2 + 24*(a^3*b*x - a^3*b)*cosh(x) - 3*(a^4*cosh(x)^8 + a^4*sinh(x)^8 + 8*a^3*b*cosh(x)^7 + 8*(a^4*cosh(x
) + a^3*b)*sinh(x)^7 + 4*(a^4 + 6*a^2*b^2)*cosh(x)^6 + 4*(7*a^4*cosh(x)^2 + 14*a^3*b*cosh(x) + a^4 + 6*a^2*b^2
)*sinh(x)^6 + 8*(3*a^3*b + 4*a*b^3)*cosh(x)^5 + 8*(7*a^4*cosh(x)^3 + 21*a^3*b*cosh(x)^2 + 3*a^3*b + 4*a*b^3 +
3*(a^4 + 6*a^2*b^2)*cosh(x))*sinh(x)^5 + 8*a^3*b*cosh(x) + 2*(3*a^4 + 24*a^2*b^2 + 8*b^4)*cosh(x)^4 + 2*(35*a^
4*cosh(x)^4 + 140*a^3*b*cosh(x)^3 + 3*a^4 + 24*a^2*b^2 + 8*b^4 + 30*(a^4 + 6*a^2*b^2)*cosh(x)^2 + 20*(3*a^3*b
+ 4*a*b^3)*cosh(x))*sinh(x)^4 + a^4 + 8*(3*a^3*b + 4*a*b^3)*cosh(x)^3 + 8*(7*a^4*cosh(x)^5 + 35*a^3*b*cosh(x)^
4 + 3*a^3*b + 4*a*b^3 + 10*(a^4 + 6*a^2*b^2)*cosh(x)^3 + 10*(3*a^3*b + 4*a*b^3)*cosh(x)^2 + (3*a^4 + 24*a^2*b^
2 + 8*b^4)*cosh(x))*sinh(x)^3 + 4*(a^4 + 6*a^2*b^2)*cosh(x)^2 + 4*(7*a^4*cosh(x)^6 + 42*a^3*b*cosh(x)^5 + 15*(
a^4 + 6*a^2*b^2)*cosh(x)^4 + a^4 + 6*a^2*b^2 + 20*(3*a^3*b + 4*a*b^3)*cosh(x)^3 + 3*(3*a^4 + 24*a^2*b^2 + 8*b^
4)*cosh(x)^2 + 6*(3*a^3*b + 4*a*b^3)*cosh(x))*sinh(x)^2 + 8*(a^4*cosh(x)^7 + 7*a^3*b*cosh(x)^6 + 3*(a^4 + 6*a^
2*b^2)*cosh(x)^5 + 5*(3*a^3*b + 4*a*b^3)*cosh(x)^4 + a^3*b + (3*a^4 + 24*a^2*b^2 + 8*b^4)*cosh(x)^3 + 3*(3*a^3
*b + 4*a*b^3)*cosh(x)^2 + (a^4 + 6*a^2*b^2)*cosh(x))*sinh(x))*log(2*(a*cosh(x) + b)/(cosh(x) - sinh(x))) + 8*(
3*a^4*x*cosh(x)^7 + 21*(a^3*b*x - a^3*b)*cosh(x)^6 - 9*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^2*b^2)*x)*cosh(x)^5 + 3*a
^3*b*x - 5*(11*a^3*b + 22*a*b^3 - 3*(3*a^3*b + 4*a*b^3)*x)*cosh(x)^4 - 3*a^3*b - (6*a^4 + 112*a^2*b^2 + 50*b^4
- 3*(3*a^4 + 24*a^2*b^2 + 8*b^4)*x)*cosh(x)^3 - 3*(11*a^3*b + 22*a*b^3 - 3*(3*a^3*b + 4*a*b^3)*x)*cosh(x)^2 -
3*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^2*b^2)*x)*cosh(x))*sinh(x))/(a^9*cosh(x)^8 + a^9*sinh(x)^8 + 8*a^8*b*cosh(x)^
7 + 8*a^8*b*cosh(x) + a^9 + 8*(a^9*cosh(x) + a^8*b)*sinh(x)^7 + 4*(a^9 + 6*a^7*b^2)*cosh(x)^6 + 4*(7*a^9*cosh(
x)^2 + 14*a^8*b*cosh(x) + a^9 + 6*a^7*b^2)*sinh(x)^6 + 8*(3*a^8*b + 4*a^6*b^3)*cosh(x)^5 + 8*(7*a^9*cosh(x)^3
+ 21*a^8*b*cosh(x)^2 + 3*a^8*b + 4*a^6*b^3 + 3*(a^9 + 6*a^7*b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^9 + 24*a^7*b^2 +
8*a^5*b^4)*cosh(x)^4 + 2*(35*a^9*cosh(x)^4 + 140*a^8*b*cosh(x)^3 + 3*a^9 + 24*a^7*b^2 + 8*a^5*b^4 + 30*(a^9 +
6*a^7*b^2)*cosh(x)^2 + 20*(3*a^8*b + 4*a^6*b^3)*cosh(x))*sinh(x)^4 + 8*(3*a^8*b + 4*a^6*b^3)*cosh(x)^3 + 8*(7*
a^9*cosh(x)^5 + 35*a^8*b*cosh(x)^4 + 3*a^8*b + 4*a^6*b^3 + 10*(a^9 + 6*a^7*b^2)*cosh(x)^3 + 10*(3*a^8*b + 4*a^
6*b^3)*cosh(x)^2 + (3*a^9 + 24*a^7*b^2 + 8*a^5*b^4)*cosh(x))*sinh(x)^3 + 4*(a^9 + 6*a^7*b^2)*cosh(x)^2 + 4*(7*
a^9*cosh(x)^6 + 42*a^8*b*cosh(x)^5 + a^9 + 6*a^7*b^2 + 15*(a^9 + 6*a^7*b^2)*cosh(x)^4 + 20*(3*a^8*b + 4*a^6*b^
3)*cosh(x)^3 + 3*(3*a^9 + 24*a^7*b^2 + 8*a^5*b^4)*cosh(x)^2 + 6*(3*a^8*b + 4*a^6*b^3)*cosh(x))*sinh(x)^2 + 8*(
a^9*cosh(x)^7 + 7*a^8*b*cosh(x)^6 + a^8*b + 3*(a^9 + 6*a^7*b^2)*cosh(x)^5 + 5*(3*a^8*b + 4*a^6*b^3)*cosh(x)^4
+ (3*a^9 + 24*a^7*b^2 + 8*a^5*b^4)*cosh(x)^3 + 3*(3*a^8*b + 4*a^6*b^3)*cosh(x)^2 + (a^9 + 6*a^7*b^2)*cosh(x))*
sinh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \coth{\left (x \right )} + b \operatorname{csch}{\left (x \right )}\right )^{5}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*coth(x)+b*csch(x))**5,x)

[Out]

Integral((a*coth(x) + b*csch(x))**(-5), x)

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Giac [A]  time = 1.20338, size = 182, normalized size = 1.86 \begin{align*} \frac{\log \left ({\left | a{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{5}} - \frac{25 \, a^{3}{\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 104 \, a^{2} b{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 48 \, a^{3}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 168 \, a b^{2}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 64 \, a^{2} b{\left (e^{\left (-x\right )} + e^{x}\right )} + 96 \, b^{3}{\left (e^{\left (-x\right )} + e^{x}\right )} + 48 \, a^{3} - 32 \, a b^{2}}{12 \,{\left (a{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b\right )}^{4} a^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*coth(x)+b*csch(x))^5,x, algorithm="giac")

[Out]

log(abs(a*(e^(-x) + e^x) + 2*b))/a^5 - 1/12*(25*a^3*(e^(-x) + e^x)^4 + 104*a^2*b*(e^(-x) + e^x)^3 - 48*a^3*(e^
(-x) + e^x)^2 + 168*a*b^2*(e^(-x) + e^x)^2 - 64*a^2*b*(e^(-x) + e^x) + 96*b^3*(e^(-x) + e^x) + 48*a^3 - 32*a*b
^2)/((a*(e^(-x) + e^x) + 2*b)^4*a^4)