3.7.53 \(\int \frac {1}{(a \coth (x)+b \text {csch}(x))^5} \, dx\) [653]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.11, antiderivative size = 98, normalized size of antiderivative = 1.00, 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 = {4477, 2747, 711} \begin {gather*} \frac {4 b}{a^5 (a \cosh (x)+b)}+\frac {\log (a \cosh (x)+b)}{a^5}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*Coth[x] + b*Csch[x])^(-5),x]

[Out]

-1/4*(a^2 - b^2)^2/(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 711

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]

Rule 2747

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 4477

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]

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 {\text {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 {\text {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 = 0.22, size = 138, normalized size = 1.41 \begin {gather*} \frac {-3 a^4+2 a^2 b^2+25 b^4+12 b^4 \log (b+a \cosh (x))+12 a^4 \cosh ^4(x) \log (b+a \cosh (x))+48 a^3 b \cosh ^3(x) (1+\log (b+a \cosh (x)))+12 a^2 \cosh ^2(x) \left (a^2+9 b^2+6 b^2 \log (b+a \cosh (x))\right )+8 a b \cosh (x) \left (a^2+11 b^2+6 b^2 \log (b+a \cosh (x))\right )}{12 a^5 (b+a \cosh (x))^4} \end {gather*}

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)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(197\) vs. \(2(94)=188\).
time = 1.53, size = 198, normalized size = 2.02

method	result	size
risch	\(-\frac {x}{a^{5}}+\frac {4 \left (6 a^{3} b \,{\mathrm e}^{6 x}+3 a^{4} {\mathrm e}^{5 x}+27 a^{2} b^{2} {\mathrm e}^{5 x}+22 \,{\mathrm e}^{4 x} a^{3} b +44 a \,b^{3} {\mathrm e}^{4 x}+3 a^{4} {\mathrm e}^{3 x}+56 a^{2} b^{2} {\mathrm e}^{3 x}+25 b^{4} {\mathrm e}^{3 x}+22 \,{\mathrm e}^{2 x} a^{3} b +44 a \,b^{3} {\mathrm e}^{2 x}+3 a^{4} {\mathrm e}^{x}+27 a^{2} b^{2} {\mathrm e}^{x}+6 a^{3} b \right ) {\mathrm e}^{x}}{3 a^{5} \left ({\mathrm e}^{2 x} a +2 \,{\mathrm e}^{x} b +a \right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{a^{5}}\)	\(174\)
default	\(\frac {\frac {8 a^{3} \left (3 a^{2}+2 a b -b^{2}\right )}{3 \left (a -b \right )^{2} \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+a +b \right )^{3}}+\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+a +b \right )-\frac {2 a^{2}}{\left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+a +b \right )^{2}}-\frac {2 a}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+a +b}-\frac {4 a^{4} \left (a^{2}+2 a b +b^{2}\right )}{\left (a -b \right )^{2} \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+a +b \right )^{4}}}{a^{5}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{5}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{5}}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (94) = 188\).
time = 0.30, size = 285, normalized size = 2.91 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2564 vs. \(2 (94) = 188\).
time = 0.41, size = 2564, normalized size = 26.16 \begin {gather*} \text {Too large to display} \end {gather*}

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))

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a \coth {\left (x \right )} + b \operatorname {csch}{\left (x \right )}\right )^{5}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.43, size = 135, normalized size = 1.38 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (\frac {b}{\mathrm {sinh}\left (x\right )}+a\,\mathrm {coth}\left (x\right )\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/sinh(x) + a*coth(x))^5,x)

[Out]

int(1/(b/sinh(x) + a*coth(x))^5, x)

________________________________________________________________________________________